Hyperbolic actions and 2nd bounded cohomology of subgroups of $\mathsf{Out}(F_n)$. Part II: Finite lamination subgroups
Michael Handel Lee Mosher




 







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thmTheorem[section]

theorem[thm]Theorem

lemma[thm]Lemma

*theoremATheorem A
*theoremBTheorem B
*theoremCTheorem C
*theoremDTheorem D
*theoremETheorem E
*theoremFHyperbolic action theorem (for automorphism subgroups)
*theoremGTheorem G
*theoremGPlusTheorem G^+
*theoremDWPDTheorem D (Special Version)
*theoremDWWPDTheorem D (General Version)
*corollary*Corollary
*disintegrationtheoremDisintegration Theorem



cor[thm]Corollary
corollary[thm]Corollary
proposition[thm]Proposition
*proposition*Proposition
question[thm]Question
*restate
*mainthmMain Theorem

conj[thm]Conjecture
prop[thm]Proposition

sublemma[thm]Sublemma
fact[thm]Fact
*WeakWA1propositionWeak version of Proposition <ref>
*TheoremSevenPointOneTheorem 7.1 of 
*TheoremSevenPointTwoTheorem 7.2 of 



definition

definition[thm]Definition 

*defn*Definition
defns[thm]Definitions
notation[thm]Notation
notations[thm]Notations
com[thm]Remark
ex[thm]Example
exs[thm]Examples
remark[thm]Remark
remark

remarks
remarks


  
*Remarksremarks. remarks
 
 
 
 
 
 






















 



























































 

 




















 











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Hyperbolic actions and 2nd bounded cohomology 
of subgroups of (F_n) 
 Part II: Finite lamination subgroups
    Michael Handel and Lee Mosher The first author  was supported by the National Science Foundation under Grant No. DMS-1308710 and by PSC-CUNY under grants in Program Years 46 and 47. The second author was supported by the National Science Foundation under Grant No. DMS-1406376.
    December 30, 2023
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This is the second part of a two part work in which we prove that for every finitely generated subgroup Γ(F_n), either Γ is virtually abelian or its second bounded cohomology H^2_b(Γ;) contains an embedding of ℓ^1. Here in Part II we focus on finite lamination subgroups Γ — meaning that the set of all attracting laminations of elements of Γ is finite — and on the construction of useful hyperbolic actions of those subgroups.
 
 




§ INTRODUCTION


In this two part work, we study the second bounded cohomology of finitely generated subgroups of (F_n), the outer automorphism group of a free group of finite rank n. The main theorem of the whole work is an “H^2_b-alternative” analogous to similar results for subgroups of mapping class groups  and for groups acting in certain ways on hyperbolic spaces <cit.>:



For every finitely generated subgroup Γ(F_n), either Γ is virtually abelian or H^2_b(Γ;) has an embedded copy of ℓ^1 and so is of uncountable dimension.


In Part I , Theorem A was reduced to Theorem C which is the main result here in Part II. Theorem C explains how to produce useful hyperbolic actions for a certain class of finite lamination subgroups of (F_n). To fully state Theorem C we first briefly review the mathematical objects relevant to its statement: hyperbolic actions and their WWPD elements; subgroups of (F_n) which have (virtually) abelian restriction to any invariant proper free factor; and the dichotomy of finite lamination subgroups versus infinite lamination subgroups of (F_n).

Consider a group action G on a hyperbolic space. Recall that γ∈ G is loxodromic if it acts on the Gromov boundary  with a unique repeller-attractor pair (_- γ, _+ γ) ∈× - Δ. Recall also that G is nonelementary if (following ) there exist independent loxodromic elements δ,γ∈ G, meaning that the sets {_-δ,_+δ}, {_-γ,_+γ} are disjoint. Given a loxodromic element γ∈Γ, the WWPD property for γ was first defined in <cit.>, and that property has several equivalent reformulations that are collected together in <cit.> and are denoted WWPD (1)–(4). In particular, WWPD (2) says that the G-orbit of the ordered pair (_- γ, _+ γ) is a discrete subset of the space of distinct ordered pairs × - Δ. 


Let (F_n) denote the finite index normal subgroup consisting of all outer automorphisms whose induced action on H_1(F_n;/3) is trivial. A subgroup Γ has (virtually) abelian restrictions  (Definition <ref>) if for each proper free factor A  F_n whose conjugacy class [A] is fixed by each element of Γ, the natural restriction homomorphism Γ↦(A) has (virtually) abelian image. As explained in Part I , the property of Γ having (virtually) abelian restrictions plays a role in our theory analogous to the role played by irreducible subgroups of mapping class groups in the theory of Bestvina and Fujiwara .

The decomposition theory of Bestvina, Feighn and Handel  associates to each ϕ∈(F_n) a finite set Ł(ϕ) of attracting laminations. Associated to a subgroup Γ(F_n) is the set Ł(Γ) = _ϕ∈Γ Ł(Γ). If Ł(Γ) is finite then Γ is a finite lamination subgroup, otherwise Γ is an infinite lamination subgroup. In Part I we reduced Theorem A to two results about subgroups of  with (virtually) abelian restrictions: Theorem B concerning infinite lamination subgroups which was proved there in Part I; and Theorem C concerning finite lamination subgroups, which is proved here in Part II. Each of Theorems B and C concludes with the existence of a hyperbolic action of a finite index normal subgroup possessing a sufficiently rich collection of WWPD elements. The reduction argument in Part I combines those two conclusions with the Global WWPD Theorem of <cit.> to prove Theorem A. 

Here is the main theorem of this paper:



For any finitely generated, non virtually abelian, finite lamination subgroup Γ such that Γ has virtually abelian restrictions, there exists a finite index normal subgroup N Γ and an action N on a hyperbolic space, such that the following hold:


  * 
Every element of N acts either elliptically or loxodromically on ;

  * 
The action N is nonelementary;

  * 

Every loxodromic element of the commutator subgroup [N,N] is a strongly axial, WWPD element with respect to the action N.


See below, after the Hyperbolic Action Theorem, for the definition of “strongly axial”.







  
Remarks: WPD versus WWPD. In lectures on this topic we stated a version of Theorem C ItemThmC_WWPD with a stronger conclusion, saying that in the group ⊷(N ↦()), every loxodromic element of the commutator subgroup satisfies WPD. That conclusion requires a stronger hypothesis, saying roughly that “virtually abelian restrictions” holds for a broader class of Γ-invariant subgroups of F_n than just free factors. This makes the proof and the applications of the theorem considerably more intricate, which was not necessary for the application to Theorem A in Part I, and so we have settled for the version of Theorem C presented here.



 §.§ Methods of proof of Theorem C
 
The first step of Theorem C will be to reduce it to a theorem which produces hyperbolic actions of certain subgroups of automorphism groups of free groups. The key step of the reduction argument, carried out in Proposition <ref>, is the construction of “automorphic lifts”: for each subgroup Γ(F_n) that satisfies the hypotheses of Theorem C, there exists a free factor A < F_n of rank 2 ≤ k ≤ n-1, such that its conjugacy class [A] is Γ-invariant, and such that the natural homomorphism Γ↦(A) lifts to a homomorphism Γ↦(A) whose image in (A) is not virtually abelian. In Section <ref> we shall show, using an automorphic lift for which the free factor factor A has minimal rank, how to reduce Theorem C to the following statement, in which we have identified A ≈ F_k and then rewritten k as n. Recall the canonical isomorphism F_n ≈(F_n) which associates to each γ∈ F_n the inner automorphism i_γ(δ)=γδγ^.




Consider a subgroup (F_n) with n ≥ 2, and denote = ⊷(↦(F_n)) and J = (↦) = (F_n), giving the following commutative diagram of short exact sequences:



    1 [r]      J   [r]^⊂[d]^-⊂   @->>[r] [d]^-⊂   [r] [d]^-⊂    1 
    
    1 [r]     F_n ≈(F_n) [r]_⊂   (F_n) @->>[r]    (F_n) [r]     1




If  is abelian, if  is not virtually abelian, and if no proper, nontrivial free factor of F_n is fixed by the action of  on the set of subgroups of F_n, then there exists a finite index normal subgroup  and an action  on a hyperbolic space such that the following properties hold:


  * 
Every element of  acts either elliptically or loxodromically on ;

  * 
The action  is nonelementary;

  * 
Every loxodromic element of J is a strongly axial, WWPD element with respect to the action . 


To say that a loxodromic element ϕ∈ G of a hyperbolic action G  X is strongly axial means that it has a strong axis, which is a quasi-isometric embedding ℓ→ X for which there exists a homomorphism τ(_±ϕ) → such that for all ψ∈(_±ϕ) and s ∈ we have ψ(ℓ(s)) = ℓ(s+τ(ψ)).



The proof of the Hyperbolic Action Theorem begins in Section <ref> by reducing to the case .

We then choose a maximal, proper, -invariant free factor system  in F_n, and break the proof into cases depending on the “co-edge number” of , which is the minimum integer k ≥ 1 such that  is represented by a subgraph H ⊂ G of a marked graph G for which the complement G ∖ H has k edges. The “one-edge” case, where the co-edge number of  equals 1, is handled in Section <ref> using an action of  on a simplicial tree that is naturally associated to the free factor system . 

The “multi-edge” case, where the co-edge number of  is ≥ 2, takes up the majority of the paper from Section <ref> to the end. For a full introduction to the multi-edge case, see Section <ref>. In brief, one finds ϕ∈ which is fully irreducible relative to the free factor system , and one uses the top  stratum of a good relative train track representative of ϕ

to produce a certain hyperbolic suspension space . The construction of , the formulation and verification of flaring properties of , and the proof of hyperbolicity of  by applying the Mj-Sardar combination theorem <cit.>, are found in Sections <ref> and <ref>. Then one constructs an isometric action  by applying the theory of abelian subgroups of (F_n) <cit.>; see Sections <ref>, <ref> and <ref>. The pieces are put together, and the multi-edge case of the Hyperbolic Action Theorem is proved, in Section <ref>.



  
Prerequisites from the theory of (F_n). We will assume that the reader is familiar with certain basic topics of (F_n) that have already been reviewed in Part I of this paper , with detailed references found there. We list these some of these topics here with original citations:


  <cit.> Marked graphs, topological representatives, and relative train track maps . Free factor systems and attracting laminations .


  <cit.> Properties of = ((F_n) ↦ GL(n,/3)) (<cit.> and ).

  <cit.> Elements and subgroups of (F_n) which are fully irreducible relative to a free factor system  of (F_n). The co-edge number of a free factor system  in F_n.

  <cit.> Weak attraction theory. The nonattracting subgroup system _Λ^±_ϕ associated to ϕ∈(F_n) and one of its lamination pairs Λ^±_ϕ  (<cit.> and ).

Where needed in this paper, we will conduct reviews of other basic concepts.












§ LIFTING TO AN AUTOMORPHISM GROUP


In this section, the first thing we do is to study the structure of finitely generated, finite lamination subgroups Γ which are not abelian but have virtually abelian restrictions. The motivating question of this study is: If A  F_n is a proper, nontrivial free factor whose conjugacy class [A] is fixed by Γ, can the natural restriction map Γ↦(A) be lifted to a homomorphism Γ↦(A)? And can this be done so that the image is still not virtually abelian? Sections <ref>–<ref> are devoted to constructions of such “automorphic lifts”. Using this construction, in Section <ref> we prove the implication (Hyperbolic Action Theorem)(Theorem C). 



After that, in Section <ref>, we consider any free splitting F_n  T, and we study a natural subgroup of (F_n) which acts on T in a manner that extends the free splitting action of F_n ≈(F_n). That study is used in Section <ref> to prove one of the two major cases of the Hyperbolic Action Theorem.















 §.§ Definition of automorphic lifts



Recall that for any group and subgroup H  G which is its own normalizer (e.g. a free factor), letting [H] (G) be the stabilizer of the conjugacy class of H, the natural restriction homomorphism [H] ↦(H), denoted ϕ↦ϕ H, is well-defined by choosing Φ∈(G) representing ϕ and preserving H, and then taking ϕ H to be the outer automophism class of Φ H ∈(H) (see e.g.  Fact 1.4).

Throughout the paper we use the theorem that virtually abelian subgroups of  are abelian <cit.>. We sometimes write “(virtually) abelian” as a reminder that one may freely include or ignore the adverb “virtually” in front of the adjective “abelian” in the context of a subgroup of _F(/3) for any finite rank free group F. One has this freedom, for example, in the following definition (see <cit.>):



A subgroup Γ has (virtually) abelian restrictions if for any proper free factor A  F_n such that Γ[A], the restriction homomorphism Γ↦(A) has (virtually) abelian image.



Let Γ be a 

subgroup which is not (virtually) abelian and which has (virtually) abelian restrictions. An automorphic lift of Γ is a homomorphism ρΓ↦(A), where A  F_n is a proper free factor and Γ[A], such that the group = ⊷(ρ) is not virtually abelian, and such that the following triangle commutes

    (A) [d] 
    Γ[ur]^ρ[r]     (A)

In this diagram the horizontal arrow is the natural restriction homomorphism [A] ↦(A) with domain restricted to Γ, and the vertical arrow is the natural quotient homomorphism. To emphasize the role of A we will sometimes refer to an automorphic lift of Γ rel A. Adopting the notation of the Hyperbolic Action Theorem, we set = ⊷(ρ), = ⊷(↦(A)) = ⊷(Γ↦(A)), and J = (↦) = (A), thus obtaining the commutative diagram shown in Figure <ref>.

We note two properties which follow from the definition:


  *  is abelian;

  * The free group J has rank ≥ 2.

The first holds because A < F_n is proper and Γ has (virtually) abelian restrictions (see Definition <ref>). The second is a consequence of the first combined with the defining requirement that  is not virtually abelian, for otherwise the free group J is abelian and so  is virtually solvable, but (A) injects into (F_n) and solvable subgroups of (F_n) are virtually abelian by . We put no further conditions on the rank of J, it could well be infinite. When referring to J, each of its elements will be thought of ambiguously as an element of the free factor A  F_n or as the corresponding element of the inner automorphism group (A); this ambiguity should cause little trouble, by using the canonical isomorphism A ↔(A) given by δ↔ i_δ where i_δ(γ)=δγδ^. 

The rank of the automorphic lift Γ↦(A) is defined to be (A), and note that (A) ≥ 2 because otherwise (A) is finite in which case each of its subgroups is virtually abelian. 

This completes Definition <ref>.


Here is the first of two main results of Section <ref>. 


 If Γ is a finitely generated, finite lamination subgroup which is not (virtually) abelian but which has (virtually) abelian restrictions, then there exists an automorphic lift Γ↦(A). 


The proof is found in Section <ref>, preceded by Lemma <ref> in Section <ref>. 

When n=2, one recovers from Proposition <ref> the simple fact that every finite lamination subgroup of (F_2) is virtually abelian, for otherwise the intersection with  would have an automorphic lift to (A) for some proper free factor A, from which it would follow that 2 ≤(A) ≤ n-1=1. Of course this fact has a simple proof, expressed in terms the isomorphism (F_2) ↦(H_1(F_2;)) ≈ GL(2,), which we leave to the reader.




 §.§ A sufficient condition to be abelian.


In this section we prove Lemma <ref> which gives a sufficient condition for a finitely generated, finite lamination subgroup Γ to be abelian. The negation of this condition then becomes a property that must hold when Γ is not abelian.





  Let Γ be a finitely generated, finite lamination subgroup. If = {[A_1],…,[A_I]} is a maximal proper Γ-invariant , if each restriction Γ A_i (A_i) is abelian for i=1,…,I, and if  has co-edge number ≥ 2 in F_n, then Γ is abelian.


 By Theorem C of , there exists η∈Γ which is fully irreducible rel , meaning that there does not exist any η-invariant proper free factor system that strictly contains . By relative train track theory, there is a unique lamination pair Λ^±_η∈Ł^±(η) for η which is not carried by  (<cit.> Section 3; also see  Fact 1.55). The nonattracting subgroup system of Λ^±_η has one of two forms, either _(Λ^±_η) = or _(Λ^±_η) = {[C]} where C  F_n is a certain maximal infinite cyclic subgroup that is not carried by  (see <cit.>; also see , Definitions 1.2 and Corollary 1.9). 


For every nonperiodic line ℓ that is not carried by , evidently ℓ is not carried by [C], and so ℓ is not carried by _(Λ^±_η). By applying Theorem H of , it then follows that ℓ is weakly attracted to either Λ^+_η by iteration of η or to Λ^-_η by iteration of η^. From this it follows that the two laminations Λ^+_η, Λ^+_η^=Λ^-_η are the unique elements of Ł(Γ) not carried by , for if there existed ψ∈Γ with attracting lamination Λ^+_ψ∈Ł(Γ) - {Λ^±_η} not supported by  then the generic lines of Λ^+_ψ would be weakly attracted either to Λ^+_η by iteration of η or to Λ^-_η by iteration of η^, and in either case the set of laminations η^k(Λ^+_ψ) = Λ^+_η^k ψη^-k∈Ł(Γ), k ∈, would form an infinite set, contradicting that Γ is a finite lamination group.

Since Ł(Γ) is finite, for each ϕ∈Γ each element of Ł(Γ) has finite orbit under the action of of ϕ. Since Γ, it follows by <cit.> that each element of Ł(Γ) is fixed by ϕ. In particular, Γ(Λ^+_η). By <cit.>, there exists a homomorphism _Λ^+_η(Λ^+_η) → having the property that for each ϕ∈(Λ^+_η), the inequality _Λ^+_η(ϕ)  0 holds if and only if Λ^+_η∈Ł(ϕ). 

Consider the following homomorphism, the range of which is abelian:

    ΩΓ  →           ⊕  Γ A_1   ⊕⋯⊕  Γ A_I 
    Ω(ϕ) = PF_Λ^+_η(ϕ)    ⊕  ϕ A_1   ⊕⋯⊕  ϕ A_I

We claim that that the kernel of Ω is also abelian. This claim completes the proof of the lemma, because every solvable subgroup of (F_n) is virtually abelian , and every virtually abelian subgroup of  is abelian <cit.>. 

To prove the claim, by Proposition 5.5 of , every subgroup of the group (PF_Λ^+_η(ϕ)) consisting entirely of  elements is abelian, and so we need only check that each element of (Ω) is . By <cit.>, every  element of  is , and so we need only check that each ϕ∈(Ω) is , equivalently Ł(ϕ) = ∅. Suppose to the contrary that there exists Λ^+_ϕ∈Ł(ϕ), with dual repelling lamination Λ^-_ϕ∈Ł(ϕ^). Since ϕ A_i is trivial in (A_i) for each component [A_i] of , neither of the laminations Λ^±_ϕ is supported by . Since Ł(ϕ) ⊂Ł(Γ), it follows as shown above that {Λ^+_ϕ,Λ^-_ϕ} = {Λ^+_η,Λ^-_η}, and so PF_Λ^+_η(ϕ)  0 a contradiction.







 §.§ Constructing automorphic lifts: proof of Proposition <ref>



Let Γ be a finitely generated, finite lamination subgroup that is not (virtually) abelian and has (virtually) abelian restrictions. Choose a maximal proper Γ-invariant free factor system = {[A_1],…,[A_I]}, and so each restricted group denoted _i = Γ [A_i] (A_i) is abelian. Since Γ, it follows that each component [A_i] of  is fixed by Γ (, Lemma 4.2). 

The group Γ is not abelian by <cit.>, and so by Lemma <ref> the extension ⊏{[F_n]} is a one-edge extension. We may therefore choose a marked graph pair (G,H) representing  so that G ∖ H = E is a single edge, and we may choose H so that its components are roses, with each endpoint of E being the rose vertex of a component of H. The number of components of  equals the number of components of H, that number being either one or two, and we cover those cases separately. 

Case 1:  has two components, say = {[A_1],[A_2]} where F_n = A_1 ∗ A_2. We construct a commutative diagram as follows:

    (A_1) ⊕(A_2) [r]^q    (A_1) ⊕(A_2) 
       Γ[ul]^α=α_1 ⊕α_2   [u]_ρ=ρ_1⊕ρ_2[dl]_ω[d]^⊂
    (F_n,A_1,A_2) [uu]^r [r]    (F_n)

(F_n,A_1,A_2) is the subgroup of (F_n) that preserves both A_1 and A_2. The homomorphism r is induced by restricting (F_n,A_1,A_2) to (A_1) and to (A_2). Evidently r is injective, since an automorphism of F_n is determined by its restrictions to the complementary free factors A_1,A_2. The homomorphisms denoted by the top and bottom arrows of the diagram are induced by canonical homomorphisms from automorphism groups to outer automorphism groups. For i=1,2 the homomorphism ρ_i is the composition Γ[A_i] ↦(A_i) where the latter map is the natural restriction homomorphism.

We must construct ω, α_1, α_2. We may choose the marked graph pair (G,H) representing the free factor system  to have the following properties: the two rose components H_1, H_2 of the subgraph H have ranks equal to (A_1), (A_2) respectively; the edge E is oriented and decomposes into oriented half-edges E=E_1 E_2; the common initial point of E_1,E_2 is denoted w; their respective terminal vertices are the rose vertices v_i ∈ H_i; and there is an isomorphism π_1(G,w) ≈ F_n which restricts to isomorphisms π_1(E_i  H_i, w) ≈ A_i for i=1,2. Given ϕ∈Γ, let f  G → G be a homotopy equivalence that represents ϕ, preserves H_1 and H_2, and restricts to a locally injective path on E = E_1 E_2. By Corollary 3.2.2 of  we have f(E) = u̅_1 E^± 1 u_2 for possibly trivial paths closed paths u_i in H_i, i=1,2, and in fact the plus sign occurs and so f(E) = u̅_1 E u_2, because ϕ∈Γ. After pre-composing f with a homeomorphism of G isotopic to the identity that restricts to the identity on H_1  H_2 and that moves the point w ∈ E to f^(w) ∈ E, we may also assume that f(w)=w, and so f(E_1) = E_1 u_1 and f(E_2) = E_2 u_2. Define α_i(ϕ) ∈(A_i) ≈(π_1(E_i  H_i,v_i)) to be the automorphism induced by f  E_i  H_i, and then use the isomorphism F_n = A_1 * A_2 to define ω(ϕ). Note that ω(ϕ) is the unique lift of ϕ∈(F_n) to (F_n) which preserves A_1 and A_2, because any two such lifts differ by an inner automorphism i_c that preserves both A_1 and A_2, implying by malnormality that c ∈ A_1  A_2 and so is trivial. It follows from uniqueness that ω, α_1, and α_2 are homomorphisms, and hence so is α. Commutativity of the diagram is straightforward from the construction. The homomorphism ω is injective because it is a lift of the inclusion Γ(F_n). Since r and ω are injective, by commutativity of the diagram α is also injective. 

At least one of the two maps α_i Γ→(A_i) is an automorphic lift because at least one of the corresponding images _i = ⊷(α_i) < (A_i) is not virtually abelian: if both were virtually abelian, then α(Γ) _1 ⊕_2 would be virtually abelian, but α is injective and so Γ would be virtually abelian, a contradiction. 



Case 2:  has a single component, say ={[A]}, and so F_n = A * b for some b ∈ F_n. The proof in this case is similar to Case 1, the main differences being that in place of direct sum we use fiber sum, and the marked graph pair (G,H) representing  will have connected subgraph H.

We shall construct the following commutative diagram:

    ^2(A,A^b) [rr]^q      (A) 
          [u]_ρ[ull]^α[dll]^ω[d]^⊂
    (F_n,A,A^b) [uu]^r[rr]       (F_n)

In this diagram we use the following notations: the conjugate A^b = b A b^; the restricted inner automorphism i_b A → A^b where i_b(a)=b a b^; the adjoint isomorphism _b (A) →(A^b) where _b(Φ) = i^_b  Φ   i^_b; the canonical epimorphism q_A (A) →(A); and the epimorphism q_A^b = q_A _b^(A^b) →(A). Also, the fiber sum of q_A and q_A^b is the following subgroup of (A) ⊕(A^b):

    ^2(A,A^b) = {(Φ,Φ') ∈(A) ⊕(A^b)  q_A(Φ)=q_A^b(Φ')}

Define the homomorphism q by q(Φ,Φ') = q_A(Φ)=q_A^b(Φ'). Define (F_n,A,A^b) (F_n) to be the subgroup that preserves both A and A^b. The homomorphism r is jointly induced by the two restriction homomorphisms r_A, r_A^b from (F_n,A,A^b) to (A), (A^b) because the two compositions q_A  r_A, q_A^b r_A^b(F_n,A,A^b) →(A) are evidently the same. Note that r is injective, for if Φ∈(F_n,A,A^b) restricts to the identity on each of A and A^b, then Φ(a)=a and Φ(a^b)=a^b for all a ∈ A, and it follows that b^Φ(b) commutes with all a ∈ A; since (A) ≥ 2, we have Φ(b)=b, and hence Φ is trivial.

To construct the homomorphism ω, we may in this case choose the marked graph pair (G,H) representing  so that: H is a rose whose rank equals (A)=n-1; as before E=E_1 E_2 with w the common initial point of E_1,E_2; the terminal endpoints of both E_1 and E_2 equal the rose vertex v ∈ H; and there is an isomorphism π_1(G,w) ≈ F_n which restricts to π_1(E_1  H,w) ≈ A and E_2 E_1 ≈ b. It follows that π_1(E_2  H,w) ≈ A^b. For each ϕ∈, applying Corollary 3.2.2 of  as in the previous case, we find Φ = ω(ϕ) ∈(F_n,A,A^b) that represents ϕ and that is represented by a homotopy equivalence f_ϕ G → G that preserves H, fixes w, and takes E_i ↦ E_i u_i for possibly trivial paths u_1,u_2 in H based at v. The map ω is a homomorphism because Φ is the unique element of (F_n,A,A^b) that represents ϕ, for if c ∈ F_n and if the inner automorphism i_c preserves both A and A^b then by malnormality of A and A^b we have c ∈ A ∩ A^b, and again by malnormality we have that c is trivial. It follows that ω is injective. Since r is injective it follows that α = r ω is injective. Denote α(ϕ) = (α_A(ϕ),α_A^b(ϕ)) ∈^2(A,A^b). Obviously the compositions q_A   α_A, q_A^b  α_A^b→(A) are the same, and hence there is an induced homomorphism ρ→(A) which is topologically represented by f_ϕ H. This completes the construction of the above diagram, and commutativity is evident. 

As in the previous case, we will be done if we can show that at least one of the two homomorphisms α_A Γ_0 →(A) or α_A^bΓ_0 →(A^b) has image that is not virtually abelian, but if both are virtually abelian then ⊷(α) is contained in a virtually abelian subgroup of ^2(A,A^b), which by injectivity of α implies that Γ is virtually abelian, a contradiction.





 §.§ Proof that the Hyperbolic Action Theorem implies Theorem C



Let Γ be a finitely generated, finite lamination subgroup which is not (virtually) abelian and which has (virtually) abelian restrictions. 

By applying Proposition <ref>, there exists a free factor A  F_n such that Γ[A], and there exists an automorphic lift ρΓ↦(A); we may assume that (A) is minimal amongst all choices of A and ρ. We adopt the notation of Figure <ref> in Section <ref>, matching that notation with the Hyperbolic Action Theorem by choosing an isomorphism A ≈ F_k where k = (A). Most of the hypotheses of the Hyperbolic Action Theorem are now immediate: = ⊷(ρ) (A) is not virtually abelian by definition of automorphic lifts; also = ⊷(↦(A)) is abelian.

We must check the one remaining hypothesis of the Hyperbolic Action Theorem, namely that no proper, nontrivial free factor B  A is preserved by the action of . Assuming by contradiction that B is preserved by , consider the restriction homomorphism σ↦(B). We claim that the composition σρΓ→(B) is an automorphic lift of Γ. Since (B)<(A), once this claim is proved, it contradicts the assumption that ρΓ→(A) is an automorphic lift of minimal rank. The canonical isomorphism (F_k) ↔ F_k, denoted i_δ↔δ, restricts to an isomorphism between J = (F_k) and some subgroup of F_k. If i_δ∈ J then i_δ preserves B, and since B is malnormal in A it follows that δ∈ B. Thus σ restricts to an injection from J to (B). Also, the group J is a free group of rank ≥ 2, for if it were trivial or cyclic then  would be virtually solvable and hence, by , virtually abelian, a contradiction. Since the image of the map σρΓ(B) contains σ(J), it follows that ⊷(σρ) is not virtually abelian. Since  preserves the A-conjugacy class of B, and since B is malnormal in F_n, it follows Γ preserves the F_n-conjugacy class of B. Tracing through the definitions one easily sees that the composed homomorphism Γ(B) ↦(B) is equal to the composition of Γ_(F_n)[B] ↦(B), where the latter map is the natural restriction homomorphism. Thus σρΓ→(B) is an automorphic lift of Γ, completing the proof of the claim.

Applying the conclusions of the Hyperbolic Action Theorem using the free group A ≈ F_k and (F_k), we obtain a finite index normal subgroup N and a hyperbolic action N satisfying conclusions ItemThmF_EllOrLox, ItemThmF_Nonelem and ItemThmF_WWPD of that theorem. The subgroup N = ρ^( N) Γ is a finite index normal subgroup of Γ, and by composition we have a pullback action N ↦ N. By the Hyperbolic Action Theorem ItemThmF_EllOrLox, each element of N acts elliptically or loxodromically on , and so the same holds for each element of N, which is Theorem C ItemThmC_EllOrLox. By item ItemThmF_Nonelem of the Hyperbolic Action Theorem, the action N is nonelementary, and so the same holds for the action N, which is Theorem C ItemThmC_Nonelem. Since the image of the homomorphism N ↦ N ↦ is abelian, it follows that the image in  of the commutator subgroup [N,N] is contained in J = (↦), and hence the image of [N,N] in N is contained in J  N. By conclusion ItemThmF_WWPD of the Hyperbolic Action Theorem, each loxodromic element of J  N is a strongly axial WWPD element with respect to the action N. Applying <cit.> which says that WWPD is preserved under pullback, and using the evident fact that the strongly axial property is preserved under pullback, it follows that each loxodromic element of [N,N] is a strongly axial, WWPD element of the action N, which is the statement of Theorem C ItemThmC_WWPD. 





 §.§ Automorphic extensions of free splitting actions


From the hypothesis of the Hyperbolic Action Theorem, our interest is now transferred to the context of a finite rank free group F_n — perhaps identified isomorphically with some free factor of a higher rank free group — and of a subgroup (F_n) that has the following irreducibility property: no proper, nontrivial free factor of F_n is preserved by . To prove the Hyperbolic Action Theorem one needs actions of such groups  on hyperbolic spaces. In this section we focus on a natural situation which produces actions on trees.



  
Free splittings. Recall that a free splitting of F_n is a minimal, simplicial action F_n  T on a simplicial tree T such that the stabilizer of each edge is trivial. Two free splittings F_n  S,T are simplicially equivalent if there exists an F_n-equivariant simplicial isomorphism S ↦ T; we sometimes use the notation [T] for the simplicial equivalence class of a free splitting F_n  T. Formally the action F_n  T is given by a homomorphism α F_n ↦(T), which we denote more briefly as α F_n  T. In this formal notation, (T) refers to the group of simplicial self-isomorphisms of T, equivalently the self-isometry group of T using the geodesic metric given by barycentric coordinates on simplices of T. We note that an element of (T) is determined by its restriction to the vertex set, in fact it is determined by its restriction to the subset of vertices of valence ≥ 3. Two free splittings are equivalent if there is an F_n-equivariant simplicial isomorphism between them, the equivalence class of a free splitting F_n  T is denoted [T], and the group (F_n) acts naturally on the set of equivalence classes of free splittings.

Given a free splitting F_n  T, the set of conjugacy classes of nontrivial vertex stabilizers of a free splitting is a free factor system of F_n called the vertex group system of T denoted as (T). The function which assigns to each free splitting T its vertex group system (T) induces a well-defined, (F_n)-equivariant function 

    [T] ↦(T)

from the set of simplicial equivalence classes of free splittings to the set of free factor systems. 

Every free splitting F_n  T can be realized by some marked graph pair (G,H) in the sense that T is the F_n-equivariant quotient of the universal cover G where each component of the total lift H ⊂ G is collapsed to a point. One may also assume that each component of H is noncontractible, in which case the same marked graph pair (G,H) topologically represents the vertex group system of T.



  
Twisted equivariance (functional notation). Given two free splittings F_n  S,T and an automorphism Φ∈(F_n), a map h  S → T is said to be Φ-twisted equivariant if

    h(γ· x) = Φ(γ) · h(x)   for all x ∈ S, γ∈ F_n.

The special case when Φ = is simply called equivariance. 

A twisted equivariant map behaves well with respect to stabilizers, as shown in the following simple fact:


For any free splittings F_n  S,T, for each Φ∈(F_n), and for each Φ-twisted equivariant map f  S → T, we have


  * 
Φ((x)) (f(x)) for all x ∈ S.

  * 
If in addition the map f  S → T is a simplicial isomorphism, then the inclusion of item ItemSublemmaPoints is an equality: Φ((x)) = (f(x)). Furthermore, γ∈ F_n acts loxodromically on S if and only if Φ(γ) acts loxodromically on T, in which case their axes A^S_γ⊂ S and A^T_Φ(γ) ⊂ T satisfy A^T_Φ(γ) = f(A^S_γ).





  
Remark. One could approach this proof by first working out the equivariant case (Φ =), and then reducing the twisted case to the equivariant case by conjugating the action F_n  T using Φ. We instead give a direct proof.

 To prove ItemSublemmaPoints, for each x ∈ S and γ∈ F_n we have

    γ∈Φ((x))    Φ^(γ) ∈(x) Φ^(γ) · x = x 
        f(Φ^(γ) · x) = f(x) 
       Φ(Φ^(γ)) · f(x) = f(x)   (by twisted equivariance)
       γ· f(x) = f(x) 
       γ∈(f(x))

and so Φ((x)) (Φ· x). 

To prove ItemSublemmaAxes, consider the inverse simplicial automorphism f^ T → S. The implication in the second line may be inverted by applying f^ to both sides of the equation in the second line. For the rest of ItemSublemmaAxes, it suffices to prove the “only if” direction, because the “if” direction can then be proved using that f^ is Φ^-twisted equivariant. Assuming γ is loxodromic in S with axis A^S_γ, consider the line f(A^S_γ) ⊂ T. Calculating exactly as above one shows that the equation Φ((A^S_γ)) = (f(A^S_γ)) holds. We may assume that γ is a generator of the infinite cyclic group (A^S_γ), and so Φ(γ) = Φγ = (f(A^S_γ)). Since the stabilizer of the line f(A^S_γ) is the infinite cyclic group Φ(γ), it follows that Φ(γ) is loxodromic and its axis A^T_Φ(γ) is equal to f(A^S_γ).




  
Free splittings of co-edge number 1.
Recall the co-edge number of a free factor system  of F_n (see for example <cit.>), which is the minimal number of edges of G ∖ H amongst all marked graph pairs (G,H) such that H is a representative of . There is a tight relationship between free splittings with a single edge orbit and free factor systems with co-edge number 1. To be precise:


 <cit.>
When the (F_n)-equivariant function [T] ↦ = (T) is restricted to free splittings T with one edge orbit and free factor systems  with co-edge number 1, the result is a bijection, and hence 

    [T] = ()

whenever T and  correspond under this bijection.

Under the bijection in Fact <ref>, the number of components of  equals the number of vertex orbits of T which equals 1 or 2.  If ={[A]} has a single component then (A)=n-1, there is a free factorization F_n = A * B where (B)=1, and the quotient graph of groups T / F_n is a circle with one edge and one vertex. On the other hand if = {[A_1],[A_2]} has two components then (A_1) + (A_2)=n, A_1,A_2 can be chosen in their conjugacy classes so that there is a free factorization F_n = A_1 * A_2, and the quotient graph of groups T / F_n is an arc with one edge and two vertices.




  
The stabilizer of a free splitting and its automorphic extension
Consider a free splitting α F_n  T and its stabilizer subgroup [T] (F_n). Let [T] (F_n) be the preimage of [T] under the standard projection homomorphism (F_n) ↦(F_n), and so we have a short exact sequence

    1 ↦ F_n ≈(F_n) [T] ↦[T] ↦ 1

From this setup we shall define in a natural way an action [T]  T which extends the given free splitting action α F_n  T. We proceed as follows. 

For each Φ∈(F_n) we have the composed action α∘Φ F_n  T. Assuming in addition that Φ∈[T], in other words that Φ is a representative of some element of [T], it follows the actions α and α∘Φ F_n  T are equivalent, meaning that there exists a simplicial automorphism h ∈(T) such that h α(γ) = α(Φ(γ))  h. When this equation is rewritten in action notation it simply says that h satisfies Φ-twisted equivariance: h(γ· x) = Φ(γ) · h(x) for all γ∈ F_n, x ∈ S. Suppose conversely that for some Φ∈(F_n) there exists a Φ-twisted equivariant h ∈(T). Since h conjugates the action α F_n  T to the action α∘Φ F_n  T, it follows that ϕ∈[T] and Φ∈[T]. This proves the equivalence of ItemLTUInStabT, ItemLTUForall and ItemLTUExists in the following lemma, which also contains some uniqueness information regarding the conjugating maps h.





For each free splitting α F_n  T and each ϕ∈(F_n) the following are equivalent:


  * 
ϕ∈[T]

  * 
For each Φ∈(F_n) representing ϕ, there exists a Φ-twisted equivariant isomorphism h  T → T.

  * 
For some Φ∈(F_n) representing ϕ, there exists a Φ-twisted isomorphism h  T → T.

Furthermore, 


  * 
If ϕ satisfies the equivalent conditions ItemLTUInStabT, ItemLTUForall, ItemLTUExists then for each Φ∈[T] representing ϕ, the Φ-twisted equivariant isomorphism of h_Φ is uniquely determined by Φ, and is denoted

    h_Φ T → T


  * 
For each γ∈ F_n with corresponding inner automorphism i_γ(δ)=γδγ^, the two maps h_i_γ T → T and α(γ)  T → T are equal.

 

Remark: In item ItemLTUInner, note that h_i_γ is defined because i_γ∈(F_n) [T].  



The uniqueness statement ItemLTUUnique is a special case of a more general uniqueness statement that we will make use of later:


For any two free splittings F_n  S,T and any Φ∈(F_n), there exists at most one Φ-equivariant simplicial isomorphism h  S ↦ T. In particular, taking Φ =, there exists at most one equivariant simplicial isomorphism h  S ↦ T.


 Suppose that a Φ-twisted equivariant simplicial isomorphism h  S ↦ T exists. For each γ∈ F_n, it follows by Φ-twisted equivariance that γ acts loxodromically on S with axis A^S_γ if and only if Φ(γ) acts loxodromically on T with axis A^T_γ = f(A^Sγ) (by Lemma <ref>). The map that h induces on the set of axes of loxodromic elements is therefore uniquely determined. It follows that the restriction of h to the set of vertices v ∈ T of valence ≥ 3 is uniquely determined by Φ, because v may be expressed in the form {v} = A^S_β A^S_γ A^S_δ for a certain choice of β,γ,δ∈ F_n, and hence 

    {h(v)}   = h(A^S_β)  h(A^S_γ)  h(A^S_δ) 
       = A^T_Φ(β) A^T_Φ(γ) A^T_Φ(δ)

Since h is uniquely determined by its restriction to the vertices of valence ≥ 3, it follows that h is uniquely determined amongst simplicial isomorphisms.


   The uniqueness clause ItemLTUUnique is an immediate consequence of Lemma <ref>. Item ItemLTUInner follows from the uniqueness clause ItemLTUUnique, because for each γ∈ F_n the map h = α(γ) clearly satisfies i_γ twisted equivariant: α(γ) α(δ) = α(i_γ(δ)) α(γ) for all δ∈ F_n.













Consider now the function α[T] ↦[T] defined by α(Φ) = h_Φ as given by Lemma <ref>. This defines an action α[T]  T, because for each Φ,Ψ∈[T] both sides of the action equation h_Φ∘ h_Ψ = h_ΦΨ
clearly satisfy ΦΨ-twisted equivariance, and hence the equation holds by application of the uniqueness clause ItemLTUUnique of Lemma <ref>. 

The following lemma summarizes this discussion together with the evident generalization to subgroups of [T], and rewrites the twisted equivariance property using action notation instead of functional notation.






Associated to each free splitting F_k  T there is a unique isometric action [T]  T which assigns to each Φ∈[T] the unique simplicial isomorphism T ↦ T as stated in Lemma <ref>, denoted in action notation as x ↦Φ· x, satisfying the following:


  Twisted equivariance (action notation): For all Φ∈(F_n), γ∈ F_n, x ∈ T,

Φ· (γ· x) = Φ(γ) · (Φ· x)

More generally, the restriction to any subgroup [T] of the action [T]  T is the unique isometric action T such that satisfies twisted equivariance.



    Remark:
 In the twisted equivariance equation Φ· (γ· x) = Φ(γ) · (Φ· x), the action dot “·” is used ambiguously for both the action of F_n on T and the action of [T] on T. The meaning of any particular action dot should be clear by context. Furthermore, in contexts where the two meanings overlap they will always agree. For example, Lemma <ref> ItemLTUInner says that the action dots always respect the standard isomorphism F_n ≈(F_n) given by δ≈ i_δ.



The next lemma will be a key step of the proof of the loxodromic and WWPD portions of the Hyperbolic Action Theorem (see remarks after the statement).  In brief, given a free splitting F_n  T and a certain subgroup J ⊂ F_n, the lemma gives a criterion for verifying that the restriction of the free splitting action F_n  T to a certain group J ⊂ F_n is nonelementary. To understand the statement of the lemma, the reader may note that by applying Lemma <ref>, the entire setup of the lemma — including hypotheses ItemJActionsAgree and ItemHatHTwistedEquiv — is satisfied for any free splitting F_n  T and any subgroup [T].







Let F_n  T be a free splitting with vertex set V, and let V^ be the subset of all v ∈ V such that (v) is nontrivial. Let (F_n) be a subgroup with normal subgroup J = (F_n). Let J  T denote the restriction to J of the given free splitting action (F_n) ≈ F_n  T. Let V^ be an action that satisfies the following:


  * 
The two actions J  V^, one obtained by restricting to J the given action V^, the other by restricting the action J  T to the subset V^, are identical. 

  * 
The following twisted equivariance condition holds:

    Φ· (γ· x) = Φ(γ) · (Φ· x),   for all Φ∈, γ∈ F_n, x ∈ V^




If no subgroup _F_n(v) (v ∈ V^(T)) is fixed by the whole group , and if the free group J has rank ≥ 2, then the action J  T is nonelementary.


Remarks. Lemma <ref> is applied in Section <ref> when proving the one-edge case of the Hyperbolic Action Theorem, in which case we have a group [T] for which the hypotheses ItemJActionsAgree and ItemHatHTwistedEquiv hold automatically (by Lemma <ref>).

Lemma <ref> is also applied in Section <ref> when proving the multi-edge case of the Hyperbolic Action Theorem. When Lemma <ref> is applied in that case, the group (F_n) is not contained in [T], and the given action V^ does not extend to an action of  on T itself. Nonetheless  will have a kind of “semi-action” on T which extends the given free splitting action J  T, and this will be enough to give an action V^(T) that also satisfies ItemJActionsAgree and ItemHatHTwistedEquiv.

 

 By hypothesis ItemJActionsAgree the action J  T has trivial edge stabilizers and each point of V-V^ has trivial stabilizer. It follows that for each nontrivial α∈ J, either α is elliptic and its fixed point set (α) ⊂ T is a single point of V^, or α is loxodromic with repeller–attractor pair (_-α,_+α) ⊂ T × T and _J(_-α) = _J(_+α) = _J{_-α,_+α} is an infinite cyclic group. 

We claim there is no point v ∈ V^(T) such that (α)={v} for each nontrivial α∈ J. Otherwise that point v is unique, and its stabilizer B_v = _F_n(v) is the unique nontrivial vertex stabilizer fixed by the action of J on subgroups of F_n, because the bijection v ↔ B_v is J-equivariant (by Lemma <ref>). It follows that each Φ∈ also fixes B_v, because J = Φ J Φ^ fixes the subgroup Φ(B_v)  F_n which is also a nontrivial vertex stabilizer, namely Φ(B_v) is equal to the stabilizer of Φ· v by hypothesis ItemHatHTwistedEquiv; by uniqueness of B_v we therefore have Φ(B_v)=B_v. Since this holds for all Φ∈, we have contradicted the hypothesis of the lemma, thus proving the claim.

The proof that the action J  T is nonelementary now follows a standard argument. Some γ∈ J is loxodromic, for otherwise by applying the claim it follows that J has nontrivial elliptic elements α,β with (α) (β) ∈ T but in that case γ=αβ is loxodromic (see for example <cit.>). Since (J) ≥ 2, there exists δ∈ J - _J{_-γ,_+γ}. It follows that γ and δγδ^ are independent loxodromic elements of J.



















































 §.§ Hyperbolic Action Theorem, One-edge case: Proof.



Adopting the notation of the Hyperbolic Action Theorem, we may assume that . Choose  to be a maximal, proper, -invariant free factor system of F_n, and so  is fully irreducible relative to the extension ⊏{[F_n]}, meaning that there is no free factor system  with strict nesting ⊏⊏{[F_n} such that  is invariant under any finite index subgroup of , equivalently (since ) such that  is invariant under . 

The proof of the Hyperbolic Action Theorem breaks into two cases: 

The one-edge case: The co-edge number of  equals 1.

The multi-edge case: The co-edge number of  is ≥ 2. 


Proof of the Hyperbolic Action Theorem in the one-edge case. Assuming that ⊏{[F_n]} is a one-edge extension, using that (), and applying Fact <ref>, there exists a free splitting F_n  T with one edge orbit whose vertex stabilizer system forms the free factor system , and we have equality of stabilizer subgroups () = [T]. It follows that [T]. Applying Lemma <ref>, consider the resulting action [T]  T. We show that the restricted action T satisfies the conclusions of the Hyperbolic Action Theorem (using =T and =). Since every isometry of a tree is either elliptic or loxodromic, Conclusion ItemThmF_EllOrLox of the Hyperbolic Action Theorem holds. 

For proving Conclusion ItemThmF_Nonelem of the Hyperbolic Action Theorem we shall apply Lemma <ref> to the action T, so we must check its hypotheses. Hypotheses ItemJActionsAgree and ItemHatHTwistedEquiv of Lemma <ref> hold by applying Lemma <ref> to the subgroup [T]. Also, by the hypothesis of the Hyperbolic Action Theorem, the subgroup (F_n) does not preserve any proper, nontrivial free factor of F_n, in particular it does not preserve any nontrivial vertex stabilizer of the free splitting F_n  T. Finally, by hypothesis of the Hyperbolic Action Theorem,  is abelian and  is not virtually abelian, and so it follows that J has rank ≥ 2, for otherwise the group (F_n) (F_k+1) would be solvable and hence virtually abelian (by ), a contradiction. The conclusion of Lemma <ref> therefore holds, saying that the restricted action J  T is nonelementary. The action T is therefore nonelementary, which verifies conclusion ItemThmF_Nonelem of the Hyperbolic Action Theorem.  

Conclusion ItemThmF_WWPD of the Hyperbolic Action Theorem says that each loxodromic element of J is a strongly axial, WWPD element for the action T, and this is an immediate consequence of the next lemma (which will also be used in the multi-edge case of the Hyperbolic Action Theorem). 

In formulating this lemma, we were inspired by results of Minasyan and Osin <cit.> producing WPD elements of certain group actions on trees.

 

Let G  T be a group action on a simplicial tree equipped with the simplicial metric on T that assigns length 1 to each edge. If J  G is a normal subgroup such that the restricted action J  T has trivial edge stabilizers, 
then each loxodromic element of J is a strongly axial, WWPD element of the action G  T.


 Given an oriented line A ⊂ T, let the stabilizers of A under the actions of G and of J be denoted

    _G(A)    = {γ∈ G γ(A)=A}
    _J(A)    = J _G(A)

Consider a loxodromic μ∈ J. Clearly the axis A_μ⊂ T of μ is a strong axis of μ. Orient A_μ so that μ translates in the positive direction, with repelling/attracting endpoints _- A_μ, _+ A_μ∈ T. Since J  T has trivial edge stabilizers, the group _J(_- A_μ) = _J(_+ A_μ) = _J(_- A_μ,_+A_μ) is infinite cyclic. 


By <cit.>, for μ to satisfy WWPD with respect to the action G  T is equivalent to saying that the ordered pair _± A_μ = (_-A_μ,_+A_μ) is an isolated point in its orbit G ·_± A_μ⊂ T × T - Δ (this is the property denoted “WWPD (2)” in <cit.>). We may assume that μ is a generator for the infinite cyclic group _J(_± A_μ); if this is not already true then, without affecting the WWPD property for μ, we may replace μ with a generator. Letting ℓ_μ>0 denote the integer valued length of a fundamental domain for the action of μ on A_μ, it follows that any edge path in A_μ of length ℓ_μ is a fundamental domain for the _J(_± A_μ) on A_μ. Choose a subsegment α⊂ A_μ of length ℓ_μ+1. There is a corresponding neighborhood U_α⊂ T × T - Δ consisting of all endpoint pairs of oriented lines in T containing α as an oriented subsegment. Consider γ∈ G such that γ(_± A_μ) ⊂ U_α. It follows that μ' = γμγ^∈ J has axis A_μ' = γ(A_μ) and that A_μ A_μ' contains α, and hence A_μ A_μ' has length ≥ℓ_μ+1. Also, the map γ A_μ→ A_μ' takes the restricted orientation of the subsegment A_μ A_μ'⊂ A_μ to the restricted orientation on the subsegment γ(A_μ A_μ') ⊂ A_μ'. Let the edges of the oriented segment A_μ A_μ' be parameterized as E_0 E_1 E_2 … E_J, J ≥ℓ_μ. Since μ and μ' both have translation number ℓ_μ it follows that μ(E_0)=μ'(E_0)=E_ℓ_μ, and so μ^μ' ∈_J(E_0). Since J has trivial edge stabilizers it follows that μ=μ' and so γ∈_J{_- A_μ,_+ A_μ} = μ. And having shown that γ preserves orientation, it follows that γ(_± A_μ) = _± A_μ. This shows that _± A_μ is isolated in its orbit G ·_± A_μ, being the unique element in the intersection U_α (G ·_± A_μ).


This completes the proof of the Hyperbolic Action Theorem in the one-edge case.




§ HYPERBOLIC ACTION THEOREM, MULTI-EDGE CASE: INTRODUCTION.
 


The proof of the Hyperbolic Action Theorem in the multi-edge case will take up the rest of the paper. In this section, we give a broad outline of the methods of proof, followed by some motivation coming from well-known constructions in geometric group theory. 



 §.§ Outline of the multi-edge case.



We will make heavy use of the theory of abelian subgroups of (F_n) developed by Feighn and Handel . In very brief outline here are the main features of that theory we will need. 



Disintegration groups. (, and see Section <ref>) Any element of (F_n) has a uniformly bounded power which is rotationless, meaning roughly that each of various natural finite permutations induced by that element are trivial. Any rotationless ϕ∈(F_n) has a particularly nice relative train track representative called a . For any  f  G → G there is an associated abelian subgroup (f) (F_n) that contains ϕ and is called the “disintegration subgroup” of f. The idea of the disintegration group is to first disintegrate or decompose f into pieces, one piece for each non-fixed stratum H_r, equal to f on H_r and to the identity elsewhere. Then one re-integrates those pieces to form generators of the group (f), by choosing a list of non-negative exponents, one per non-fixed stratum, and composing the associated powers of the pieces of f. However, in order for this composition to be continuous and a homotopy equivalence, and for (f) to be abelian, the exponents in that list cannot be chosen independently. Instead two constraints are imposed: the non-fixed strata are partitioned into a collection of “almost invariant subgraphs” on each of which the exponent must be constant; and certain linear relations are required amongst strata that wrap around a common twist path of f.

The following key theorem about disintegration groups lets us study an abelian subgroup of (F_n) (up to finite index) by working entirely in an appropriate disintegration group:


[<cit.>]
For every rotationless abelian subgroup (F_n) there exists ϕ∈ such that for every  f  G → G representing ϕ, the intersection (f) has finite index in .


The proof of the Hyperbolic Action Theorem in the multi-edge case. The detailed proof is carried out in Section <ref>, based on material whose development we soon commence. Here is a sketch.

By a finite index argument we may assume that every element of the abelian group  is rotationless. Let  be a maximal, proper, -invariant free factor system of F_n. Being in the multi-edge case means that  has co-edge number ≥ 2. From the Disintegration Theorem we obtain ϕ∈, and we apply the conclusions of that theorem to a  representative f  G → G of ϕ having a proper core filtration element G_t representing . We may assume that (f), by replacing  with its finite index subgroup (f). From the construction of the disintegration group (f), any core filtration element properly contained between G_t and G would represent a free factor system which is (f)-invariant, hence -invariant, contradicting maximality of . Thus G_t is the maximal proper core filtration element, and ϕ is fully irreducible relative to . Since  has co-edge number ≥ 2, the top stratum H_u is an  stratum. By maximality of G_t, every stratum strictly between G_t and G is either an -linear edge with terminal endpoint attached to G_t or a zero stratum enveloped by H_u.

The hard work of the proof breaks into two major phases, the first of which is:



  Sections <ref>, 
<ref>: Construction and hyperbolicity of . 


The hyperbolic metric space  needed for verifying the conclusions of the Hyperbolic Action Theorem is constructed in terms of the  f  G → G (see Section <ref> for further motivation of the construction). First we describe  in the simpler case that f has no height u indivisible Nielsen path. Starting with f  G → G, lift to the universal cover to obtain f  G → G. Let G_u-1⊂ G be the total lift of G_u-1. Collapse to a point each component of G_u-1, obtaining a simplicial tree T. The map f  G → G induces a map f_T  T → T. Let  be the bi-infinite mapping cylinder of f_T, obtained from T  [0,1] by identifying (x,n,1) ∼ (f_T(x),n+1,0) for each x ∈ T and n ∈. 

The construction of  is more complex when H_u is geometric, meaning that f has a unique (up to inversion) height u indivisible Nielsen path ρ, and ρ is closed, forming a circuit c in G (, and see ). Each line c ⊂ G obtained by lifting c projects to a line in T called a geometric axis, and the projections to T of the lifts of ρ in c are fundamental domains for that axis. In the bi-infinite mapping cylinder as defined above, the portion of the mapping cylinder corresponding to each geometric axis is a quasiflat, contradicting hyperbolicity. Thus we do not take  to be the bi-infinite mapping cylinder itself, instead  is constructed from the bi-infinite mapping cylinder by coning off each geometric axis in T  n for each n ∈, using one cone point per geometric axis, attaching arcs that connect the cone point to the endpoints of the fundamental domains of that axis.

In Section <ref>, we use the Mj-Sardar combination theorem <cit.> to prove hyperbolicity of . 
The Mj-Sardar theorem, a generalization of the Bestvina-Feighn Combination Theorem <cit.>, requires us to verify a flaring hypothesis. To do this, in Section <ref> we study relative flaring properties of  f  G → G, specifically: how f flares relative to the lower filtration element G_u-1; and in the geometric case, how f flares relative to the Nielsen path ρ. Then in Section <ref>, we study flaring properties of the induced map f_T  T → T (and, in the geometric case, flaring properties of the induced map obtained by coning off each geometric axis of T). The required flaring hypothesis on  itself can then be verified in Section <ref>, allowing application of the Mj-Sardar theorem to deduce hyperbolicity of .



The other major phase of the proof is:


  Sections <ref>, <ref>: Use the theory of disintegration groups to obtain an isometric action  with appropriate WWPD elements.

To do this, one uses that there is a number λ > 0 and a homomorphism 

    PF_Λ(f) →

such that for each ψ∈(f), the lamination Λ is an attracting lamination for ψ if and only if PF(ψ)>0, in which case the stretch factor is λ^(ψ). One would like to think of H_u as an  stratum for ψ having Perron-Frobenius eigenvalue λ^(ψ), but this does not yet make sense because we do not yet have an appropriate topological representative of ψ on G. We define a subgroup and sub-semigroup 

    _0(f)    = (PF_Λ) 
    _+(f)    = PF_Λ^[0,∞)

and we then lift the sequence of inclusions _0(f) ⊂_+(f) ⊂(f) (F_n) to a sequence of inclusions
to 

    D_0(f) ⊂ D_+(f) ⊂ D(f) (F_n)

The hard work in Section <ref> is to use the theory of disintegration groups to construct a natural action of the semigroup _+(f) on T, in which each element Ψ∈_+(f) acts on T by stretching each edge by a uniform factor equal to λ^(ψ), and such that for each Ψ∈_+(f) the resulting map x ↦Ψ· x of T is Ψ-twisted equivariant. When restricted to the subgroup _0 we obtain an action on T by twisted equivariant isometries, which allows us to identify the action _0  T with the one described in Lemma <ref>.

Then what we do in Section <ref> is to suspend the semigroup action of _+(f) on T (and, in the geometric case, on the graph obtained by coning off geometric axes), obtaining an isometric action (f). By restriction we obtain the required action .

Finally,


  Section <ref>: Put the pieces together and verify the conclusions of the Hyperbolic Action Theorem.

The basis of the WWPD conclusions in the multi-edge case of the Hyperbolic Action Theorem is Lemma <ref>, which has already played the same role for the one-edge case. 




 §.§ Motivation: Suspension actions and combination theorems.
 

The construction of the hyperbolic suspension action (f) may be motivated by looking at some familiar examples in a somewhat unfamiliar way. 

Example: Mapping torus hyperbolization after Thurston.
Consider a pseudo-Anosov homeomorphism f  S → S of a closed, oriented hyperbolic surface S of genus ≥ 2 with associated deck action π_1 S  S = ^2. The map f uniquely determines an outer automorphism ϕ∈(π_1 S). The associated extension group Γ(π_1 S) is the inverse image of the infinite cyclic group ϕ(π_1 S) under the natural homomorphism (π_1 S) ↦(π_1 S). A choice of Φ∈(π_1 S) representing ϕ naturally determines a semidirect product structure Γ≈π_1(S) _Φ. 

The deck transformation action π_1 S ^2 extends naturally to an action Γ^2, whereby if Ψ∈Γ projects to ψ = ϕ^k then the action of Ψ on ^2, denoted F_Ψ^2 →^2, is the lift of f^k whose induced action on the circle at infinity ^2 agrees with the induced action of γ. Equivalently, F_Ψ is the unique Ψ-twisted equivariant lift of f^k. Although the deck action π_1 S ^2 is by isometries, the extended action Γ^2 is not by isometries. 

However, there is a way to suspend the action Γ^2 obtaining an isometric action Γ, as follows. The suspension space  is the bi-infinite mapping cylinder obtained as the quotient of ^2 ×× [0,1] under the identifications (x,n,1) ∼ (F_Φ(x),n+1,0). One may check (and we shall do in Section <ref> in a different context) that there is an action Γ which is generated by letting π_1 S act as the deck group on ^2 ≈^2 {0}⊂ and extending naturally over the rest of , and by letting Φ according to the formula Φ· (x,n,t)=(x,n-1,t). One may also check that the hyperbolic metrics on the slices ^2 × n × 0 extend to a path metric on , uniquely up to quasi-isometry, such that the action Γ is by isometries.

Returning to the land of the familiar, the group Γ may be identified with the fundamental group of the 3-dimensional mapping torus M_f of the surface homeomorphism f  S → S. By Thurston's hyperbolization theorem applied to M_f, the suspension space  may be identified up to Γ-equivariant quasi-isometry with the universal cover M_f ≈^3 equipped with its deck transformation action by the group Γ≈π_1 M_f. 

Example: Mapping torus hyperbolization after . Consider now an irreducible train track representative f  G → G of a nongeometric, fully irreducible outer automorphism ϕ∈(F_n). Again we have a natural extension group Γ(F_n) of ϕ∈(F_n), with a semidirect product structure Γ = F_n _Φ determined by a choice of Φ∈(F_n) representing ϕ.

Unlike in the previous situation where we started with a surface homeomorphism, here we have started with a non-homeomorphic topological representative f  G → G of ϕ, and so the deck action F_n  G does not extend to an action Γ G. But it does extend to an action of the semigroup Γ_+ which is the inverse image under (F_n) ↦(F_n) of the semigroup ϕ_+ = ϕ^i  i ∈{0,1,2,…}: for each Ψ∈Γ_+ mapping to ϕ^i with i ≥ 0, the associated map F_Ψ G → G is the unique Ψ-twisted equivariant lift of f^i. Although the deck action F_n  G is by isometries, the semigroup action Γ_+  G is not by isometries.

But there is a way to suspend the semigroup action Γ_+  G to an isometric action Γ where  is the bi-infinite mapping cyclinder of F_Φ G → G, defined exactly as above, namely the quotient space  of G ×× [0,1] where (x,n,1) ∼ (F_Φ(x),n+1,0) with an appropriate metric. What is done in <cit.> is to apply properties of the train track map f  G → G to prove a flaring hypothesis, allowing one to apply Bestvina–Feighn combination theorem <cit.> to conclude that  is Gromov hyperbolic, thus proving that Γ is a hyperbolic group.

Flaring methods. Our proof of the multi-edge case will use the combination theorem of Mj and Sarder <cit.>, a generalization of the Bestvina and Feighn combination theorem <cit.>. A common feature of the above examples that is shared in the construction of this paper is that  is a “metric bundle over ”: there is a Lipschitz projection map π↦ such that the minimum distance from π^(s) to π^(t) equals s-t for all s,t ∈, and each point x ∈ is contained in the image of a geodesic section σ→ of the map π. In studying the large scale geometry of  it is important to study quasigeodesic sections σ→ and their flaring properties. In our context it is convenient to translate these concepts into dynamical systems parlance: each such quasigeodesic section turns out to be a “pseudo-orbit” of a suspension semiflow on  whose true orbits are the geodesic sections (see the closing paragraphs of Section <ref>). The combination theorems of <cit.> and its predecessor <cit.> share key hypotheses regarding the asymptotic “flaring” behavior of such pseudo-orbits (see Definition <ref>). We remark, though, that those combination theorems hold in much more general settings, e.g. certain kinds of metric bundles over more general metric base spaces.

In Section <ref> we study flaring of pseudo-orbits in the context of a relative train track map, which is then used in Section <ref> to extend to further contexts building up to the suspension space , its metric bundle ↦, and its pseudo-orbits, allowing application of the Mj-Sardar combination theorem.




§ FLARING IN A TOP  STRATUM



We assume the reader is familiar with the theory of  representatives of elements of (F_n), developed originally in . More narrowly, we shall focus on the terminology and notation of a  having a top  stratum, as laid out in Section 2.2 of Part I , under the heading  properties of . Here is a brief review, in order to fix notations for what follows in the rest of this paper; see <cit.> for detailed citations drawn primarily from . At present we need no more about  than is described here; when needed in Section <ref> we will give a more thorough review of .







We fix ϕ∈(F_n) and a relative train track representative f  G → G with associated f-invariant filtration ∅ = G_0 ⊂ G_1 ⊂⋯⊂ G_u = G satisfying the following:


  * 
 The top stratum H_u is  with Perron-Frobenius transition matrix M_u having top eigenvalue λ > 1. The attracting lamination of ϕ corresponding to H_u is denoted Λ^+ or just Λ, and its dual repelling lamination is Λ^-.

  * 
There exists (up to reversal) at most one indivisible periodic Nielsen path ρ of height u, meaning that ρ is not contained in G_u-1. In this case ρ and its inverse ρ̅ are Nielsen paths, there is a decomposition ρ = αβ where α,β are u-legal paths with endpoints at vertices, and (α,β) is the unique illegal turn in H_u. At least one endpoint of ρ is disjoint from G_u-1; we assume that ρ is oriented with initial point p ∉G_u-1 and terminal point q. The initial and terminal directions are distinct fixed directions in H_u. The stratum H_u is geometric if and only if ρ exists and is a closed path in which case p=q.

  * 
From the matrix M_u one obtains an eigenlength function l_(σ) defined on all paths σ in G having endpoints at vertices, and having the following properties: 


  * for each edge E ⊂ G we have l_(E)=0 if and only if E ⊂ G_u-1; 

  * in general if σ = E_1 … E_K then l_(σ) = l_(E_1) + ⋯ + l_(E_K); 

  * for all edges E ⊂ G we have l_(f(E)) = λ  l_(E). 

  * If ρ = αβ exists as in ItemCTiNP then l_(α)=l_(β) = 1/2 l_(ρ). 


  * 
If γ is a path with endpoints at vertices of H_u or a circuit crossing an edge of H_u then for all sufficiently large i the path f^i_#(γ) has a splitting with terms in the set {edges of H_u}{ρ,ρ̅}{paths in G_u-1 with endpoints in H_u}. 


In what follows, there are three cases to consider: the ageometric case where ρ does not exist; the parageometric case where ρ exists but is not closed; and the geometric case where ρ exists and is closed. Occasionally we will need to consider these cases separately. But for the most part we will attempt to smoothly handle all three cases simultaneously, using the following conventions:


  * In the ageometric case, where ρ does not exist, the notation ρ should simply be ignored (for an exception see the “Notational Convention” following Corollary <ref>);

  * In the parageometric case, where ρ exists but is not closed, the notations ρ^i for i≥ 2 should be ignored.

This completes Notations <ref>.



The main result of this section is Proposition <ref>, a uniform flaring property for the top  stratum H_u of f. This result generalizes <cit.> which covers the special case that f is a nongeometric train track map, that is, u=1 and if ρ exists then it is not closed. There are two features which distinguish our present general situation from that special case. First, our flaring property is formulated relative to the penultimate filtration element G_u-1 of G. Second, if the height u indivisible Nielsen path ρ exists then our flaring property is formulated relative to ρ. Taken together, these two features require very careful isolation and annihilation of the metric effects of paths in G_u-1 and of the path ρ. This task is particularly tricky in the geometric case where ρ exists and is closed, in which situation we must also annihilate the metric effects of all iterates ρ^n.




 §.§ The path functions L_u and L_.



Throughout this paper, and following standard terminology in the literature of relative train track maps, a (finite) path in a graph G is by definition a locally injective continuous function σ [0,1] → G. We often restrict further by requiring that the endpoints σ(0),σ(1) be vertices in which case σ is a concatenation of edges without backtracking. A more general concatenation of edges in which locally injectivity may fail, hence backtracking may occur, will be called an “edge path”.

Generalizing the eigenlength function l_ of Notations <ref> ItemCTEigen, a path function on a graph is simply a function l(·) which assigns, to each finite path σ having endpoints at vertices, a number l(σ) ∈ [0,), subject to two metric-like properties:


  * l(·) is symmetric, meaning that l(σ)=l(σ̅). 

  * l(·) assigns length zero to any trivial path. 

We do not require that l(σ) > 0 for all nontrivial paths σ. We also do not require additivity: the value of l(·) on a path σ need not equal the sum of its values over the 1-edge subpaths of σ. We also do not require any version of the triangle inequality, but see Lemma <ref> and the preceding discussion regarding coarse triangle inequalities for the path functions of most interest to us.

Henceforth in this section we adopt adopt Notations <ref>. 


The “big L” path functions L_u, L_ are defined in terms of auxiliary “little l” path functions l_u, l_ by omitting certain terms. The function l_ is already defined in Notations <ref> ItemCTEigen. 

Consider a path σ in G with edge path decomposition σ = E_1 … E_K. Define l_u(σ) to be the number of terms E_k contained in H_u. Now define L_u(σ) and L_(σ) by summing l_u(E_k) and l_(E_k) (respectively) only over those terms E_k which do not occur in any ρ or ρ̅ subpath of σ. Note that for each edge E ⊂ G the “little l” eigenvector equation l_(f(E)) = λ  l_(E) implies:


  “Big L” eigenvector equation:

    L_(f(E)) = λ  L_(E)   for all edges E ⊂ G.


To see why this holds, if E ⊂ G_u-1 both sides are zero. For E ⊂ H_u this follows from the fact that f(E) is H_u-legal and hence has no ρ or ρ̅ subpath.

The following “quasicomparability” result is an obvious consequence of the fact that the Perron-Frobenius eigenvector of the transition matrix M_u has positive entries:



There is a constant K = K_<ref>(f) ≥ 1 such that for all edges E of H_u we have 1/K L_(E) ≤ L_u(E) ≤ K · L_(E). 

The next lemma says in part that ρ and ρ̅ subpaths never overlap. 


 
For any path σ in G with endpoints, if any, at vertices, there is a unique decomposition σ = …σ_i  … σ_j  … with the following properties:


  * Each term σ_i is an edge or a copy of ρ or ρ̅.

  * Every subpath of σ which is a copy of ρ or ρ̅ is a term in the decomposition. 

 
Note that we do not assume σ is finite in this statement. 



Leaving the proof of Lemma <ref> for the end of Section <ref>, we give applications. For a path σ in G with endpoints at vertices, let K_ρ(σ) denote the number of ρ or ρ̅ subpaths in the Lemma <ref> decomposition of σ.



For any finite path σ with endpoints at vertices, letting its Lemma <ref> decomposition be σ = σ_1 …σ_B, we have

    L_u(σ)    = 
    
    l_u(σ) - K_ρ(σ) · l_u(ρ)    in general
    
    l_u(σ)    if ρ does not exist
    and similarly
    L_(σ)    = 
    
    l_(σ) - K_ρ(σ) · l_(ρ)    in general
    
    l_(σ)    if ρ does not exist


Notational convention when ρ does not exist:

Motivated by Corollary <ref>, we extend the meaning of the notations l_u(ρ) and l_(ρ) to the case that ρ does not exist by defining  

    l_u(ρ)=l_(ρ)=0
 




For any finite path σ in G with endpoints at vertices, there is a unique decomposition σ = μ_0  ν_1  μ_1  … ν_A  μ_A with the following properties:


  * If ρ does not exist then A=0 and σ=μ_0.

  * If ρ exists then each ν_a is an iterate of ρ or ρ̅ (the iteration exponent equals 1 if ρ is not closed), and each μ_a contains no ρ or ρ̅ subpath.

  * 
If ρ exists, and if 1 ≤ a < a+1 ≤ A-1, then at least one of the subpaths μ_a,μ_a+1 contains an edge of H_u; if in addition ρ is closed then each μ_a subpath contains an edge of H_u.


Note that in the context of Corollary <ref> the subpaths μ_a are nondegenerate for 1 ≤ a ≤ A-1, but the subpaths μ_0 and μ_A are allowed to be degenerate.

   Everything is an immediate consequence of Lemma <ref> except perhaps item ItemEveryOtherHasHu, which follows from the fact that at least one endpoint of ρ is disjoint from G_u-1 (Notations <ref> ItemCTiNP). 


The following is an immediate consequence of Lemma <ref> and Corollary <ref>, with Lemma <ref> applied term-by-term for the “Furthermore” part:

 

For any finite path σ with endpoints at vertices, and using the Corollary <ref> decomposition of σ, we have

    L_u(σ) = ∑_a=1^A L_u(μ_a) = ∑_a=1^A l_u(μ_a),         
    L_(σ) = ∑_a=1^A L_(μ_a) = ∑_a=1^A l_(μ_a)

Furthermore, letting K = K_<ref>(f) ≥ 1, we have

    1/K L_(σ) ≤ L_u(σ) ≤ K · L_(σ)


The latter inequality of Corollary <ref> gives us the freedom to switch back and forth between L_ and L_u, using L_u in more combinatorial situations and L_ in more geometric situations through much of the rest of the paper.




  
Coarse triangle inequalities for L_u and L_. The path functions l_u and l_ satisfy a version of the triangle inequality: for any two paths γ,δ with endpoints at vertices such that the terminal vertex of γ equals the initial vertex of δ we have

    l_u[γδ] ≤ l_u(γ)+l_u(δ),    l_[γδ] ≤ l_(γ) + l_(δ)

where [·] denotes the operation that straightens an edge path rel endpoints to obtain a path. For L_u and L_ the best we can get are coarse triangle inequalities:


There exists a constant C=C_<ref> such that for any finite paths γ,δ in G with endpoints at vertices such that the terminal point of γ coincides with the initial point of δ we have

    L_u[γδ]    ≤ L_u(γ) + L_u(δ) + C 
    
    L_[γδ]    ≤ L_(γ) + L_(δ) + C
 


 Consider the Lemma <ref> decompositions of γ and δ. We may write γ=γ_1γ_2γ_3, δ=δ_1δ_2δ_3 so that [γδ] = γ_1 γ_2 δ_2 δ_3 and so that the following hold: γ_1 is a concatenation of terms of the Lemma <ref> decomposition of γ, and γ_2 is either degenerate or an initial subpath of a single ρ or ρ̅ term of that decomposition; δ_2 is either degenerate or a terminal subpath of a single ρ or ρ̅ term of the Lemma <ref> decomposition of δ, and δ_3 is a concatenation of terms of that decomposition. It follows that

    L_u[γδ]    ≤ L_u(γ_1) + l_u(γ_2) + l_u(δ_2) + L_u(δ_3) 
       ≤ L_u(γ) + L_u(δ) + 2 l_u(ρ) 
    and similarly
    L_[γδ]    ≤ L_(γ) + L_(δ) + 2 l_(ρ)





  
Quasi-isometry properties of f. The next lemma describes quasi-isometry properties of the relative train track map f  G → G with respect to the path functions L_u and L_.

 
There exist constants D=D_<ref> > 1, E=E_<ref> > 0 such that for any finite path γ in G with endpoints at vertices, the following hold:

    L_u(f_#(γ)) ≤ D · L_u(γ) + E      and   L_u(γ) ≤ D · L_u(f_#(γ)) + E 
    
    L_(f_#(γ)) ≤ D · L_(γ) + E      and   L_(γ) ≤ D · L_(f_#(γ)) + E





 Once D,E are found satisfying ItemQiU, by applying Corollary <ref> we find D,E satisfying ItemQiPF, and by maximizing we obtain D,E satisfying both.


If ρ exists let P be its endpoint set, a single point if H_u is geometric and two points otherwise; and if ρ does not exist let P=∅. Each point of P is fixed by f. By elementary homotopy theory (see e.g. Lemma <ref> ItemGPHomInvExists in Section <ref>), the map of pairs f  (G,P) → (G,P) has a homotopy inverse f̅ (G,P) → (G,P) in the category of topological pairs. By homotoping f̅ rel vertices, we may assume that f̅ takes vertices to vertices and edges to (possibly degenerate) paths. If ρ exists then, using that ρ = f_#(ρ), and applying f̅_# to both sides, we obtain f̅_#(ρ)=f̅_#(f_#(ρ)) = ρ. 

We prove the following general fact: if g  (G,P) → (G,P) is a homotopy equivalence of pairs which takes vertices to vertices and takes edges to paths, and if g_#(ρ)=ρ assuming ρ exists, then there exist D ≥ 1, E ≥ 0 such that for each path δ with endpoints at vertices we have

    L_u(g_#(δ)) ≤ D · L_u(δ) + E

This obviously suffices for the first inequality of ItemQiU taking g=f and δ=γ. It also suffices for the second inequality of ItemQiU taking g = f̅ and δ=γ because in that case we have g_#(ρ) = f̅_#(ρ) = f̅_#(f_#(ρ)) = ρ.

To prove ItemQiG, consider the Corollary <ref> decomposition γ = μ_0  ν_1  μ_1  …ν_A  μ_A. Applying Corollary <ref> we have

    L_u(γ) = l_u(μ_0) + ⋯ + l_u(μ_A)

We also have

    g_#(γ) = [g_#(μ_0)  ν_1   g_#(μ_1)  … ν_A   g_#(μ_A)]

By inductive application of Lemma <ref> combined with the fact that each L_u(ν_a)=0, it follows that

    L_u(g_#(γ))    ≤ L_u(g_#(μ_0)) + ⋯ + L_u(g_#(μ_A)) + 2   A   C_<ref>
       ≤ l_u(g_#(μ_0)) + ⋯ + l_u(g_#(μ_A)) + 2   A    C_<ref>
       ≤ D'(l_u(μ_0) + ⋯ + l_u(μ_A)) + 2   A    C_<ref>

where D' is the maximum number of H_u edges crossed by g(E) for any edge E ⊂ G. If ρ does not exist then A=0 and we are done. If ρ exists then, by Corollary <ref> ItemEveryOtherHasHu, if 1 ≤ a ≤ a+1 ≤ A-1 then at least one of l_u(μ_a),l_u(μ_a+1) is ≥ 1. It follows that A ≤ 2(l_u(μ_0)+⋯+l_u(μ_A)) + 3 and therefore

    L_u(g_#(γ)) ≤ (D' + 4   C_<ref>) L_u(γ) + 6   C_<ref>







  
Proof of Lemma <ref>. The lemma is equivalent to saying that two distinct ρ or ρ̅ subpaths of σ cannot overlap in an edge. Lifting to the universal cover G this is equivalent to saying that for any path σ⊂ G, if μ and μ' are subpaths of σ each of which projects to either ρ or ρ̅, and if μ and μ' have at least one edge in common, then μ = μ'.

As a first case, assume that μ and μ' both project to ρ or both project to ρ̅; up to reversal we may assume the former. Consider the decomposition ρ = αβ where α,β are legal and the turn (α̅, β) is illegal and contained in H_u. There are induced decompositions μ= αβ and μ' = α' β'. If the intersection μ∩μ' contains the height u illegal turn of μ or μ' then μ = μ' so assume that it does not. After interchanging μ and μ' we may assume that μ∩μ' ⊂β. Projecting down to ρ we see that β = β_1 β_2 and α = α_1 α_2 where β_2=α_1 is the projection of μ∩μ'. This implies the initial directions of α̅_1 and of β_2 (which are respectively equal to the terminal and initial directions of ρ) are fixed, distinct, and in H_u, and so for all k ≥ 0 the initial directions of f^k_#(α̅_1) and of f^k_#(β_2) are also fixed, distinct, and in H_u. Using the eigenlength function l_ (Notations <ref> ItemCTEigen) we have l_(α)=l_(β) and l_(α_1)=l_(β_2) and so l_(α_2)=l_(β_1). Thus when the path f^k_#(α_1) f^k_#(α_2)f^k_#( β_1)f^k_#(β_2) is tightened to form f^k_#(ρ) = ρ for large k, the subpaths f^k_#(α_2) and f^k_#( β_1) cancel each other out, and so the concatenation f^k_#(α_1) f^k_#(β_2) tightens to ρ. But this contradicts that the two terms of the concatenation are u-legal and that the turn {f^k_#(α̅_1),f^k_#(β_2)} taken at the concatenation point is also u-legal as seen above.

The remaining case is that orientations on the projections of μ and μ' do not agree. In this case there is either an initial or terminal subpath of ρ that is its own inverse, which is impossible. 




 §.§ Flaring of the path functions L_u and L_



In this section we continue to adopt Notations <ref>. 





For each path function l on G and each η≥ 0 we define a relation 

β∼γ
on paths β,γ in G with endpoints at vertices: this relation means that there exist paths α,ω with endpoints at vertices such that γ = [αβω] and such that l(α),l(γ) ≤η. Note that this relation is symmetric because β = [α̅γω̅]. When we need to emphasize the dependence of the relation on l and η we will write more formally β ηl γ.




A path function l on G is said to satisfy the flaring condition with respect to f  G → G if for each μ > 1, η≥ 0 there exist integers R ≥ 1 and A ≥ 0 such that for any sequence of paths β_-R,β_-R+1,…,β_0,…,β_R-1,β_R in G with endpoints at vertices, the flaring inequality 

    μ· l(β_0) ≤max{ l(β_-R), l(β_R) }
 
holds if the following two properties hold:



Threshold property: l(β_0) ≥ A;



Pseudo-orbit property: β_r ηl f_#(β_r-1) for each -R < r ≤ R.



Remark. These two properties are translations into our present context of two requirements in the original treatment of flaring by Bestvina and Feighn <cit.>: the threshold property corresponds to the requirement of large “girth”; and the “pseudo-orbit property” corresponds to the “ρ-thin” requirement.





The path functions L_u and L_ each satisfy the flaring condition with respect to f  G → G, with constants R = R_<ref>(μ,η) and A = A_<ref>(μ,η).


Much of the work is to prove the following, which is a version of Proposition <ref> under the special situation of Definition <ref> where l=L_ and η = 0. Note that η=0 implies that β_r=f_#(β_r-1), and so the choice of γ = β_-R determines the whole sequence up to β_R.



For any ν>1 there exist positive integers N ≥ 1 and A=A_<ref>≥ 0 so that if γ is a finite path in G with endpoints at vertices and if L_u(f^N_#(γ)) ≥ A then 

    ν L_u(f^N_#(γ)) ≤max{ L_u(γ), L_u(f^2N_#(γ)) }





The proof of this special flaring condition, which takes up Sections <ref>–<ref>, is similar in spirit to <cit.>, and is based on two special cases: a “negative flaring” result, Lemma <ref> in Section <ref>; and a “positive flaring” result, Lemma <ref> estimate3 in Section <ref>. These two special cases are united in Section <ref>, using the uniform splitting property given in Lemma 3.1 of <cit.>, in order to prove the Special Flaring Condition.

Before embarking on all that, we apply Lemma <ref> to:

 Applying Corollary <ref> it is easy to see that L_u satisfies a flaring condition if and only if L_ satisfies a flaring condition. Thus it suffices to assume that the special flaring condition for L_u holds, and use it to prove the general flaring condition for L_u. The idea of the proof is standard: the exponential growth given by Proposition <ref> swamps the constant error represented by the relation ηL_u which we will denote in shorthand as ∼. The hard work is to carefully keep track of various constants and other notations.

Fixing μ>1 and η≥ 0, consider the relation ∼ given formally as ηL_u. Fix an integer R ≥ 1 whose value is to be determined — once an application of Lemma <ref> has been arranged, R will be set equal to the constant N of that lemma.

Consider a sequence of paths 

    β_-R,β_-R+1,…,β_0,…,β_R-1,β_R
 
in G with endpoints at vertices, such that we have

    β_r ∼ f_#(β_r-1)   for   -R < r ≤ R
 
Choose a vertex V ∈ G which is f-periodic, say f^K(V)=V for some integer K ≥ 1. We may assume that β_0 has endpoints at V: if not then there are paths α'_0,ω'_0 of uniformly bounded length such that β'_0 = [α'_0β_0ω'_0] has endpoints at V, and replacing β_0 with β'_0 reduces the flaring property as stated to the flaring property with uniform changes to the constants μ, η, and A.

We choose several paths with endpoints at vertices, denoted α_r,ω_r,γ_r,α'_r,ω'_r, as follows. First, using that f_#(β_r-1) ∼β_r, there exist α_r,ω_r so that L_u(α_r),L_u(ω_r) ≤η and β_r = [α_r   f_#(β_r-1)  ω_r] and hence f_#(β_r-1) = [α̅_r β_r ω̅_r]. Next, anticipating an application of Lemma <ref>, choose γ_-R so that f^R_#(γ_-R)=β_0, and then for -R ≤ r ≤ R let γ_r = f^r+R_#(γ_-R), hence γ_0=β_0. Finally, choose α'_r,ω'_r so that γ_r = [α'_r β_r ω'_r] and hence β_r = [α̅'_r γ_r ω̅'_r] (in particular α'_0,ω'_0 are trivial paths). We also require α'_r,ω'_r to be chosen so that if r > -R then we have

    f_#(β_r-1)    = f_#(α̅'_r-1 γ_r-1 ω̅'_r-1) 
       = [f_#(α̅'_r-1)  γ_r   f_#(ω̅'_r-1)] 
       = [f_#(α̅'_r-1)  α'_r  β_r  ω'_r   f_#(ω̅'_r-1)] 
       = [ [f_#(α̅'_r-1)  α'_r]_=  α̅_r β_r  [ω'_r   f_#(ω̅'_r-1)]_=  ω̅_r ]

from which we record the following identities:

    (*)     α̅_r = [f_#(α̅'_r-1) α'_r]     ω̅_r = [ω'_r f_#(ω̅'_r-1)]

To see why these choices of γ_r,α'_r,ω'_r are possible, we work with a lift to the universal cover f  G → G. First choose any lift β_-R. By induction for -R< r ≤ R the lifts α_r, ω_r, β_r are determined by the equation β_r = [α_r  f_#(β_r-1) ω_r]. Using f-periodicity of V, the f^R-preimages of the initial and terminal vertices of β_0 contain vertices of G amongst which we choose the initial and terminal vertices of γ̃_-R, respectively; it follows that f^R_#(γ_-R) = β_0. Then define γ_r =  f^r+R_#(γ_-R), define α'_r to be the path from the initial vertex of γ_r to the initial vertex of β_r, and define ω'_r to be the path from the terminal vertex of β_r to the terminal vertex of γ_r. Projecting down from G to G we obtain the paths γ_r,α'_r,ω'_r. The identities (*) follow from the evident fact that the paths α'_r, α_r may be concatenated to form a path α'_r α_r, that the two paths α'_r α_r and f_#(α'_r-1) have the same initial endpoint as f_#(γ'_r-1)=γ_r, and that those two paths have the same terminal endpoint as the initial endpoint of f_#(β_r-1); and similarly for the ω's.  

We need new upper bounds on the quantities L_u(α'_r) and L_u(ω'_r) which represent the difference between L_u(β_r) and L_u(γ_r). These bounds will be expressed in terms of the known upper bound η on L_u(α_r) and L_u(ω_r), and will be derived by applying applying Lemmas <ref> and <ref> inductively, starting from L_u(α'_0)=L_u(ω'_0)=0 and applying (*) in the induction step. These new upper bounds will then be used to derive an expression for the threshold constant A_<ref>, which will be so large so that the differences L_u(α'_r), L_u(ω'_r) become insignificant.

The new upper bounds have the form 

    L_u(α'_r), L_u(ω'_r) ≤ F_r(C,D,E,η)   for -R ≤ r ≤ R

where for each r the expression F_r(C,D,E,η) represents a certain polynomial with integer coefficients in the variables C,D,E,η and where we substitute 

    C = C_<ref>,    D=D_<ref>,    E=E_<ref>
 
The proofs of these inequalities are almost identical for α' and for ω'; we carry out the proof for the latter. To start the induction, since γ_0=β_0 the path ω'_0 degenerates to a point and so L_u(ω'_0)=0 ≡ F_0(C,D,E,η). Inducting in the forward direction on the interval 1 ≤ r ≤ R, and using from (*) that 


    ω'_r   = [ω̅_r f_#(ω'_r-1)] 
    we have
    L_u(ω'_r)    ≤ L_u(ω_r) + L_u(f_#(ω'_r-1)) + C 
       ≤η + (D · F_r-1(C,D,E,η) + E ) + C 
       ≡ F_r(C,D,E,η)

Inducting in the backward direction on the interval -R ≤ r ≤ -1, and using from (*) that 

    f_#(ω'_r) = [ω_r+1 ω'_r+1]

we have

    L_u(ω'_r)    ≤ D · L_u(f_#(ω'_r)) + E 
       ≤ D · (L_u(ω_r+1) + L_u(ω'_r+1)+C) + E 
       ≤ D (η + F_r+1(C,D,E,η)+C) + E 
       ≡ F_r(C,D,E,η)
 
To summarize, we have proved

    L_u(α'_-R), L_u(ω'_-R)    ≤ F_-R(C,D,E,η) 
    
    L_u(α'_R), L_u(ω'_R)    ≤ F_R(C,D,E,η)

From this it follows that

    L_u(γ_-R)    ≤ L_u(α'_-R) + L_u(β_-R) + L_u(ω'_-R) + 2C 
       ≤ L_u(β_-R) + 2F_-R(C,D,E,η) + 2C
    and similarly
    L_u(γ_R)    ≤ L_u(β_R) + 2F_R(C,D,E,η) + 2C


Let M = 2max{F_-R(C,D,E,η) + F_R(C,D,E,η)} + 2C, which we use below in setting the value of the threshold constant A_<ref>.

Now apply Lemma <ref>, the special flaring condition for L_u, with the constant ν = 2μ-1 >1, to obtain integers N ≥ 1 and A_<ref>. Setting R=N, from the threshold requirement L_u(β_0) ≥ A_<ref> it follows that 

    ν L_u(β_0)    = ν L_u(γ_0) 
       ≤max{L_u(γ_-R),L_u(γ_R)}
       ≤max{L_u(β_-R),L_u(β_R)} + M 
    With the additional threshold requirement L_u(β_0) ≥2 M/ν-1 we have:ν L_u(β_0)    ≤max{L_u(β_-R),L_u(β_R)} + ν-1/2 L_u(β_0) 
    μ L_u(β_0) = ν+1/2 L_u(β_0)    ≤max{L_u(β_-R), L_u(β_R)}

Thus we have proved the (general) flaring condition for L_u given any μ≥ 1, η≥ 0, using R_<ref>=N and A_<ref> = max{A_<ref>,2M/ν-1}.









 §.§ Negative flaring of L_u
 


We continue to adopt Notations <ref>. 

In the context of Lemma <ref> on Special Flaring for L_u, the next lemma establishes flaring in the “negative direction”. 

For any path σ⊂ G, define (σ) to be the maximum of l_u(τ) over all paths τ in G such that that τ is a subpath both of σ and of some leaf ℓ of Λ^- realized in G. In this definition we may always assume that ℓ is a generic leaf of Λ^-, because every finite subpath of every leaf is a subpath of every generic leaf. Notice that if σ is already a subpath of a leaf of Λ^- then (σ)=l_u(σ).
 
  There exists L_<ref>≥ 1 such that for each L ≥ L_<ref> and each a>0 there exists an integer N ≥ 1, so that the following holds: for each n ≥ N, for each finite subpath τ of a generic leaf ℓ of Λ^- realized in G such that τ has endpoints at vertices and such that (τ) = l_u(τ) = L, and for each finite path σ in G, if f^n_#(σ) = τ then (σ) ≥ aL. 
 

Intuitively, at least in the nongeometric case (see the proof of Proposition 6.0.8 of , page 609), the point of this lemma is that in any leaf of Λ^-, the illegal turns have a uniform density with respect to the path function l_u, and those turns die off at an exponential rate under iteration of f_#.

 For the proof, a height u illegal turn in a path σ is said to be out of play if it is the illegal turn in some ρ or ρ̅ subpath of σ, otherwise that turn is in play.

The generic leaf ℓ of Λ^-, as realized in G, is birecurrent, is not periodic, and is not weakly attracted to Λ^+ under iteration of f_#. It follows that ℓ does not split as a concatenation of u-legal paths and copies of ρ and ρ̅. By applying Lemma <ref> it follows that ℓ has an illegal turn that is in play. In addition, ℓ is quasiperiodic with respect to edges of H_u <cit.>: for any integer L>0 there exists an integer L'>0 such that for any finite subpaths α, β of ℓ, if l_u(α) = L and l_u(β) ≥ L' then β contains a subpath which is a copy of α. It follows that there exists an integer L_<ref>>0 so that if τ is a subpath of ℓ such that l_u(τ) ≥ L_<ref>, then τ has at least three height u illegal turns that are in play.
 
Arguing by contradiction, if the lemma fails using the value of L_<ref> just given then there exist L ≥ L_<ref>, a>0, positive integers n_i → +, finite paths τ_i in G with endpoints at vertices, and paths σ_i in G, such that τ_i is a subpath of ℓ, and l_u(τ_i) = L, and f^n_i_#(σ_i) = τ_i, and (σ_i) < a L. We derive contradictions in separate cases. 

Case 1: l_u(σ_i) has an upper bound B. Decompose σ_i as σ_i = ϵ^-_i η_i ϵ^+_i where ϵ^±_i are each either partial edges or trivial, and where η_i is a path with endpoints at vertices. There exists a positive integer d depending only on B such that for each i the path f^d_#(η_i) has a splitting into terms each of which is either a single edge in H_u, a copy of ρ or ρ̅, or a subpath of G_u-1, and therefore each height u illegal turn in the interior of f^d_#(η_i) is out of play (see  Lemma 4.25; also see  Lemma 1.53). For each i such that n_i ≥ d the paths f^n_i_#(ϵ^±_i) are each u-legal, and each height u illegal turn of f^n_i_#(η_i)= f^n_i-d_#(f^d_#(η_i)) is out of play. Since τ_i is obtained from f^n_i_#(ϵ^-_i) f^n_#(η_i) f^n_i_#(ϵ^+_i) by tightening, at most two illegal turns of τ_i are in play, a contradiction to our choice of L_<ref>. 

Case 2: l_u(σ_i) has no upper bound. Consider a line M in G which is a weak limit of a subsequence σ_i_m such that M crosses at least one edge of H_u. We apply to each such M the weak attraction results of  as follows. If H_u is non-geometric then there are two options for M: either the closure of M contains Λ^-; or M is weakly attracted to Λ^+ (see Lemma 2.18 of ). If H_u is geometric — and so a height u closed indivisible Nielsen path ρ exists — then there is a third option, namely that M is a bi-infinite iterate of ρ or ρ̅ (see Lemma 2.19 of ). 

Since no σ_i contains a subpath of a leaf of Λ^- that crosses aL edges of H_u, neither does M. This shows that the closure of M does not contain Λ^-. If M were weakly attracted to Λ^+ then for any K>0 there would exist l> 0 such that f^l_#(M), and hence f^l_#(σ_i_m) for all sufficiently large m, contains a u-legal subpath that crosses 2K+1 edges of H_u. By  Lemma 4.2.2 we can choose K so that f^l_#(σ_i_m) splits at the endpoints of the middle H_u edge of that subpath, and hence the number of edges of H_u crossed by τ_i_m = f_#^n_i_m-l(f^l_#(σ_i_m)) goes to infinity with m, contradicting that l_u(τ_i)=L.

We have shown that each of the first two options leads to a contradiction for any choice of M as above. This concludes the proof if H_u is nongeometric. 

It remains to show that if H_u is geometric then the third option can also be avoided by careful choice of M, and hence the desired contradiction is achieved. That is, we show that there exists a weak limit M of a subsequence of σ_i such that M contains at least one edge of H_u and M is not a bi-infinite iterate of the closed path ρ or ρ̅. This may be done by setting up an application of Lemma 1.11 of , but it is just as simple to give a direct proof. Lift σ_i to the universal cover of G and write it as an edge path σ_i =  E_i1 E_i2… E_iJ_i⊂ G; the first and last terms are allowed to be partial edges. Let b equal twice the number of edges in ρ. Given j ∈{1+b,…,J_i-b}, we say that E_ij is well covered if E_i,j-b, E_i,j+b are full edges and there is a periodic line ρ_ij⊂ G that projects to ρ or to ρ̅ and that contains E_i,j-b… E_ij … E_i,j+b as a subpath. Since the intersection of distinct periodic lines cannot contain two fundamental domains of both lines, ρ_ij is unique if it exists. Moreover, if both E_ij and E_i,j+1 are well covered then ρ_ij=ρ_i,j+1. It follows that if E_ij is well covered then we can inductively move forward and backward past other well covered edges of σ̃_i all in the same lift of ρ, until either encountering an edge that is not well covered, or encountering initial and terminal subsegments of σ_i of uniform length. After passing to a subsequence, one of the following is therefore satisfied:

 
  *  There exists a sequence of integers K_i such that 1 < K_i < J_i, and K_i →, and J_i - K_i →∞, and such that E_iK_i⊂ H_u is not well covered. 
 
  *  σ_i = α_i ρ^p_iβ_i where the number of edges crossed by α_i and β_i is bounded independently of i and p_i→∞.
 
If subcase good weak limit holds then the existence of a weak limit that crosses an edge of H_u and is not a bi-infinite iterate of ρ or ρ̅ follows immediately. If subcase just rho holds, τ_i is obtained from f^n_i_#(α_i) ρ^p_if^n_i_#(β_i) by tightening. Since the number of illegal turns of height u in the first and last terms is uniformly bounded, the number of edges that are cancelled during the tightening is uniformly bounded, and it follows that τ_i contains ρ^q_i as a subpath where |q_i| →∞, a contradiction to the fact that ρ is not a leaf of Λ^-. 
 



 §.§ Positive flaring and other properties of L_u



In this section we continue to adopt Notations <ref>.

In the context of Lemma <ref> on Special Flaring for L_u, item estimate3 of Lemma <ref> establishes flaring in the “positive direction”. 
Lemma <ref> also includes several useful metrical/dynamical properties of L_u, which will be used in Section <ref> where we tie together the negative and positive flaring results to prove Lemma <ref>.

The lemma and its applications make use of the path map f_ and its properties as laid out in  Section 1.1.6, particularly Lemma 1.6 which we refer to as the “ Lemma”. Roughly speaking, the path f_(σ) is defined for any finite path σ in G as follows: for any finite path τ containing σ as a subpath, the straightened image f_#(τ) contains a subpath of f_#(σ) that is obtained by deleting initial and terminal subpaths of f_#(σ) that are no longer than the bounded cancellation constant of f; the path f_(σ) is the longest subpath of f_#(σ) which survives this deletion operation for all choices of τ. 






  
The following conditions hold:


  *  

For each path σ with endpoints at vertices and any decomposition into subpaths σ = σ_0 σ_1 …σ_K with endpoints at vertices we have

∑_0^K L_u(σ_k) - K   l_u(ρ) ≤ L_u(σ ) ≤∑_0^K L_u(σ_k)

and similarly with L_ and l_ in place of L_u and l_u respectively.

  *  For all positive integers d there exist B = B_<ref>(d) > 0 and b = b_<ref>(d) > 0 so that for all finite paths σ with endpoints at vertices, if L_u(σ) ≥ B then L_u(f^d_(σ)) ≥ b   L_u(σ).


  *  There exist constants A > 0 and κ≥ 1 so that for all subpaths τ of leaves of Λ^- in G, if l_u(τ) ≥ A then 

    l_u(τ) ≤ L_u(τ) ≤κ  l_u(τ)



  *  There exists a positive integer A = A_<ref> and a constant 0< R = R_<ref> <1, so that if a path α⊂ G splits as a concatenation of u-legal paths and copies of ρ or ρ̅, and if L_u(α) ≥ A, then for all m ≥ 1 we have

    L_u(f^m_(α)) ≥ R  λ^m/2  L_u(α)

where λ is the expansion factor for f and Λ^+. 






  
Remark: The notation f^m_ is disambiguated by requiring that the exponent binds more tightly than the -operator, hence f^m_ = (f^m)_ — this is how items estimate1 and estimate3 are applied in what follows. Note that this makes the statements of estimate1 and estimate3 weaker than if f^m_ were intepreted as (f_)^m, because (f_)^m(α) is a subpath of (f^m)_(α). 


 We prove ItemAlmostAdditive for L_u when K=1; the proof for general K follows by an easy induction, and the statement for L_ is proved in the exact same manner. If there are non-trivial decompositions σ_0 = σ'_0 σ”_0 and σ_1 =σ'_1 σ”_1 such that σ”_0σ'_1 is a copy of ρ or ρ̅ then H_u-edges in σ”_0σ'_1 contribute to L_u(σ_0) + L_u(σ_1) but not to L_u(σ), and all other H_u-edges contribute to L_u(σ_0) + L_u(σ_1) if and only if they contribute to L_u(σ). In this case L_u(σ) = L_u(σ_0) + L_u(σ_1) - l_u(ρ). If there are no such non-trivial decompositions then concatenating the decompositions of σ_0 and σ_1 given by Lemma <ref> produces the decomposition of σ given by Lemma <ref> and L_u(σ) = L_u(σ_0) + L_u(σ_1). This completes the proof of ItemAlmostAdditive.
 
 We next prove estimate1. Fix the integer d ≥ 1. The path f^d_(σ) is obtained from f^d_#(σ) by removing initial and terminal segments that cross a number of edges that is bounded above, independently of σ, by the bounded cancellation constant of f^d. It follows that L_u(f^d_(σ))-L_u(f^d_#(σ)) is bounded above independently of σ, so it suffices to find B,b > 0 depending only on d so that if L_u(σ) > B then L_u(f^d_#(σ)) > b L_u(σ). Applying Lemma <ref> ItemQiU we obtain D > 1, E > 0 so that L_u(γ) ≤ D   L_u(f_#(γ)) + E for all finite paths γ with endpoints at vertices, from which it follows by induction that

    L_u(f^d_#(γ))    ≥1/D^d L_u(γ) - E( 1/D + ⋯ + 1/D^d) 
       ≥1/D^d L_u(γ) - E/D-1
    and so if L_u(γ) ≥2D^d E/D-1 then
    L_u(f^d_#(γ))      ≥1/D^d L_u(γ) - 1/2D^d L_u(γ) = 1/2D^d L_u(γ)








The first inequality of estimate2 follows immediately from Corollary <ref>, for any subpath τ of a leaf of Λ^-. To prepare for proving the second inequality, given a positive integer C let Σ_C be the set of paths in G that do not contain a subpath of the form ρ^ϵ C where ϵ=± 1. Since ρ has an endpoint that is not contained in G_u-1 ( Fact 1.43), each maximal subpath of σ of the form ρ^k or ρ̅^k that is neither initial nor terminal in σ is adjacent in σ to an H_u-edge that contributes to L_u(σ). Applying Corollary <ref> it follows that
 
 
  (*) For any fixed C, amongst those paths σ∈Σ_C for which L_u(σ) > 0, the ratio l_u(σ) / L_u(σ) has a positive lower bound that is independent of σ. 
  






For proving estimate2, the key observation is that there exists a positive integer C so that each subpath τ of Λ^- is contained in Σ_C — this is equivalent to saying that ρ^∞ is not a weak limit of lines in Λ^-, which is equivalent to saying that ρ^∞ is not a leaf of Λ^-, which follows from Lemma 3.1.15 of  and the fact that Λ^- ρ^∞. Item estimate2 therefore follows by combining this observation with (*).



For proving estimate3, we focus primarily on an analogue for L_, connecting it with L_u version stated in estimate3 by applying Corollary <ref>. From the assumption on a splitting of α we have

    L_(f^m_#(α)) = λ^m L_(α)

We shall show how to replace f^m_# by f^m_, at the expense of replacing λ by its square root, and of requiring L_(α) to exceed some threshold constant. To be precise, we have:


  Claim: There exists A' ≥ 0 such that if L_(α) ≥ A' then for all m ≥ 0 we have

    L_(f^m_(α)) ≥λ^m/2 L_(α)


This suffices to prove estimate3, because if L_u(α) ≥ K_<ref>(f) · A' = A then from Corollary <ref> it follows that L_(α) ≥ A', from which using the Claim we obtain L_(f^m_(α)) ≥λ^m/2 L_(α), and then by two more applications of Corollary <ref> we obtain

    L_u(f^m_(α)) ≥1/K_<ref>(f) L_(α) ≥1/K_<ref>(f)λ^m/2 L_(α) ≥1/(K_<ref>(f))^2λ^m/2 L_u(α)


To prove the claim, the case m=0 is evident, so suppose by induction that

    L_(f^m-1_(α)) ≥λ^(m-1)/2 L_(α)

Since f^m-1_(α) is a subpath of f^m-1_#(α), and since the latter splits into terms each of which is an edge of H_u, a copy of ρ or ρ̅, or a path in G_u-1, it follows that f^m-1_(α) may be deconcatenated in the form

    f^m-1_(α) = ζα̂ω

such that α̂ splits into terms exactly as above, and such that either ζ,ω are both trivial, or ρ exists and ζ,ω are both proper subpaths of ρ or ρ̅; it follows that l_(ζ),l_(ω) ≤ l_(ρ).  Applying item ItemAlmostAdditive it follows that

    L_(f^m-1_(α))    ≤ L_(α̂) + 2 l_(ρ) 
    
    L_(α̂)    ≥λ^(m-1)/2  L_(α) -  2 l_(ρ) 
    Using the splitting of α̂ we obtain
    L_(f_#(α̂))    = λ L_(α̂) ≥λ^(m+1)/2  L_(α) -  2  λ  l_(ρ)

The path f_(α̂) is obtained from f_#(α̂) by truncating initial and terminal segments no longer than the bounded cancellation constant of f, and since this is a finite number of paths their L_-values have a finite upper bound C_2, so by applying item ItemAlmostAdditive it follows that

    L_(f_(α̂)) ≥λ^(m+1)/2  L_(α) -  2  λ  l_(ρ) - 2 C_2

Now we apply the  Lemma. Since α̂ is a subpath of f^m-1_(α) it follows that f_(α̂) is a subpath of f_(f^m-1_(α)) ( Lemma 1.6 (3)), which is a subpath of f^m_(α) ( Lemma 1.6 (4)). Thus we have f^m_(α) = η f_(α̂) θ for some paths η,θ, and hence by item ItemAlmostAdditive we have

    L_(f^m_(α))    ≥λ^(m+1)/2  L_(α) -  2  λ  l_(ρ) - 2 C_2 - 2   l_(ρ) 
    To complete the induction we show that with appropriate threshold constant the quantity on the right is ≥λ^m/2  L_(α), equivalentlyλ^(m+1)/2  L_(α)    ≥λ^m/2  L_(α)   + (2   (λ+1)   l_(ρ) + 2 C_2_C_1) 
    λ^1/2   ≥ 1 + C_1/λ^m/2  L_(α)

Since λ>1 is suffices to show

    λ^1/2≥ 1 + C_1/L_(α)     L_(α) ≥C_1/λ^1/2 - 1

Taking the threshold constant to be 

    A' = C_1/λ^1/2 - 1

the induction is complete.





 §.§ Proof of Lemma <ref>: the Special Flaring Condition for L_u


Once this proof is complete, the proof of the General Flaring Condition stated in Proposition <ref> will also be complete, as shown in Section <ref>.



If the Special Flaring Condition for L_u fails then there exists a sequence n_k →∞, and there exist paths γ_k ⊂ G with endpoints at vertices, such that L_u(f^n_k_#(γ_k)) →∞ as k →∞, and such that 

    (*)     ν  L_u(f^n_k_#(γ_k))  ≥max{ L_u(γ_k), L_u(f^2 n_k_#(γ_k))}

Assuming this, we argue to a contradiction.

Consider the integer L_<ref>≥ 1 satisfying the conclusions of Lemma <ref>. By Lemma <ref> estimate2 there is an integer L_2 so that if μ is a subpath of Λ^- that crosses ≥ L_2 edges of H_u then L_u(μ) ≥ 1. Let L_1 = max{L_<ref>,L_2}. Choose an integer d ≥ 1 satisfying the conclusion of <cit.>, the “uniform splitting lemma”, with respect to the constant L_1. This conclusion says that for any finite path σ in G with endpoints at vertices of H_u, if ℓ^-_u(σ) < L_1 then the path f^d_#(σ) splits into terms each of which is u-legal or a copy of ρ or ρ̅. In this context we shall refer to d as the “uniform splitting exponent”.
 
Let {μ_ik} be a maximal collection of subpaths of f^n_k_#(γ_k) with endpoints at vertices that have disjoint interiors, that are subpaths of Λ^-, and that cross ≥ L_1 edges of H_u. The complementary subpaths {ν_jk} of f^n_k_#(γ_k) all satisfy (ν_jk) < L_1 and all have endpoints at vertices as well.
 
Our first claim is that 

    (i)         lim_k →∞∑_i L_u(μ_ik)/L_u(f^n_k_#(γ_k)) = 0

If not, then after passing to a subsequence we may assume that 

    ∑_i L_u(μ_ik) > ϵ_1 L_u(f^n_k_#(γ_k))

for some ϵ_1> 0 and all k. Choose subpaths σ_ik of γ_k with disjoint interiors such that f_#^n_k(σ_ik) = μ_ik. Since l_u(μ_ik) ≥ L_1 ≥ L_<ref>, and since n_k → +∞, we may apply “Negative Flaring”, Lemma <ref>, to obtain subpaths σ_ik' of σ_ik which have endpoints at vertices and which are also subpaths of Λ^- such that for all i we have

    lim_k →∞l_u(σ'_ik)/l_u(μ_ik)= ∞


The ratios l_u(σ'_ik)/L_u(σ'_ik) and l_u(μ_ik)/L_u(μ_ik) have positive upper and lower bounds independent of i and k: the upper bound of 1 follows from Corollary <ref>; and the lower bound comes from Lemma <ref> estimate2. For all i we therefore obtain 









    lim_k →∞L_u(σ'_ik)/L_u(μ_ik)= ∞

Using this limit, and using that L_u(μ_ik) ≥ 1, it follows that for all sufficiently large k we have

    L_u(γ_k) ≥∑_i (L_u(σ'_ik) - 2ρ) > ν/ϵ_1∑_i L_u(μ_ik) > ν  L_u(f^n_k_#(γ_k))

where the first inequality follows by applying Lemma <ref> ItemAlmostAdditive

to the subdivision of γ_k into the paths σ'_ik and their complementary subpaths. This contradicts (*), verifying the first claim.

Our second claim is that for any constant A (on which constraints will be placed below when this claim is applied) we have

    (ii)         lim_k →∞∑_L_u(ν_jk)≥ A L_u(ν_jk) /   L_u(f^n_k_#(γ_k))= 1

To see why, let I_k be the number of μ_ik subpaths of f^n_k_#(γ_k), let J_k be the number of ν_jk subpaths, and let K_k = I_k + J_k, and so J_k ≤ I_k + 1 and K_k ≤ 2I_k+1. By Lemma <ref> ItemAlmostAdditive applied to f^n_k_#(γ_k) we obtain

    L_u(f^n_k_#(γ_k))  ≤ ∑_j L_u(ν_jk)     +  ∑_i L_u(μ_ik) 
       ≤  L_u(f^n_k_#(γ_k)) +   K_k   l_u(ρ) 
    
    
    1 ≤∑_L_u(ν_jk) ≥ A L_u(ν_jk) /L_u(f^n_k_#(γ_k))   + ∑_i L_u(μ_ik)/L_u(f^n_k_#(γ_k))_δ_k + ∑_L_u(ν_jk)<A L_u(ν_jk)/L_u(f^n_k_#(γ_k))_ϵ_k
       ≤ 1 + K_k   l_u(ρ)/L_u(f^n_k_#(γ_k))_ζ_k

From (i) it follows that δ_k → 0 as k → +. Multiplying the inequality 
K_k ≤ 2I_k+1 by l_u(ρ) / L_u(f^n_k_#(γ_k)), and using that L_u(μ_ik) ≥ 1, it follows that 

    0 ≤ζ_k ≤ 2   l_u(ρ)  δ_k + l_u(ρ)/L_u(f^n_k_#(γ_k))

and so ζ_k → 0 as k → +. Multiplying the inequality J_k ≤ I_k + 1 by A / L_u(f^n_k_#(γ_k)), it follows that 

    0 ≤ϵ_k ≤ A δ_k + A/L_u(f^n_k_#(γ_k))

and so ϵ_k → 0 as k → +. This proves the second claim. 

In what follows we will be applying Lemma <ref> estimate3, and we will use the constants A_<ref>, R_<ref> involved in that statement.

By definition of L_1 and by the choice of the uniform splitting exponent d, since ℓ^-_u(ν_jk) < L_1 it follows that f^d_#(ν_jk) splits into terms each of which is either u-legal or a copy of ρ or ρ̅. Consider the constants B = B_<ref>(d) > 0 and b = b_<ref>(d) > 0 of Lemma <ref>. Constraining A ≥ B, we may combine (ii) with Lemma <ref> estimate1 to obtain

    (iii)   ∑_L_u(ν_jk)≥ A L_u(f^d_(ν_jk)) /L_u(f^n_k_#(γ_k))≥ b ·∑_L_u(ν_jk)≥ A L_u(ν_jk) /L_u(f^n_k_#(γ_k)) > 3b/4

for sufficiently large values of k.

By construction, the paths {ν_jk} occur in order of the subscript j as subpaths of f^n_k_#(γ_k) with disjoint interiors. By applying the  Lemma using f^n_k, it follows that the the paths f^n_k_(ν_jk) occur in order as subpaths of the path f^2n_k_#(γ_k) with disjoint interiors ( Lemma 1.6 (5)). It then follows that f^n_k-d_f^d_(ν_jk) is a subpath of f^n_k_(ν_jk) ( Lemma 1.6 (4)). Putting these together we see that the paths f^n_k-d_f^d_(ν_jk) occur in order as subpaths of the path f^2n_k_#(γ_k) with disjoint interiors. These subpaths being J_k in number, together with their complementary subpaths one has a decomposition of f^2n_k_#(γ_k) into at most 2J_k+1 paths. Ignoring the complementary subpaths, Lemma <ref> ItemAlmostAdditive therefore implies 

    L_u(f^2n_k_#(γ_k)) 
       ≥∑_ L_u(f^n_k-d_f^d_(ν_jk))    -    2   J_k   l_u(ρ) 
       ≥∑_L_u(ν_jk)≥ A L_u(f^n_k-d_f^d_(ν_jk))    -    2   J_k   l_u(ρ) 
       ≥∑_L_u(ν_jk)≥ A L_u(f^n_k-d_f^d_(ν_jk))    -    l_u(ρ)   L_u(f^n_k_#(γ_k))

where the last inequality follows for sufficiently large k by applying (i) and the inequality L_u(μ_ik) ≥ 1 to conclude that 

    L_u(f^n_k_#(γ_k)) ≥ 2∑_i L_u(μ_ik)+2 ≥ 2I_k+2 ≥ 2J_k

For sufficiently large k we therefore have

    (iv)   L_u(f^2n_k_#(γ_k))/L_u(f^n_k_#(γ_k)) > 
    ∑_L_u(ν_jk)≥ A L_u(f^n_k-d_f^d_(ν_jk))/L_u(f^n_k_#(γ_k)) - l_u(ρ)


We have already constrained A so that A ≥ B, and we now put one more constraint on A. Applying Lemma <ref> estimate1 to f^d it follows that if L_u(ν_jk) ≥ B then L_u(f^d_(ν_jk)) ≥ b   L_u(ν_jk), and so if L_u(ν_jk) ≥ A = max{B,1/b A_<ref>} it follows that L_u(f^d_(ν_jk)) ≥ A_<ref>. This allows us to apply “Positive Flaring”, Lemma <ref> estimate3, with the conclusion that, letting R = R_<ref>,

    (v)    L_u(f^n_k-d_f^d_(ν_jk)) ≥ R  λ^(n_k - d)/2 L_u(f^d_(ν_jk))

as long as L_u(ν_jk) ≥ A and as long as k is sufficiently large. 
Combining (iv) and (v), if k is sufficiently large we obtain

    L_u(f^2n_k_#(γ_k))/L_u(f^n_k_#(γ_k))    >
    R  λ^(n_k - d)/2 ∑_L_u(ν_jk)≥ A L_u(f^d_#(ν_jk)) /L_u(f^n_k_#(γ_k)) - l_u(ρ) 
    and combining this with (iii) we obtainL_u(f^2n_k_#(γ_k))/L_u(f^n_k_#(γ_k))    > 3 b R/4λ^(n_k - d)/2 - l_u(ρ) 
       > ν

where the second inequality holds for sufficiently large k. This gives us the final contradiction to (*), which completes the proof of Lemma <ref>. 



 §.§ Appendix: The graph homotopy principle


Lemma <ref> in this section was used earlier in the proof of Lemma <ref>, and it will be used later in the construction of the “homotopy semigroup action” in Section <ref>. It is an elementary result in homotopy theory; for precision we state the result in the language of category theory, and we give the complete proof.

Define the graph-point category, a subcategory of the standard homotopy category of pairs, as follows. The objects are pairs (G,P) where G is a finite graph and P ⊂ G is a finite subset. Each morphism, denoted [f]  (G,P) ↦ (H,Q), is the homotopy class rel P of a homotopy equivalence f  G → H that restricts to a bijection f  P → Q. Define the fundamental group functor from the graph-point category to the category of indexed groups as follows. To each pair (G,P) we associate the indexed family of groups π_1(G,P) = (π_1(G,p))_p ∈ P, and to each morphism [f]  (G,P) → (H,Q) we associate the indexed family of group isomorphisms 

    [f]_* π_1(G,P) →π_1(H,Q)    =   (f_* π_1(G,p) →π_1(H,f(p)))_p ∈ P

The category and functor axioms implicit in this discussion are easily checked.

Let ^(G,P) denote the group of automorphisms of (G,P) in the graph-point category. Let ^_0(G,P) ^(G,P), which we call the pure automorphism group of (G,P) in the graph-point category, denote the finite index subgroup consisting of those [f] ∈^(G,P) such that f  P → P is the identity.









  

  * 
The graph-point category is a groupoid: every morphism [f]  (G,P) → (H,Q) has an inverse morphism [g]  (H,Q) → (G,P), meaning that g ∘ f  (G,P) → (G,P) is homotopic to the identity rel P and f ∘ g  (H,Q) → (H,Q) is homotopic to the identity rel Q.

  * 
The fundamental group functor is faithful: for any pair of morphisms [f], [f']  (G,P) → (H,Q), we have [f]=[f'] if and only if the restricted maps f, f'  P ↦ Q are equal and the induced isomorphisms f^_*, f'_* π_1(G,p) →π_1(H,f(p)) are equal for all p ∈ P.

  * 
Two morphisms [f]  (G,P) → (H,Q) and [g]  (H,Q) → (G,P) are inverses if and only if their restrictions f  P → Q and g   Q → P are inverses and the isomorphisms f_* π_1(G,p) →π_1(H,f(q)) and g_* π_1(H,f(q)) →π_1(G,p) are inverses. 

  * 
The fundamental group functor restricts to an injective homomorphism defined on the pure automorphism group ^_0(π_1(G,P)) ↦⊕_p ∈ P(π_1(G,p)).
 


 Once ItemGPHomInvExists and ItemGPInverse are proved, ItemGPFaithful and ItemGPAut follow immediately.



To prove ItemGPInverse, the “only if” direction is obvious, and for the “if” direction it suffices to prove this special case: for any self-morphism [f]  (G,P) → (G,P), if f fixes each p ∈ P and induces the identity π_1(G,p) for each p ∈ P, then f is homotopic to the identity rel P. For the proof, we know that f is freely homotopic to the identity map _G, because G is an Eilenberg-MacClane space and f induces the identity on its fundamental group. Choose a homotopy h  G × [0,1] → G from f to _G. 

We claim that for each p ∈ P the closed path γ_p(t) = h(p,t) (0 ≤ t ≤ 1) is trivial in π_1(G,p). Applying this claim, we alter the homotopy h as follows: using the homotopy extension property, for each p ∈ P we may homotope the map h  G × [0,1] → G, keeping it stationary on G × 0, stationary on G × 1, and stationary outside of U_p × [0,1] for an arbitrarily small neighborhood U_p of p, to arrange that h(p × [0,1])=p; note that for a homotopy X × [0,1] → Y to be “stationary on A ⊂ X” means that the restricted map {a}× [0,1] → Y is constant for each a ∈ A. Doing this independently for each p ∈ P, we obtain a homotopy rel P from f to the identity and we are done, subject to the claim.

To prove the claim, consider a closed path δ [0,1] → G based at p, representing an arbitrary element [δ] ∈π_1(G,p). We obtain a path homotopy H_t  [0,1] → G from the path H_0 = f ∘δ to the concatenated path H_1 = γ_p * δ * γ̅_p as follows: 

    H_t(s) = γ_p(3s)      if 0 ≤ s ≤ t/3
    
    h(δ(3s   -   t/3  -   2t),t)      if t/3 ≤ s ≤ 1 - t/3
    γ_p(3-3s)      if 1 - t/3 ≤ s ≤ 1

Since for all [δ] ∈π_1(G,p) we have [δ] = [f ∘δ] = [γ_p] · [δ] · [γ_p]^, it follows that [γ_p] is in the center of π_1(G,p) ≈ F_n, hence is trivial, completing the proof of ItemGPInverse.



To prove ItemGPHomInvExists, start with any homotopy inverse g'  H → G of f. We may assume that the maps f  P → Q and g'  Q → P are inverses, because by the homotopy extension property we may homotope g' to be stationary outside of a small neighborhood of P so that for each p ∈ P the track of the homotopy on the point g'(f(p)) moves it back to p. Since g' ∘ f  G → G fixes each point in P and is homotopic to the identity, for each p ∈ P the induced map (g' ∘ f)_* π_1(G,p) →π_1(G,p) is an inner automorphism represented by some closed curve γ_p based at p, and so for each element of π_1(G,p) having the form [δ] for some closed curve δ based at p we have (g' ∘ f)_*(δ) = [γ_p * δ * γ̅_p]. Let h  (G,p) → (G,p) be the morphism obtained from the identity by a homotopy that is stationary outside a small neighborhood of P and such that the track of the homotopy on each p ∈ P is the closed curve γ̅_p; again we are applying the homotopy extension property. Letting g = h ∘ g' we may apply ItemGPInverse to conclude that the morphism [f] is an isomorphism with inverse [g].




  
Remark. Note that the proof of ItemGPInverse depends heavily on the fact that the center of F_n is trivial. The proof breaks down, for instance, if G is replaced by a torus; in fact the analogue of Lemma <ref>, where a graph is replaced by a torus and P is a two-point subset of the torus, is false. On the other hand the analogue for any K(π,1) space whose fundamental group has trivial center is true.






§ FLARING IN T^* AND HYPERBOLICITY OF .
 


Throughout this section we continue with Notations <ref> ItemCTEG–ItemCTSplitSimple regarding an outer automorphism ϕ∈(F_n) and a relative train track representative f  G → G having penultimate filtration element G_u-1 and top  stratum H_u. 

The main result of this section is the construction, carried out in Section <ref>, of the Gromov hyperbolic space  that is used in later sections for proving the multi-edge case of the Hyperbolic Action Theorem. The construction of  is based on results found in Sections <ref>–<ref>, in particular Proposition <ref> which is a re-interpretation of the flaring result of Proposition <ref> expressed in the context of a certain natural free splitting. The statement of Proposition <ref> is found in Section <ref>, after preliminary work carried out in Sections <ref>–<ref>.

Once some of the definitions have been formulated, the reader may wish to pause to consider the “Motivational Remarks” found in Section <ref> following Lemma <ref>.



 §.§ The free splitting F_n  T and its Nielsen lines.



We begin with a description of the free splitting F_n  T associated to the marked graph G and its subgraph G_u-1, together with a description of some features of T associated to height u Nielsen paths in G.


Let  denote the free factor system corresponding to the subgraph G_u-1, having the form = {[A_1],…,[A_K]} where G_u-1 has noncontractible components C_1,…,C_K and A_k  F_n is in the conjugacy class of the image of the injection π_1(C_k) π_1(G) ≈ F_n (that injection determined up to inner automorphism of F_n by appropriate choices of base points and paths between them).



 
Let F_n  T denote the free splitting corresponding to the subgraph G_u-1⊂ G. What this means is that, starting from the deck action F_n  G associated to the universal covering map G ↦ G, the tree T is obtained from G by collapsing to a point each component of the total lift G_u-1 of G_u-1. Let p  G → T denote the F_n-equivariant collapse map. Since G_u-1 is F_n-invariant, the action F_n  G induces via p an action F_n  T which is evidently a free splitting, i.e. a minimal action on a simplicial tree with trivial edge stabilizers. 

Note that the set of conjugacy classes of nontrivial vertex stabilizers of this action is precisely the free factor system  — indeed the stabilizer of a vertex v ∈ T equals the stabilizer of p^(v) ⊂ G which is nontrivial if and only if p^(v) is a component of G_u-1 covering some noncontractible component C_k ⊂ G_u-1, in which case the stabilizer of v is conjugate to A_k.


 

Fixing a standard isomorphism between the deck transformation group of the universal covering map G ↦ G and the group F_n ≈π_1(G), recall from covering space theory that the lifts G → G of f  G → G are in bijective correspondence with the automorphisms Φ∈(F_n) that represent ϕ, where the bijection Φ↔ f^Φ is given by the following relation:


  Φ-twisted equivariance in G:   f^Φ(γ· x) = Φ(γ) · f^Φ(x)   for all γ∈ F_n, x ∈ G





Since f preserves G_u-1, any of its lifts f^Φ preserves G_u-1, and hence f^Φ induces a map f^Φ_T  T → T. The Φ twisted equivariance property in G implies a corresponding property in T:


  Φ-twisted equivariance in T:   f^Φ_T(γ· x) = Φ(γ) · f^Φ_T(x)   for all γ∈ F_n, x ∈ T

When the automorphism Φ is understood or is not important to the discussion, we will often drop it from the notations f^Φ and f^Φ_T, writing simply f and f_T instead.

In the next section we will impose an additional metric constraint on f_T; see under the heading “Stretch properties of f_T”.





In the geometric and parageometric cases, where ρ,ρ̅ exist and are the unique inverse pair of Nielsen paths of height u, 
a Nielsen path in G is any lift of ρ or ρ̅, and a Nielsen path in T is any projection to T of any Nielsen path in G. In the geometric case, where ρ is closed and has distinct initial and terminal directions, a Nielsen line in G is a line which projects to a bi-infinite iterate of ρ, and a Nielsen line in T is the projection of a Nielsen line in G. 

The Nielsen set , a collection of subsets of T, is defined as follows. In the ageometric case, = ∅; in the geometric case,  is the set of Nielsen lines in T; and in the parageometric case,  is the set of Nielsen paths in T. Furthermore, for each N ∈ its basepoint set or base lattice, denoted Z(N), is defined as follows. In the geometric case, the Nielsen line N has a unique decomposition as a bi-infinite concatenation of Nielsen paths, and Z(N) is defined to be the set of concatenation points. In the parageometric case, where N is just a single Nielsen path, Z(N) is its endpoint pair.

Note that in the geometric case, basepoint sets of distinct Nielsen lines are disjoint — for all N  N' ∈ we have Z(N)  Z(N') = ∅. This follows from two facts about the base point p of ρ. First, each point p ∈ G lying over p is an endpoint of exactly two Nielsen paths in G, both contained in the same Nielsen line in G. Second, p is not contained in G_u-1 <cit.>, so no lift p is contained in G_u-1, and the projection map G ↦ T is locally injective on the complement of G_u-1.

Consider a finite path σ_ G in G with endpoints at vertices, with projections σ_G in G and σ_T in T. If σ_G = ρ^i or ρ̅^i for some integer i  0 — in the parageometric case where ρ is not closed, i must equal 1 — then we say that σ_T is a ρ^*-path in T, the superscript ^* representing the exponent. Note that ρ^* paths in T are precisely the paths of the form QQ' for which there exists N ∈ such that Q,Q' ∈ Z(N). 

























 §.§ Path functions on G and T.
 


In a tree, a finite path with initial and terminal endpoints V,W is determined by those endpoints and is denoted VW.

Each of l_u, l_, L_u, L_ is a path function on G that vanishes on paths in G_u-1 (see Section <ref>). Each lifts via the universal covering map G → G to an F_n-invariant path function on G that vanishes on paths in G_u-1, and hence projects via q  G → T to a well-defined and F_n-invariant path function on T. We re-use the notations l_u, l_, L_u, L_ for these path functions on G and on T; the context should help to avoid ambiguities. For any path β_ G in G with endpoints at vertices, letting β_G be its projection to G and β_T its projection to T, it follows from the definitions that l(β_G)=l(β_ G)=l(β_T) for any of l=l_u, l_, L_u, or L_. 

Remark. The point of “well-definedness” of, say, L_ on T is that for any vertices V,W ∈ T, if V_1,V_2 ∈ G both map to V and if W_1,W_2 both map to W then each of the paths V_1V_2 and W_1W_2 is either degenerate or is contained in G_u-1, and hence L_(V_1W_1) = L_(V_2W_2) = L_(VW).




Associated to the path functions l_u(·), l_(·), L_u(·), 
L_(·) on G and on T, we have, respectively, F_n-equivariant functions d_u(·,·), d_(·,·), D_u(·,·), D_(·,·) on pairs of vertices in G and in T. For example

    D_(V,W) = L_(VW)

In particular, tracing back through the definitions one sees that for vertices V,W ∈ T their distance d_u(V,W) simply counts the number of edges of T in the path VW.


For each of l=l_u, l_, L_u, L_ the quantity l_(E)=L_(E) is bounded away from zero as E ⊂ T varies over edges; this is an immediate consequence of the fact that as e varies over the finite set of edges of H_u, the finite set of positive numbers l(e) has a positive minimum. We record this fact as:


There exists η = η_<ref> > 0 such that for each edge E ⊂ T with endpoints V  W ∈ T, the values of d_u, d_, D_u, D_ on V,W are all ≥η. 

Each of the path functions l_u and l_ is additive, meaning that its value on an edge path is the sum of its values on individual edges. It follows that each of d_u and d_ is a path metric on T. Furthermore, d_u and d_ are quasicomparable to each other, because H_u has only finitely many edges hence T has only finitely many edge orbits under the action of F_n, and the values of d_u and d_ on the endpoint pair of each edge is positive (Lemma <ref>). The “metrics” D_u and D_ are also quasicomparable to each other, by application of Corollary <ref>. However, D_u and D_ are not actual metrics because they may violate the triangle inequality. Nonetheless D_u and D_ do satisfy a coarse version of the triangle inequality, as a consequence of Lemma <ref>, and we will refer to this by saying that D_u and D_ are coarse metrics on the vertex set of  T. We record these observations as:


d_u and d_ are quasicomparable, F_n-equivariant metrics on vertices of T. Also, D_u and D_ are quasicomparable, F_n-equivariant coarse metrics on vertices in T.




    Motivational remarks.
 The metrics d_u and d_ may fail to satisfy the desired flaring condition: if H_u is geometric then for any Nielsen line N ⊂ T, iteration of f_T=f_T^Φ produces a sequence of Nielsen lines N_i = (f_T^k)_#(N) (k ∈), and furthermore the map f^k_T takes the base lattice Z(N) to the base lattice Z(N_k) preserving path distance. Since the base lattice of a Nielsen line has infinite diameter in any invariant path metric on T, this demonstrates the failure of flaring. In hopes of averting a failure of flaring, we might instead consider using the coarse metrics D_u and D_: when two vertices V,W are contained in the base lattice of the same Nielsen line we have the equations D_u(V,W)=D_(V,W)=0 which are precisely designed to correct the failure of flaring. Although using D_u or D_ creates its own problem because they are not actually metrics, in Section <ref> we shall solve that problem by using the coning construction often employed in studies of relative hyperbolicity, coning off those paths in T which exhibit nonflaring behavior to obtain a graph T^*. Furthermore, this graph will come equipped with an actual path metric d^* such that the inclusion T  T^* is a quasi-isometry from D_ to d^*; see Proposition <ref> in Section <ref>.





Next we translate several results on path functions in Section <ref> into the context of the coarse metric D_ on T:





For any vertices V  W ∈ T there is a unique decomposition of the path VW = μ_0 ν_1 μ_1 ⋯ν_A μ_A such that the following properties hold:


  * 
If ρ does not exist then A=0.

  * 
If ρ exists then the ν_a's are precisely all of the maximal ρ^* paths of VW, and so each μ_a contains no ρ^* subpath.

  * 
If ρ exists and if 1 ≤ a < a+1 ≤ A-1 then at least one of the subpaths μ_a,μ_a+1 is nondegenerate; in the geometric case, all μ_a-subpaths are nondegenerate. 

  * 
D_(V,W) = l_(μ_0) + ⋯ + l_(μ_A)

Furthermore, given any path γ in T — finite, singly infinite, or bi-infinite — whose endpoints, if any, are at vertices, and assuming that ρ exists, there is a unique decomposition of γ as an alternating concatenation of its maximal ρ^* paths (the ν-subpaths) and paths that contain no ρ^* subpath (the μ-subpaths) such that for any two consecutive μ-subpaths at least one is nondegenerate, all μ-subpaths being nondegenerate in the geometric case.


 Items ItemAIsZero–ItemOneNotDeg are translations of Corollary <ref>, and item ItemDPFFormula is a translation of Corollary <ref>; the proofs are immediate from those results combined with the definitions. 

The “Furthermore…” clause is a quick consequence of the following observations that hold for any nested pair of finite subpaths VW⊂V'W' in T. First, every ρ^* subpath of VW is a ρ^* subpath of V'W'. Also, every maximal ρ^* subpath of VW whose d_ distances from V and from W are greater than l_(ρ) is a maximal ρ^* subpath of V'W'.




    Remark on item ItemOneNotDeg.
 Consider the paths μ_a for 1 ≤ a ≤ A-1. In the geometric case each such μ_a is nondegenerate, and this can be used to improve the constants in some applications of ItemOneNotDeg (underlying nondegeneracy of μ_a is the fact that the base point of the closed Nielsen path ρ is disjoint from G_u-1). In the parageometric case, on the other hand, one of the paths μ_a, μ_a+1 may be degenerate. This happens for μ_a only if, up to orientation reversal, the Nielsen path ρ has initial vertex p ∈ G_u-1 (the terminal vertex is necessarily disjoint from G_u-1, the fact which underlies item ItemOneNotDeg), in which case μ_a is degenerate if and only if ν_a μ_a ν_a-1 lifts to a path in G that projects to a path in G of the form ρ̅μρ where μ is a nondegenerate closed path in G_u-1 based at p.














 §.§ Constructing T^* by coning off Nielsen axes of T.
 

Embed the tree T into a graph denoted T^*, and extend the action F_n  T to an action F_n  T^*, as follows. Index the Nielsen set as = {N_j}_j ∈ J, letting Z_j be the basepoint set of N_j (Definition <ref>). For each j ∈ J we cone off Z_j by adding a new vertex P_j=P(N_j) and attaching a unique edge P_j Q for each Q ∈ Z_j. The points P_j are called cone points and the edges P_j Q are called cone edges. Since the simplicial action F_n  T takes Nielsen paths to Nielsen paths and hence induces a basepoint preserving permutation of the Nielsen set , this action extends uniquely to F_n  T^* permuting the cone points and the cone edges. 

Let V^∞⊂ T^* be the set of vertices v ∈ T^* whose stabilizer subgroup (v) is infinite. 



The formula v ↦(v) defines an injection from V^∞ to the set of nontrivial subgroups of F_n. A subgroup S  F_n is equal to (v) for some v ∈ V^∞ if and only if S is conjugate to the fundamental group of some noncontractible component of G_u-1 in π_1(G) ≈ F_n, or H_u is a geometric stratum and S is conjugate to the infinite cyclic subgroup ρ F_n.




 This fact may be extracted from Theorem F of . But a direct proof is easy; here is a sketch.

We have a partition 

    V^∞ = V^∞_u-1∐ V^∞_  where   V^∞_u-1 = V^∞ T   and   V^∞_ = {cone points} = {P_j}

The collapse map G ↦ T induces an equivariant and hence stabilizer preserving bijection between V^∞_u-1 and the set of components of G_u-1 having nontrivial stabilizer. Using covering space theory, the stabilizers of the latter components are precisely those subgroups of F_n conjugate to the fundamental group of some noncontractible component of G_u-1. Furthermore, since distinct components of G_u-1 are disjoint subtrees of G, the intersections of their stabilizers are trivial, and so the stabilizers are unequal if they are nontrivial. This completes the proof of H_u is nongeometric.

If H_u is geometric then we have equivariant and hence stabilizer preserving bijections N_j ↔ N_j ↔ P_j where N_j is the Nielsen line in G mapping to N_j under the collapse map G ↦ T. By definition the N_j are precisely those lines in G that cover the closed Nielsen path ρ. The element of π_1(G) represented by ρ (and denoted by ρ) is root free in π_1(G) because ρ has a unique u-illegal turn (Notations <ref> ItemCTiNP), and so by covering space theory the stabilizers of the lines N_j are precisely the infinite cyclic subgroups in the conjugacy class of the group ρπ_1(G) ≈ F_n. And as before, two different such lines have distinct stabilizers. 

The proof is completed by noting that ρ is not conjugate in F_n to the fundamental group of a noncontractible component of G_u-1, because ρ is a circuit not contained in G_u-1 (Notations <ref> ItemCTiNP).


 

  
 We may construct an F_n-equivariant piecewise Riemannian metric on the tree T, denoted ds, such that for vertices V,W ∈ T we have d_(V,W)=∫_VW ds. We may extend ds to an F_n-equivariant piecewise Riemannian metric denoted ds^* on T^* as follows. In the nongeometric case there is nothing to do. In the geometric case the group F_n acts freely and transitively on the set of cone edges {P_j Q j ∈ J,    Q ∈ Z(N_j)}; extend ds over a single cone edge P_j Q, then extend it over all other cone edges equivariantly to obtain ds^*; note that the length ∫_P_jQ ds^* is independent of j and Q. In the parageometric case, the group F_n acts freely on the set of cone edges, and there are two orbits of cone edges corresponding to the two endpoints of ρ; we extend ds over a single cone edge in each of the two orbits, then extend equivariantly to obtain ds^*; also, we require that length ∫_P_j Q ds^* be the same on both orbits of cone edges. Define the cone height to be the length of any cone edge in the metric ds^*.

Next let d^*(·,·) be the path metric on T^* obtained by minimizing path lengths: d^*(x,y) equals the minimum of ∫_γ ds^* over all continuous paths γ in T^* having endpoints x,y. The infimum is evidently minimized by some embedded edge path in T^* having endpoints x,y. Note that since T^* is not a tree, embedded edge paths need not be determined by their endpoints. 


Bypasses in T^*. For any ρ^* path QQ' we let QQ' denote the path QP(N) * P(N)Q' in T^*, called the bypass of QQ'. Note that a path in the graph T^* is a bypass if and only if it is a two-edge path having a cone point as its midpoint, and furthermore a bypass is completely determined by its endpoints. We thus have a one-to-one correspondence ν↔ν between the set of ρ^* paths and the set of bypasses. 

Extending the map f^Φ_T  T → T to f^Φ_T^* T^* → T^*. Following Definition <ref>, and choosing Φ∈(F_n) representing ϕ, lift f  G → G to G and project to T obtaining Φ-twisted equivariant maps

    f =  f^Φ G → G      f_T = f^Φ_T  T → T

With respect to the inclusion T  T^* we extend f^Φ_T to a Φ-twisted equivariant map 

    = ^Φ T^* → T^*

as follows. As noted, for convenience we will often suppress Φ from the notation for these maps.

The action of f_T on T induces a well-defined action on the Nielsen set , and so we can extend f_T over the set of cone points by setting (P(N)) = P(f_T(N)) for each N ∈. Furthermore, for each N ∈ the map f_T restricts to a bijection of basepoint sets f_T  Z(N) → Z(f_T(N)), and so for each Q ∈ Z(N) we can extend the endpoint map (P(N),Q) ↦ (P(f_T(N)), f_T(Q)) uniquely to an isometry between cone edges P(N),Q→P(f_T(N)),f_T(Q). 

For each edge E ⊂ T we have the following equation that follows from the “big L” eigenlength equation in Section <ref>:


  Eigenlength equation in T:   ∫_f_T(E) ds = λ∫_E ds

It follows that by equivariantly homotoping f_T relative to endpoints of edges we may arrange that f_T stretches each edge of T by a uniform factor λ. Since its extension f_T* is an isometry on edges of T^* ∖ T, it follows that f_T* is λ-Lipschitz. These conditions constitute additional constraints on the maps f_T and f_T^* which we record here:

Stretch properties of f_T and f_T^*:


  * The maps f_T,f_T^* stretch each edge E ⊂ T by a constant factor λ over the path f_T(E)=f_T^*(E).

  * The map T^* permutes cone edges, taking each cone edge isometrically to its image.

These stretch properties are the first step of Lemma <ref> to follow.



We recall here some basic definitions. Given constants k ≥ 1, c ≥ 0, a map of metric spaces f  X → Y is a (k,c)-quasi-isometric embedding if for all p,q ∈ X we have

    1/k  d(p,q)-c ≤ d(f(p),f(q)) ≤ k   d(p,q) + c

If in addition each point of Y has distance ≤ c from some point of f(X) then f is a (k,c)-quasi-isometry. If the domain X is a subinterval of  then we say that f is a (k,c)-quasigeodesic. Sometimes we conflate the constants k,c by setting k=c and using terminology like “k-quasi-isometries” etc. Sometimes we ignore k,c altogether and use terminology like “quasi-isometries” etc. 



The map f_T^* T^* → T^* is a quasi-isometry.


 Let Φ∈(F_n) be the representative of ϕ corresponding to f_T^*, and so f_T^* satisfies the Φ-twisted equivariance equation (see Definition <ref>). 

The map f_T^* is Lipschitz, by the “Stretch properties” noted above. To complete the proof it suffices to show that there is a Lipschitz map f̅_T^* T^* → T^* such that f_T^* and f̅_T^* are coarse inverses, meaning that each of the composed maps f̅_T^*∘ f_T^* and f_T^*∘f̅_T^* moves each point of T^* a uniformly bounded distance. We construct f̅_T^* by taking advantage of twisted equivariance of f_T^* combined with the fact that the action F_n  T^* has finitely many vertex and edge orbits (this is a kind of “twisted equivariant” version of the Milnor-Svarc lemma).

Consider the vertex set V^* of T^* and its partition V^* = V^0 ∐ V^∞ into points whose F_n-stabilizers are trivial and infinite, respectively. Using Φ-twisted equivariance it follows that for any vertex v ∈ V we have a subgroup inclusion Φ((v)) ⊂(f_T^*(v)). It follows that f_T^*(V^∞) ⊂ V^∞. Furthermore, that subgroup inclusion is an equation Φ((v)) = (f_T^*(v)) — this is a consequence of Lemma <ref> combined with the fact that f  G → G restricts to a homotopy equivalence of the union of noncontractible components of G_u-1 and with the fact that f_#(ρ)=ρ. It follows that the restricted map f_T^* V^∞→ V^∞ is a bijection of the set V^∞. 

Define the restriction f̅_T^* V^∞ to equal the inverse of the restriction f_T^* V^∞; this map f̅_T^* V^∞ is automatically Φ^-twisted equivariant. Define the restriction f̅_T^* V^0 as follows: choose one representative v ∈ V^0 of each orbit of the action F_n  V^0, choose f̅_T^*(v) ∈ V^0 arbitrarily, and extend over all of V^0 by Φ^-twisted equivariance. Having defined a Φ^-twisted equivariant map f̅_T^* V^* → V^*, extend f̅_T^* over each edge to stretch distance by a constant factor, and hence we obtain a Φ^-twisted equivariant map f̅_T^* T^* → T^*. Since there are only finitely many orbits of edges, f̅_T^* is Lipschitz. Since f_T^* is Φ-twisted equivariant and f̅_T^* is Φ^-twisted equivariant, it follows that the two compositions f_T^*f̅_T^* and f̅_T^* f_T^* are equivariant in the ordinary untwisted sense. Each of these compositions therefore moves each point a uniformly bounded distance, hence f_T^* and f̅_T^* are coarse inverses.




 §.§ Geometry and dynamics on T^*.
 

In this section we prove several propositions regarding T^*, including Proposition <ref> which is our interpretation of flaring in T^*. The proofs will follow after stating all of the propositions.



The inclusion of the vertex set of T into the vertex set of T^* is a quasi-isometry from the coarse metric D_ to the metric d^*: there exist constants K = K_<ref>≥ 1, C=C_<ref>≥ 0 such that for all vertices V,W ∈ T we have

    1/K  d^*(V,W) - C ≤ D_(V,W) ≤ K   d^*(V,W) + C





Given an end ξ∈ T, for any vertices V,W ∈ T the intersection of the two rays Vξ, Wξ is a subray of each. It follows that as we hold ξ fixed and let V vary, one of two alternatives holds: each ray Vξ has infinite d^* diameter in which case we say that ξ is infinitely far in T^*; or each ray Vξ has finite d^* diameter in which case we say that ξ is finitely far in T^*.

Recall (e.g. <cit.>) that a path f  I → T^* defined on an interval I ⊂ is a reparameterized quasigeodesic if there exists a monotonic, surjective function μ J → I defined on an interval J ⊂ such that the composition f ∘μ J → T^* is a quasigeodesic. 





The graph T^* with the metric d^* is Gromov hyperbolic. Each geodesic segment, ray, or line in T is a reparameterized quasigeodesic in T^* with respect to d^*, with uniform quasigeodesic constants. Furthermore, there is an injection that assigns to each ξ∈ T which is infinitely far in T^* a point ξ^* ∈ T^*, so that for any two points ξη∈ T both of which are infinitely far in T^*, the line ξη⊂ T is the unique line in T which is a reparameterized quasigeodesic line in T^* with ideal endpoints ξ^*,η^*.















We now state our re-interpretation of the earlier flaring result Proposition <ref> in the setting of T^*. 
Given η>0, and given a sequence of vertices V_r ∈ T^* defined for r in some interval of integers a ≤ r ≤ b, we say that this sequence is an η-pseudo-orbit of  if d^*((V_r),V_r+1) ≤η for all a ≤ r < r+1 ≤ b.





For each μ > 1, η≥ 0 there exist integers R ≥ 1 and A ≥ 0 such that for any pair of η-pseudo-orbits V_r and W_r of f_T^* defined on the integer interval -R ≤ r ≤ R, if d^*(V_0,W_0) ≥ A then

    μ· d^*(V_0,W_0) ≤max{d^*(V_-R,W_-R),d^*(V_R,W_R)}





  
Proof of Proposition <ref>: Quasicomparibility in T^*. 
In the ageometric case where ρ does not exist then T=T^* and D_ = d^* and we are done. Henceforth we assume that ρ exists. 

Letting h denote the cone height, it follows that each bypass has length 2h. 

Given vertices V,W ∈ T, using Proposition <ref> we obtain a decomposition of the path VW and an accompanying formula for D_(V,W):

    VW   = μ_0  ν_1  μ_1  … ν_A-1 μ_A-1 ν_A  μ_A 
    
    D_(V,W)    = l_(μ_0) + ⋯ + l_(μ_A)

Each ν_i is a subpath of some element N ∈ of the Nielsen set and the endpoints of ν_i are distinct points in Z(N), and so the corresponding bypass ν_i is defined. We thus obtain a path in T^* and a length calculation as follows:

    VW   =  μ_0  ν_1  μ_1  … ν_A-1 μ_A-1 ν_A  μ_A 
    (VW)    = l_(μ_0) + ⋯ + l_(μ_A) + 2  h  A

Applying Proposition <ref> with its constant η = η_<ref>, and applying Proposition <ref> ItemOneNotDeg, it follows that if 0 ≤ a < a+1  ≤ A then at least one of l_(μ_a), l_(μ_a+1) is ≥η, and hence

    l_(μ_0) + ⋯ + l_(μ_A) ≥  (A-1)  η / 2

We therefore have

    d^*(V,W)    ≤(VW) 
       ≤ l_(μ_0) + ⋯ + l_(μ_A) + 2h( 2 (l_(μ_0) + ⋯ + l_(μ_A))/η  + 1 ) 
       = (1 + 4h/η) D_(V,W)  + 2h

This proves the first inequality using any K ≥ 1 + 4h/η and C = 2h/K.

















We turn to the opposite inequality. Before starting, we shall normalize the choice of cone height to be h = 1/2 l_(ρ) and so each bypass ν has length l_(ρ). Proving Proposition <ref> with h normalized in this fashion implies the proposition for any value of h, because the normalized version of d^* is bi-Lipschitz equivalent to any other version.






























Given vertices V,W ∈ T, choose an embedded edge path γ^* ⊂ T^* with endpoints V,W satisfying two optimization conditions:


  (i) ∫_γ ds^* is minimal, and so d^*(V,W) = ∫_γ ds^*.

  (ii) Subject to (i), ∫_γ T ds^* = ∫_γ T ds is minimal.

There is a unique decomposition of γ as an alternating concatenation of subpaths in T and bypasses:

    γ^* = μ_0  ν_1  μ_1  … μ_A-1ν_A  μ_A

from which we obtain the formula

    d^*(V,W) = l_(μ_0) + ⋯ + l_(μ_A) + A   l_(ρ)

We claim that each μ_a has no ρ^*-subpath. Otherwise we can decompose μ_a = μ' ν' μ” where ν' is a Nielsen path in T. In the case that each of μ',μ” is nondegenerate, or that a=0 and μ” is nondegenerate, or that a=A and μ' is nondegenerate, construct a path γ' from γ by replacing ν' with the corresponding bypass; from the choice of normalization we have ∫_γ' ds^* = ∫_γ ds^* but ∫_γ'  T ds < ∫_γ T ds, a contradiction. The other cases all lead to a path γ' exhibiting the same contradiction, and are described as follows. In the case that a ≥ 1 and μ' is degenerate, replace the subpath ν_aν' of γ with the unique bypass having the same endpoints. And in the case that a ≤ A-1 and μ” is degenerate, replace the subpath ν' ν_a+1 with the unique bypass having the same endpoints. And in the last remaining case, where a=A=0 and μ',μ” are both degenerate, we have ν'=γ^* and we let γ' = ν' be the corresponding bypass. 

Consider the concatenated edge path in T denoted

    μ_0  ν_1  μ_1  … μ_A-1 ν_A  μ_A

which is obtained from γ^* by replacing each bypass ν_a with the corresponding ρ^*-subpath ν_a ⊂ T. Straightening this concatenation in T produces the path VW to which we may inductively apply the coarse triangle inequality for L_ given in Lemma <ref>, with the conclusion that

    D_(V,W)    = L_(VW) 
       ≤ L_(μ_0) + L_(ν_1) + L_(μ_1) + ⋯
                +L_(μ_A-1) + L_(ν_A) + L_(μ_A) + (2A-1) C_<ref>
    Since μ_a has no ρ^± i subpath we have L_(μ_a)=l_(μ_a), and since L_(ρ^± i) = 0 we have L_(ν_a)=0, and therefore
    D_(V,W)    ≤ l_(μ_0) + ⋯ + l_(μ_A) + 2AC_<ref>
       ≤ l_(μ_0) + ⋯ + l_(μ_A) + 2C_<ref>/l_(ρ)· A l_(ρ)  
       ≤ K d^*(V,W)


for any K ≥max{1,2 C_<ref> / l_(ρ)}. This completes the proof of Proposition <ref>. 



  
Proof of Proposition <ref>: Hyperbolicity of T^*. If ρ does not exist, i.e. in the ageometric case, since T^*=T is a tree we are done. The parageometric case is similarly easy, but it is also subsumed by the general proof when ρ does exist for which purpose we will apply a result of Kapovich and Rafi <cit.>. 

Let T^** be the graph obtained from T by attaching an edge Q Q' for each unordered pair Q  Q' ∈ Z(N) for each element N ∈ of the Nielsen set. We put the simplicial metric on T^**, assigning length 1 to each edge. We have a map T^**→ T^* extending the inclusion T  T^*, defined to take each attached edge Q Q'⊂ T^** to the corresponding bypass QQ'. This map is evidently a quasi-isometry from the vertices of T^** to the vertices of T^*, and this quasi-isometry commutes with the inclusions of T into T^* and into T^**. The conclusions in the first two sentences of Proposition <ref>, namely hyperbolicity of T^* and the fact that geodesics in T are uniform reparameterized quasigeodesics in T^*, will therefore follow once we demonstrate the same conclusions with T^** in place of T^*. Those conclusions for T^** are identical to the conclusions of <cit.> applied to the inclusion map T  T^**: the graph T^** is hyperbolic; and each arc VW⊂ T is uniformly Hausdorff close in T^** to each geodesic in T^** with the same endpoints V,W. So we need only verify that the inclusion map T  T^** satisfies the hypotheses of <cit.> with respect to the simplicial path metrics on T and T^** that assign length 1 to each edge.

One hypothesis <cit.> is that T be hyperbolic, which holds for all path metrics on trees. Another hypothesis is that the inclusion T  T^** be a Lipschitz graph map which means that it takes each edge of T to a bounded length edge path of T^*, but this is immediate since the inclusion is an isometry onto its image. 

The remaining hypotheses of <cit.> are numbered (1), (2), (3), the first two of which are trivially satisfied using that the inclusion map from vertices of T to vertices T^** is surjective (this is why we use T^** instead of T^*). The last hypothesis (3) says that there exists an integer M > 0 so that for any vertices V  W ∈ T,   if V,W are connected by an edge in T^** then the diameter in T^** of the path VW⊂ T is uniformly bounded. The only case that needs attention is when VW is not a single edge in T but V,W are connected by an edge in T^**. This happens only if VW is a ρ^* subpath of some element N ∈ of the Nielsen set, and so VW is a concatenation of a sequence of Nielsen paths Q_k-1Q_k in N, where the concatenation points form a consecutive sequence in Z(N) VW of the form

    V=Q_0,Q_1,…,Q_K=W

Each pair Q_i,Q_j is connected by an edge Q_iQ_j⊂ T^** (i ≤ j =0,…,K). Since each vertex of T along VW is contained in one of the Nielsen paths Q_k-1Q_k and hence its distance to one of Q_k-1 or Q_k is at most l_u(ρ) / 2, it follows that the diameter of VW in T^** is bounded above by M = 1 + l_u(ρ).



It remains to prove the “Furthemore” sentence. Given ξ∈ T, for any vertex V ∈ T the ray Vξ is a reparameterized quasigeodesic in T^*. If ξ is infinitely far in T^* then the reparameterization of Vξ defines a quasigeodesic ray in T^* which therefore limits on a unique ξ^* ∈ T^*. This point ξ^* is well-defined because for any other vertex W ∈ T the intersection of the two rays Vξ, Wξ has finite Hausdorff distance in T^* from each of them, hence the reparameterizations of those two rays limit on the same point in T^*. For any two points ξη∈ T that are both infinitely far in T^*, consider the line ξη. Choose a vertex V ∈ T, and choose sequences x_i in Vξ and y_i in Vη so that in T the sequence x_i limits to ξ and y_i limits to η. It follows that in T^* the sequence x_i limits to ξ^* and y_i limits to η^*. Since d^*(x_i,y_i) →∞ and since ξη is a reparameterized quasigeodesic, this is only possible if ξ^* η^*, proving injectivity of the map ξ↦ξ^*. To prove the required uniqueness property of ξη, consider any other two points ξ' η' ∈ T. If one or both of ξ' or η' is finitely far in T^* then the reparameterization of ξ'η' is a quasigeodesic ray or segment, hence has infinite Hausdorff distance in T^* from the quasigeodesic line ξη. If both of ξ',η' are infinitely far in T^*, and if ξη and ξ'η' have finite Hausdorff distance in T^*, then it follows that {ξ^*,η^*}={ξ'^*,η'^*}, and hence by injectivity we have {ξ,η} = {ξ',η'} and therefore ξη=ξ'η'. 






  
Proof of Proposition <ref>: Flaring in T^*. We denote the constants of Proposition <ref> in shorthand as K=K_<ref>, C=C_<ref>.

Fix μ>1 and η≥ 0. Consider R ≥ 1, to be specified, and a pair of η-pseudo-orbits V_r, W_r for f_T^* defined for -R ≤ r ≤ R. Choose vertices V_r, W_r ∈ G projecting to V_r, W_r respectively. In G denote the path β_r = V_r W_r, and let β_r be its projection to G. Also, for -R<r≤ R denote α_r = V_r,  f(V_r-1) and ω_r =  f(W_r-1), W_r, which are the unique paths such that β_r = [α_r  f_#(β_r-1) ω_r]. Let α_r,ω_r be their projections to G. By assumption we have d^*(f_T^*(V_r-1),V_r) ≤η and d^*(f_T^*(W_r-1),W_r) ≤η, and applying Proposition <ref> it follows that

    L_(α_r),L_(ω_r) ≤ K  η + C ≡η'

and so f_#(β_r-1) η'L_β_r. 

Let μ' = 2K^2μ.

By Proposition <ref> the path function L_ satisfies the flaring condition of Definition <ref> with respect to f, from which using μ' and η' we obtain constants R' ≥ 1, A' ≥ 1. We now specify R = R'. Also let 

    A = max{KA' + C, 2K^2C + 2C/μ}

Applying Proposition <ref> it follows that if d^*(V_0,W_0) ≥ A then L_(β_0) ≥ A'. The flaring condition of L_ therefore applies with the conclusion that

    μ' L_(β_0)    ≤max{L_(β_-R), L_(β_R)}
    and soμ  d^*(V_0,W_0)    ≤ K μ  L_(β_0) + K C μ
       ≤K μ/μ'max{L_(β_-R),L_(β_R)} + K C μ
    Applying Proposition <ref> again we haveμ  d^*(V_0,W_0) 
             ≤K^2 μ/μ'max{d^*(V_-R,W_-R),d^*(V_R,W_R)} + K^2Cμ+C 
       ≤1/2max{d^*(V_-R,W_-R),d^*(V_R,W_R)} + 1/2μ  A 
       ≤1/2max{d^*(V_-R,W_-R),d^*(V_R,W_R)} + 1/2μ  d^*(V_0,W_0) 
    1/2μ  d^*(V_0,W_0)    ≤1/2max{d^*(V_-R,W_-R),d^*(V_R,W_R)}

which completes the proof.




 §.§ Construction of .



We continue to fix the choice of Φ∈(F_n) representing ϕ, and we consider the corresponding Φ-twisted equivariant map  f_T^* = f^Φ_T^* T^* → T^*. Our definition of the suspension space  will formally depend on this choice (but see remarks at the end of the section regarding dependence on Φ of the constructions to follow).





 We define  to be the suspension space of T^*, namely the quotient of T^* ×× [0,1] modulo the gluing identification (x,k,1) ≈ (f_T^*(x),k+1,0) for each k ∈ and x ∈ T^*. Let [x,k,r] ∈ denote the equivalence class of (x,k,r). We have a well-defined and continuous projection map p → given by p[x,k,r] = k+r. For each closed connected subset J ⊂ we denote _J = p^(J) which we refer to as a slice of . In the special case of a singleton s ∈ we refer to _s = p^(s) as a fiber. Each fiber may be regarded as a copy of T^*, in the sense that if k = ⌊ s ⌋ and r = s-k then we obtain a homeomorphism j_s  T^* → S_s given by j_s(x)=[x,k,r]; in the special case that s is an integer we have j_s(x)=[x,s,0]. 
 
We have an action F_n which is induced by the action F_n  T^* as follows: for each γ∈ F_n and each [x,k,r] ∈ we have

    γ· [x,k,r] = [Φ^k(γ) · x, k, r]

This action is well-defined because, using Φ-twisted equivariance of f_T^* T^* → T^*, we have

    γ· [x,k,1]    = [Φ^k(γ) · x,k,1] = [f_T^*(Φ^k(γ) · x),k+1,0] =  
       =  [Φ(Φ^k(γ)) · f_T^*(x),k+1,0] 
       = [Φ^k+1(γ) · f_T^*(x),k+1,0] 
       = γ· [f_T^*(x),k+1,0]

Note that the homeomorphism j_0  T^* →_0 is equivariant with respect to F_n actions. For generally, for each integer k the homeomorphism j_k  T^* →_k is Φ^-k-twisted equivariant, because

    j_k(γ· x) = [γ· x,k,0] = Φ^-k(γ) · [x,k,0] = Φ^-k(γ) · j_k(x)


We have a semiflow × [0,) →, which is partially defined by the formula 

    [x,k,s] · t = [x,k,s+t]     (x ∈ T^*,  k ∈,  0 ≤ s ≤ 1,  0 ≤ t ≤ 1-s)

and which uniquely extends to all t ≥ 0 by requiring the semiflow equation (p · t) · u = p · (t+u) to hold for all t,u ≥ 0. In particular _s · t = _s+t for all s ∈, t ≥ 0. For each b ∈ we define the first hitting map h_b _(-,b]↦_b by letting ξ∈_(-∞,b] flow forward from ξ∈_p(ξ) to _b along the flow segment ξ· [0,b-p(ξ)], thus obtaining the formula h_b(ξ) = ξ· (b-p(ξ)). 

This completes Definition <ref>.




We define a piecewise Riemannian metric on . Recall the piecewise Riemannian metric ds^* on T^* (Section <ref>). For each edge E ⊂ T^* and each integer n we define a Riemannian metric d_E on E × n × [0,1] (≈ E × [0,1] and not depending on n), in two cases. In the case E ⊂ T, use the metric d_E^2=λ^2t (ds^*)^2 + dt^2; note that f_T^* stretches the length of E by a constant factor λ, and hence the gluing map (x,n,1) ↦ (f_T^*(x),n+1,0) takes E × n × 1 isometrically onto f_T^*(E) × (n+1) × 0. In the case where E ⊂ T^* ∖ T, equivalently we are in the geometric or parageometric case and E is a cone edge, use the metric d_E^2 = (ds^*)^2 + dt^2; note that E'=f_T^*(E) is also a cone edge and that f_T^* maps E isometrically to E', and so once again the gluing map (x,n,1) ↦ (f_T^*(x),n+1,0) takes E × n × 1 isometrically onto f_T^*(E) × (n+1) × 0. These metrics on all of the rectangles E × n × [0,1] glue up isometrically along their common boundaries in , as follows. First, for any two edges E,E' ⊂ T^* having a common endpoint x = E  E' =  E  E', the restrictions to [0,1] ≈ x × n × [0,1] of the metric d_E on E × [0,1] ≈ E × n × [0,1] and the metric d_E' on E' × [0,1] ≈ E' × n × [0,1] are both equal to dt. Fixing n and letting E ⊂ T^* vary, this allows us to glue up all of the rectangles E × n × [0,1] to get a well-defined piecewise Riemannian metric on T^* × n × [0,1]. Next, as noted above the gluing map from T^* × n × 1 to T^* × (n+1) × 0 maps each E × n × 1 isometrically onto f_T^*(E) × (n+1) × 0; letting n vary this allows us to glue up T^* × n × [0,1] to get the desired piecewise Riemannian metric on .

By minimizing path lengths using the above piecewise Riemannian metric on , we obtain a geodesic metric d_(·,·) on . We may similarly define a geodesic metric d_J(·,·) on any slice _J, by minimizing path length with respect to the restricted piecewise Riemannian metric on _J. In the special case of a fiber _s, letting n = ⌊ s ⌋, for each edge E ⊂ T^* with image edge j_s(E) ⊂_s the metrics d^* on E and d_s in j_s(E) are related so that if E ⊂ T then j_s stretches the metric by a factor of λ^s-n, whereas if E ⊂ T^* ∖ T then j_s preserves the metric. In particular the map j_s  T^* →_s is a λ^s-n bilipschitz homeomorphism; it is therefore an isometry if s is an integer, and a λ-bilipschitz homeomorphism in general.

This completes Definition <ref>.


For use at the very end of the proof of the Hyperbolic Action Theorem, when verifying the WWPD conclusions, we shall need the following metric property of :



For any two fibers _s,_t ⊂ and any x ∈_s, y ∈_t we have d_(x,y) ≥s-t. 



The lemma follows by noting that for each edge E ⊂ T^*, the projection function p  E × [0,1] → [0,1] has the property that for each tangent vector v ∈ T(E × [0,1]) we have Dp(v)≥v, using the Riemannian metric d_E on the right hand side of the inequality. 


The following metric property will be used later to study how the inclusion map of fibers _i can distort distance.



For each integer m there exist constants k_m ≥ 1, c_m ≥ 0 such that for each integer a and each s ∈ J = [a,a+m] the inclusion map i _s _J is a k_m,c_m quasi-isometry.


 We construct a cycle of equivariant Lipschitz maps as follows:

    _s [r]^i   _J [d]^h_a+m
    _a [u]^h_r   _a+m[l]^h̅

The inclusion map i is 1-Lipschitz. The equivariant maps h_a+m and h_r are first hitting maps, each of which is λ^m Lipschitz. The map h̅ will be an equivariant coarse inverse to the equivariant map h_a+m_a →_a+m, for the construction of which we consider the commutative diagram

    T^* [r]^j_a[d]_f^m_T^*   _a [d]^h_a+m
    
    T^* [r]_j_a+m@/^3pc/@.>[u]^f̅^m_T^*   _a+m@/_3pc/@.>[u]_h̅

In this diagram the map f^m_T^* is Φ^m-twisted equivariant, and the map h_a+m is (untwisted) equivariant. The top and bottom maps are the instances k=a and k=a+m of the Φ^-k-twisted equivariant isometry j_k  T^* →_k, whose inverse j^_k _k → T^* is Φ^k twisted equivariant. By Lemma <ref> the map f_T^* has a Φ^-1-twisted equivariant Lipschitz coarse inverse f̅_T^*. It follows that f^m_T^* has a Φ^-m-twisted equivariant Lipschitz coarse inverse f̅^m_T^* whose Lipschitz and coarse inverse constants depend on m. We may therefore fill in the diagram with the map h̅ = j_a ∘f̅^m_T^*∘ j_a+m^ which makes the diagram commute and which is an untwisted equivariant Lipschitz coarse inverse for h_a+m, with Lipschitz and coarse inverse constants depending on m. 

Going round the cycle from _s to itself and from _J to itself we obtain two equivariant self-maps both of which move points uniformly bounded distances depending on m. The maps i _s →_J and h_r ∘h̅∘ h_a+m_J →_s are therefore Lipschitz coarse inverses and hence quasi-isometries with constants depending on m.




  
Remark. The definition of the suspension space  depends ostensibly on the choice of representative Φ∈(F_n) of ϕ, but in fact different choices of Φ produces suspension spaces which are F_n-equivariantly isometric, as can easily be checked using the fact that distinct choices of Φ differ by an inner automorphism of F_n.




 §.§ Proof of hyperbolicity of .



We prove that  is a hyperbolic metric space by applying the Mj-Sardar combination theorem <cit.>, a descendant of the Bestvina–Feighn combination theorem <cit.>. The hypotheses of the Mj-Sardar theorem consist of an opening hypothesis and four numbered hypotheses which we must check. The last of those four is the “flaring condition” which we prove by application of Proposition <ref>.





Opening hypothesis of <cit.>: A metric bundle. This hypothesis says that  is a metric bundle over  with respect to the map p →. We must therefore verify that the map p → satisfies the definition of a metric bundle as given in  <cit.>. First, p must be Lipschitz map, and each fiber _s must be a geodesic metric space with respect to the path metric induced from , each of which we have already verified. Also, item 2) of <cit.>, when translated into our setting, requires that for each interval [a,b] ⊂ such that b-a ≤ 1, and for each s ∈ [a,b] and ξ∈_s, there exists a path in  of uniformly bounded length that passes through ξ and has endpoints on _a and _b respectively. To obtain this path choose η∈_a so that η· (s-a)=ξ and take the path t ↦η· t defined on a ≤ t ≤ b, whose length equals b-a ≤ 1.

The remaining verification needed for  to be a metric bundle is that the set of inclusions _s (s ∈) is uniformly proper, meaning that these inclusions are uniformly Lipschitz — in fact they are all 1-Lipschitz by construction — and that there exists a single nondecreasing “gauge” function δ [0,) → [0,) for these inclusions, having the following property:


  (*) for any s ∈, any x,y ∈_s, and any D ≥ 0, if d_(x,y) ≤ D then d_s(x,y) ≤δ(D). 

To define δ consider x,y ∈_s connected by a geodesic path γ in  with (γ) = d_(x,y) ≤ D. The projection p γ in  has length ≤ D and so, letting m = ⌊ D+2 ⌋, there are integers a and b=a+m such that ⊷(p γ) ⊂ [a,b], implying that ⊷(γ) ⊂_[a,b] which implies in turn that d_(x,y) = d_[a,b](x,y). Applying Lemma <ref> we have 1/k_m  d_s(x,y) - c_m ≤ d_[a,b](x,y) and hence d_s(x,y) ≤ k_m d_(x,y) + k_m   c_m. We may assume that k_m and c_m are nondecreasing functions of m and hence k_⌊ D ⌋ and c_⌊ D ⌋ are nondecreasing functions of D, and so using δ(D) = k_⌊ D ⌋(D + c_⌊ D ⌋) we are done.

We record for later use the following:



For each s ∈ the inclusion _s is uniformly proper. In particular, the inclusion T^* = _0 is uniformly proper.














Hypotheses (1), (2) of <cit.>: Base and fiber hyperbolicity. These hypotheses require that the base space  is hyperbolic which is evident, and that the fibers _s are hyperbolic with uniform hyperbolicity constant. Proposition <ref> gives us a constant δ' ≥ 0 such that T^* is δ'-hyperbolic. Since each fiber _s is λ-bilipschitz homeomorphic to T^*, it follows that _s is hyperbolic with a uniform hyperbolicity constant δ depending only on δ' and λ.



Hypothesis (3) of <cit.>: Barycenters. This hypothesis says that the barycenter maps ^3 _s →_s are uniformly coarsely surjective as s varies. We review what this means from <cit.>. Given a δ-hyperbolic geodesic metric space X with Gromov boundary X, consider 
the triple space 

    ^3 X = {(ξ_1,ξ_2,ξ_3) ∈ ( X)^3 ξ_i ξ_j  if  i  j}

The barycenter map ^3 X → X is a coarsely well-defined map as follows. There exists constants K ≥ 1, C ≥ 0 depending only on δ such that for any two points ξ_1 ξ_2 ∈ X there exists a K,C-quasigeodesically embedded line in X having endpoints ξ_1,ξ_2; we use the notation ξ_1ξ_2 for any such quasigeodesic line. By <cit.> there exist constants D,L ≥ 0 depending only on δ such that for each triple ξ=(ξ_1,ξ_2,ξ_3) ∈^3 X there exists a point b_ξ∈ X which comes within distance D of any of the lines ξ_iξ_j, i  j ∈{1,2,3}, and for any other such point b'_ξ the distance between b_ξ and b'_ξ is ≤ L. Once the constants K, C, D, L have been chosen, any such point b_ξ is called a barycenter of ξ, and any map ^3 X → X taking each triple ξ to a barycenter b_ξ is called a barycenter map for X.

To say that the barycenter maps ^3_s →_s are uniformly coarsely surjective means that there exists a “coboundedness constant” E ≥ 0 such that for each s ∈ the image of each barycenter map ^3_s →_s comes within distance E of each point of _s. For the hyperbolic space T^*, the action F_n  T^* has a fundamental domain τ⊂ T^* of bounded diameter, and so E” = (τ) is a coboundedness constant for any F_n-equivariant barycenter map ^3 T^* → T^*, hence there is a uniform coboundedness constant E' for all barycenter maps ^3 T^* → T^*. Since each of the fibers _s comes equipped with a λ-bilipschitz homeomorphism j_s  T^* →_s, their barycenter maps have a uniform coboundedness constant E = λ E'.



Hypothesis (4) of <cit.> aka <cit.>. Here is a slight restatement of the hypothesis specialized to our present context.



  Flaring 1: For all k_1 ≥ 1, ν_1 > 1, there exist integers A_1, R ≥ 0 such that for any s ∈ and any two k_1-quasigeodesics 

    γ_1,γ_2  [s-R,s+R] →

which are sections of the projection map p — meaning that p γ_i is the identity on the interval [s-R,s+R] — the following implication holds:

  (F_1) if d_s(γ_1(s),γ_2(s)) ≥ A_1 then

    ν_1  ·  d_s(γ_1(s),γ_2(s)) ≤max{ d_s-R(γ_1(s-R),γ_2(s-R)), d_s+R(γ_1(s+R),γ_2(s+R)) }


This statement tautologically implies the flaring hypothesis given in <cit.>, the difference being that in the latter statement the quantifier order starts out as “For all k ≥ 1 there exist μ > 1 and integers A,r ≥ 0 such that…” with the remainder of the statement unchanged (a simple geometric estimation argument yields the converse implication, but we will not need this).

For proving Flaring 1 we first reduce it to a “discretized” version taking place solely in the fibers _r for integer values of r, as follows:



  Flaring 2: For all k_2 ≥ 1, ν_2 > 1, there exist integers A_2, R ≥ 0 such that for any integer m ∈ and any two k_2-quasigeodesic maps

    δ_1,δ_2 {m-R,…,m,…,m+R}→

which are sections of the projection map p, the following implication holds:

  (F_2) if d_m(δ_1(m),δ_2(m)) ≥ A_2 then




    ν_2  ·  d_m(δ_1(m),δ_2(m)) ≤max{d_m-R(δ_1(m-R),δ_2(m-R)), d_m+R(δ_1(m+R),δ_2(m+R)) }


To show that Flaring 2 implies Flaring 1, choose k_1,ν_1. Consider any integer R ≥ 0, any s ∈, and any pair γ_1,γ_2  [s-R,s+R] → of k_1-quasigeodesic sections of the projection map p. Let m = ⌊ s ⌋ and let t = s - m. The semiflow restricts to λ-bilipschitz homeomorphisms h_r _r ↦_r+t defined by h_r[x,r,0] = [x,r,t] for any integer r, having the property that the distance from each [x,r,0] to its h_r-image [x,r,t] in  is at most 1. It follows that the functions

    δ_j {m-R,…,m+R}→,   δ_j(r) = h_r^(γ_j(r+t))

are k_2-quasi-isometric sections of p, where the constant k_2 depends only on k_1 and λ. Applying Flaring 2, for any ν_2 > 1 there exist integers A_2,R ≥ 0 such that the implication (F_2) holds. Again using that the maps h_r are λ-bilipschitz homeomorphisms, if we take ν_2 = ν_1 ·λ^2 and A_1 = A_2 ·λ then the hypothesis of (F_1) implies the hypothesis of (F_2) and the conclusion of (F_2) implies the conclusion of (F_1). 

It remains to verify Flaring 2 by applying Proposition <ref> which we restate here for convenience in a form matching that of Flaring 2: 


  Flaring 3: For each μ > 1, η≥ 0 there exist integers A_2 ≥ 0, R ≥ 1 such that for any pair of η-pseudo-orbits V_r and W_r of f_T^* T^* → T^* defined for integers -R ≤ r ≤ R, the following implication holds:

  (F_3) if d^*(V_0,W_0) ≥ A_2 then 



    μ· d^*(V_0,W_0) ≤max{ d^*(V_-R,W_-R) , d^*(V_R,W_R) }


For the proof that Flaring 3Flaring 2, for each r ∈ we have a commutative diagram

    T^* [r]^f_T^*[d]_j_r-1    T^* [d]^j_r
    _r-1[r]_h_r   _r

where h_r is the first hitting map. Note that in this diagram the bottom is equivariant, the top is Φ-twisted equivariant, the left is a Φ^1-r-twisted equivariant isometry, and the right is Φ^-r-twisted equivariant isometry.

Choose k_2 ≥ 1, ν_2>1, consider integers R ≥ 0 and m, and consider a pair of k_2-quasigeodesic maps δ_i {m-R,…,m+R}→ which are sections of p, for i=1,2. It follows that if m-R ≤ r-1 < r ≤ m+R then d_(δ_i(r-1),δ_i(r)) ≤ 2 k_2 (recall that “k_2-quasigeodesic” is synonymous with “(k_2,k_2)-quasigeodesic”). For each x ∈_r-1 we have d_(h_r(x),x)) ≤ 1 and hence d_(h_r(δ_i(r-1)),δ_i(r)) ≤ 2k_2 + 1. For r ∈{-R,…,+R} denote V_r = j_m+r^(δ_1(m+r)), W_r = j_m+r^(δ_2(m+r)). It follows that d_*(f_T^*(V_r-1),V_r), d_*(f_T^*(W_r-1,W_r) ≤ 2k_2+1, and so the sequences (V_r) and (W_r) are both η-pseudo-orbits of f_T^*, defined for m-R ≤ r ≤ m+R and with η = 2k_2+1. Since j_m is an isometry we have d_m(δ_1(m),δ_2(m))=d^*(V_0,W_0), and so the hypothesis of (F_2) implies the hypothesis of (F_3). Similarly since j_m-R and j_m+R are isometries, the conclusion of (F_3) implies the conclusion of (F_2) using μ=ν_2. We have therefore proved that Flaring 3Flaring 2.



This completes the verification of the hypotheses of the Mj-Sardar combination theorem, and so the space  is therefore hyperbolic.






§ ABELIAN SUBGROUPS OF (F_N)


This section reviews some background material needed for the rest of the paper. Section <ref> contains basic material from  regarding automorphisms and outer automorphisms (see also  for a comprehensive overview). Section <ref> reviews elements of the theory of abelian subgroups of (F_n) developed in <cit.>, focussing on disintegration subgroups. 









 §.§ Background review






  §.§.§ More about 
 


In Notations <ref> we reviewed features of a  f  G → G with associated f-invariant filtration G_1 ⊂⋯⊂ G_u under the assumption that the top stratum H_u is . In studying disintegration groups we shall need some further defining properties and derived properties of  (for this section only the reader may ignore the assumption that the top stratum is ). We shall refer to  for material specific to , to  for more general material, and to  for certain “compilation” results with multiple sources in  or . One may also consult  for a comprehensive overview.



  
General properties of strata. <cit.>   Each stratum H_i = G_i ∖ G_i-1 is either an irreducible stratum meaning that its transition matrix M_i is an irreducible matrix, or a zero stratum meaning that M_i is a zero matrix. Each irreducible stratum satisfies one of the following:

H_i is an  stratum: <cit.> The matrix M_i is a k × k Perron-Frobenius matrix for some k ≥ 2, having eigenvalue λ>1; or










H_i is an  stratum: <cit.> H_i = E_i is a single edge with an orientation such that f(E_i) = E_i u where u is either trivial or a closed path in G_i-1 having distinct initial and terminal directions. 




An  stratum H_i is a fixed stratum if u is trivial, a linear stratum if u is a Nielsen path (equivalently u is a periodic Nielsen path), and a superlinear stratum otherwise. 



  
Properties of -linear strata: An -linear stratum H_i = E_i will also be referred to as a linear edge of G. The linear edges of G have the following features:

Twist path and twist coefficient: <cit.>   For each linear edge E_i we have f(E_i) = E_i w_i^d_i for a unique closed Nielsen path w_i which is root free meaning that w_i is not an iterate of any shorter closed path (equivalently, if p is the base point of w_i, then the element of the group π_1(G,p) represented by w_i is root free). We say that w_i is the twist path of E_i and that the integer d_i  0 is its twist coefficient. If E_j  E_i is another linear edge having twist path w_j, and if w_i and w_j determine the same conjugacy class up to inversion, then w_i=w_j and d_i  d_j.

 Nielsen paths: <cit.>    For each  edge E_i, if there is an indivisible Nielsen path contained in G_i but not in G_i-1 then E_i is a linear edge, and every such Nielsen path has the form E_i w_i^k E_i for some k  0.

Exceptional paths: <cit.>   These are paths of the form E_i w^k E_j where E_i  E_j are linear edges having the same twist path w=w_i=w_j and having twist coefficients d_i,d_j of the same sign.




  
Properties of  strata: These properties were stated in Notations <ref> with respect to the top  stratum H_u, but we go over them again here for an arbitrary -stratum H_i, and with a somewhat different emphasis. 

Lines: <cit.>   Recall the spaces of lines in F_n and in G and the canonical homeomorphism between them:

    (F_n)     = {2 point subsets of F_n} / F_n 
    (G)      = {bi-infinite paths in G} / reparameterization

where the topology on (F_n) is induced by the Hausdorff topology on compact subsets of F_n, and the topology on (G) has a basis element for each finite path γ consisting of all bi-infinite paths having γ as a subpath. The homeomorphism (F_n) ↔(G) is induced by the universal covering map G → G and the natural bijection F_n ≈ G. We refer to this homeomorphism by saying that a line in F_n is realized by the corresponding line in G.
 
Attracting laminations: <cit.>, <cit.>   Associated to H_i is its attracting lamination Λ_i ⊂(F_n) which is the set of all lines in F_n whose realization ℓ in G has the property that for each finite subpath γ of ℓ and each edge E ⊂ H_i there exists k ≥ 1 such that γ is a subpath of f^k_#(E). For distinct  strata H_i, H_j (i  j) the corresponding laminations Λ_i,Λ_j are distinct. The set of laminations Ł(ϕ)={Λ_i} is independent of the choice of  representative f  G → G. 

EG Nielsen paths: <cit.>,  Fact 1.42.   Up to inversion there exists at most one indivisible periodic Nielsen path ρ contained in G_i but not in G_i-1. Its initial and terminal directions are distinct, and at least one endpoint of ρ is not contained in G_i-1. 

Geometricity: , Fact 2.3.   The stratum H_i is geometric if and only if ρ exists and is a closed path.

Fixed circuits: , Fact 1.39.   If σ is a circuit fixed by f_# then σ is a concatenation of fixed edges and indivisible Nielsen paths.




  
Properties of zero strata: <cit.>

Each zero stratum H_i ⊂ G has the following properties: 

Envelopment: There exist indices s<i<r such that H_s is an irreducible stratum, H_r is an  stratum, each component of G_r is noncontractible, and each H_j with s<j<r is a zero stratum and a contractible component of G_r-1. We say that the zero strata H_j with s<j<r are enveloped by H_r, and we denote H^z_r to be the union of H_r with its enveloped zero strata. The filtration element G_s is the union of the noncontractible components of G_r-1, and H^z_r = G_r ∖ G_s. 

Taken paths.  These are the paths μ in H_i for which there exists an edge E of some irreducible stratum H_q with q>i, and there exists k ≥ 1, such that μ is a maximal subpath in H_i of the path f^k_#(E); we say more specifically that the path μ is q-taken. If H_r is the  stratum that envelopes H_i then every edge E ⊂ H_i is an r-taken path, from which it follows that the endpoints of E are vertices of H_r not contained in G_r-1.



  
Properties of complete splittings: <cit.>, <cit.>

A splitting of a path γ in G is a concatenation expression γ = γ_1 ·…·γ_J such that for all k ≥ 1 we have f^k_#(γ) = f^k_#(γ_1) ·…· f^k_#(γ_J).  The characteristic property of a  — short for “completely split relative train track map” — is the following:

Complete splitting: For each edge E there is a unique splitting f(E) = σ_1 ·…·σ_n which is complete, meaning that each term σ_i is either an edge in an irreducible stratum, an  indivisible Nielsen path, an  indivisible Nielsen path, an exceptional path, or a maximal taken subpath in a zero stratum.




  §.§.§ Principal automorphisms and rotationless outer automorphisms.
 

Consider an automorphism Φ∈(F_n),  with induced boundary homeomorphism denoted Φ F_n → F_n, and with fixed subgroup denoted (Φ)  F_n. Denote the sets of periodic points and fixed points of Φ as (Φ) and (Φ) ⊂ F_n, respectively. Consider ξ∈(Φ) of period k ≥ 1. We say ξ is an attractor for Φ if it has a neighborhood U ⊂ F_n such that Φ^k(U) ⊂ U and the collection {Φ^ki(U)  i ≥ 0} is a neighborhood basis of ξ.  Also, ξ is a repeller for Φ if it is an attractor for Φ^.

Within (Φ) and (Φ) denote the sets of attracting, repelling, and nonrepelling points, respectively, as 

    _+(Φ),  _-(Φ),  _N(Φ)    ⊂(Φ) 
    _+(Φ),  _-(Φ),  _N(Φ)    ⊂(Φ)


For each c ∈ F_n associated inner automorphism i_c(a)=cac^ we use the following special notations 

    c = (  i_c) = {_-c, _+c},   {_- c} = _-(  i_c),   {_+ c} = _+(  i_c)

The following equivalences are collected in <cit.> based on results from <cit.> and <cit.>.


For each Φ∈(F_n) and each nontrivial c ∈ F_n we have

    Φ∈(c)     i_c   commutes with  Φ  (Φ)   is invariant under  î_c 
        c ⊂(Φ)































  
Principal automorphisms. <cit.>.   We say that Φ is a principal automorphism if _N(Φ)≥ 2, and furthermore if _N(Φ)=2 then _N(Φ) is neither equal to c for any nontrivial c ∈ F_n, nor equal to the set of endpoints of a lift of a generic leaf of an attracting lamination of the outer automorphism representing Φ. For each ϕ∈(F_n) let P(ϕ) ⊂(F_n) denote the set of principal automorphisms representing ϕ. 


For each ϕ∈(F_n) let P^±(ϕ) denote[In  the definition of P^±(ϕ) was incorrectly stated as P(ϕ)  P(ϕ^). The definition given here should replace the one in . No further changes are required in , because the arguments there which use P^±(ϕ) are all written using the current, correct definition.] the set of all Φ∈(F_n) representing ϕ such that either Φ or Φ^ is principal, equivalently, 

    P^±(ϕ) = P(ϕ)  (P(ϕ^))^




The symmetry equation P^±(ϕ^)=(P^±(ϕ))^ is useful in situations where one is trying to prove a certain property of Φ∈ P^±(ϕ) that is symmetric under inversion of Φ: one may reduce to the case that Φ is principal by replacing ϕ∈(F_n) and Φ by their inverses in the case where Φ is not already principal. We use this reduction argument with little comment in many places.



  
Rotationless outer automorphisms.
<cit.>.   The concept of forward rotationless outer automorphisms is defined in , where it is proved that the forward rotationless outer automorphisms are precisely those outer automorphisms which have  representatives. Here we use the stricter property of rotationless defined in , which is symmetric under inversion, and which is better adapted to the study of abelian subgroups. 

We say that ϕ∈(F_n) is rotationless if two conditions hold. First, for each Φ∈ P(ϕ) we have (Φ)=(Φ) (in “forward rotationless” this condition is replaced by the weaker _N(Φ)=_N(Φ)). Second, for each integer k ≥ 1 the map Φ↦Φ^k is a bijection between P^±(ϕ) and P^±(ϕ^k) — recall from <cit.> that injective is always true, and so bijective holds if and only if surjective holds. 




  
Expansion factor homomorphisms. <cit.>   Consider ϕ∈(F_n) and an attracting lamination Λ∈Ł(ϕ). Under the action of (F_n) on the set of lines (F_n), consider the subgroup (Λ) (F_n) that stabilizes Λ. The expansion factor homomorphism of Λ is the unique surjective homomorphism _Λ(Λ) → such that for each ψ∈(Λ) we have _Λ(ψ) ≥ 1 if and only if Λ∈Ł(ψ). Furthermore, there exists μ > 1 such that if ψ∈(Λ) and if _Λ(ψ) > 1 then any relative train track representative f  G → G of ψ has an -aperiodic stratum corresponding to Λ on which the Perron-Frobenius eigenvalue of the transition matrix equals μ^_Λ(ψ).





  
Twistors (aka Axes). 

<cit.>. Recall that two elements a,b ∈ F_n are said to be unoriented conjugate if a is conjugate to b or to b^. The ordinary conjugacy class of a is denoted [a] and the unoriented conjugacy class is denoted [a]_u. For any marked graph G, nontrivial conjugacy classes of F_n correspond bijectively with circuits S^1 ↦ G up to orientation preserving homeomorphisms of the domain S^1. Note that a is root free in F_n if and only if the oriented circuit S^1 ↦ G representing [a] does not factor through a nontrivial covering map of S^1, and a,b are unoriented conjugate if and only if the oriented circuits representing [a],[b] differ by an arbitrary homeomorphism of S^1.

Consider ϕ∈(F_n) and a nontrivial, unoriented, root free conjugacy class μ = [c]_u, c ∈ F_n. For any two representatives Φ_1 Φ_2 ∈(F_n) of ϕ, the following are equivalent: 

    Φ_1, Φ_2 ∈(c)     c = (Φ_1) (Φ_2)

Furthermore, if these equivalent conditions hold then Φ_2^Φ^_1 = i_c^d for some integer d  0. If there exists a pair Φ_1 Φ_2 ∈ P(ϕ) such that the above equivalent conditions hold, then μ is said to be a twistor for ϕ, and the number of distinct elements of the set P(ϕ) (c) is called the multiplicity of μ as a twistor for ϕ; these properties are independent of the choice of c ∈ F_n representing μ. 

The number of twistors of ϕ and the multiplicity of each twistor is finite, as follows. First, for any  f  G → G representing ϕ, for μ to be a twistor it is equivalent that some  linear edge E twists around μ, meaning that E has twist path w such that either w or w^ represents μ. Furthermore, the multiplicity of μ is one more than the number of linear edges that twist around μ <cit.>.




  §.§.§ Rotationless abelian subgroups




  
Principal sets for rotationless abelian subgroups.   <cit.>. 
An abelian subgroup A (F_n) is rotationless if each of its elements is rotationless, equivalently A has a rotationless generating set. Every abelian subgroup contains a rotationless subgroup of index bounded uniformly by a constant depending only on n, namely the subgroup of k^th powers where k ≥ 1 is an integer such that the k^th power of every element of (F_n) is rotationless <cit.>.

Assuming A (F_n) is rotationless, a subset ⊂∂ F_n with ≥ 3 elements is a principal set for A if each ψ∈ A is represented by Ψ∈(F_n) such that ⊂(Ψ) (which determines Ψ uniquely amongst representatives of Ψ) and such that Ψ∈ P^±(ψ). When the principal set  is fixed, the map ψ↦Ψ defines a homomorphism s  A →(F_n) that is a section of the canonical map (F_n) →(F_n) over the subgroup A. Also, the set

    = ∩_ψ∈ A(s(ψ))
 
is the unique principal set which is maximal up to inclusion subject to the property that  contains ; this set  defines the same section s  A →(F_n) as  <cit.>.



 








  
Comparison homomorphisms of rotationless abelian subgroups. <cit.>. 
 Consider a rotationless abelian subgroup A. 

Consider also two principal sets _1,_2 for A that define lifts s_1,s_2  A →(F_n). Let _1,_2 be the maximal principal sets containing _1,_2 respectively, and so s_1,s_2 are also the lifts defined by _1,_2. Suppose that s_1  s_2  A →(F_n) (this if and only if _1 _2), and suppose also that _1 ∩_2 ∅ and hence _1 ∩_2 ∅. It follows that the set Y_1 ∩ Y_2 is fixed by distinct automorphisms representing the same element of A, and so _1 _2 = T^±_c for some nontrivial root free c ∈ F_n. In this situation there is an associated comparison homomorphism ω A → which is characterized by the equation

    s_2(ψ) = i_c^ω(ψ) s_1(ψ)   for all ψ∈ A.

The number of distinct comparison homomorphisms A → is finite.



  
The coordinate homomorphism. <cit.>.   Every abelian subgroup A (F_n) is a finite lamination group, that is, the set Ł(A) = _ϕ∈ AŁ(ϕ) is finite. If in addition A is rotationless then each Λ∈Ł(A) is ψ-invariant for all ψ∈ A, and so 

    A ⋂_Λ∈Ł(A)(Λ)

We thus obtain a finite collection of expansion factor homomorphisms defined on A, namely 

    _Λ A →,   Λ∈Ł(A)

Choosing an enumeration Ω_1,…,Ω_N of the expansion homomorphisms and the comparison homomorphisms, we obtain the coordinate homomorphism 

    Ω A →^N

which is injective.





 §.§ Disintegration subgroups



In this section we fix a rotationless ϕ∈(F_n) and a  representative f  G → G with corresponding f-invariant filtration G_1 ⊂⋯⊂ G_u = G (that is all we need from Notations <ref> for this section).


Using the  structure of f (see Section <ref>) one builds up to the definition of the disintegration subgroup associated to f, a certain abelian subgroup (f) (F_n) constructed in <cit.> by an explicit combinatorial procedure that we review here, using which one also obtains a description of the coordinate homomorphism Ω(f) →^N (see Section <ref>). Noting that (f) depends on the choice of f, all definitions in this section depend on f.










  §.§.§ QE-paths and QE-splittings
 

We describe structures associated to the collection of linear edges of G, augmenting the structures described under the heading “Properties of -linear strata” in Section <ref>.

<cit.> 
  Two linear edges E_i,E_j ⊂ G with a common twist path w are said to be linearly equivalent, abbreviated to LIN-equivalent, and the associated LIN-equivalence class is denoted _w. Recall that distinct elements E_i  E_j ∈_w have distinct twist coefficients d_i  d_j. 

<cit.>   Given a twist path w and distinct linear edges E_i  E_j ∈_w, a path of the form E_i w^p E_j is called a quasi-exceptional path or QE-path (it is an exceptional path if and only if the twist coefficients d_i,d_j have the same sign). For every completely split path σ, any QE-subpath E_i w^p E_j of σ is a concatenation of terms of the complete splitting of σ: either d_i,d_j have the same sign and E_i w^p E_j is an exceptional path and hence a single term; or the terms consist of the initial E_i, the terms of the complete splitting of w^p, and the terminal E_j. No two QE-subpaths can overlap in an edge, and so there is a uniquely defined QE-splitting of σ which is obtained from the complete splitting by conglomerating each QE-subpath of σ into a single term. 

Consider a twist path w. For each E_i,E_j ∈_w, the associated quasi-exceptional family is the set of paths

    E_i w^* E_j = {E_i w^p E_j  p ∈}

Also, associated to w itself is its linear family, the set

    _w   ⋃_   E_i  E_j ∈_w E_i w^* E_j
 
Every quasi-exceptional path belongs to a unique quasi-exceptional family, and every  linear edge or quasi-exceptional path belongs to a unique linear family. 



  §.§.§ Almost invariant subgraphs
 

The intuition behind the “almost invariant subgraphs” of G is that they are a partition of the collection of non-fixed strata determined by the following requirement: if one perturbs f  G → G by letting its restrictions f  H_i to non-fixed strata be replaced by iterates f^a_i H_i with varying integer exponents a_i ≥ 0, and if one wishes to do this perturbation so that the resulting outer automorphisms commute with the outer automorphism represented by f, then the exponents a_i should be constant on the edges contained in each almost invariant subgraph.

<cit.>   Define an equivalence relation on the non-fixed irreducible strata {H_i} of G as follows. Given H_i, a path μ is a short path for H_i if μ an edge in H_i, or if H_i is  and μ is a taken connecting path in a zero stratum enveloped by H_i. Define a relation amongst the non-fixed irreducible strata denoted H_i  H_j, meaning that there exists a short path μ for H_i such that some term of the QE-splitting of f_#(μ) is an edge of H_j; note that if H_i  H_j ⋯ H_k then i ≥ j ≥⋯≥ k. Let B be the directed graph whose vertex set is the set of non-fixed irreducible strata, with a directed edge for each relation H_i  H_j. Let B_1,…, B_S denote the connected components of the graph B. For each s = 1,…,S define the almost invariant subgraph X_s ⊂ G to be the union of the strata H_i comprising the vertices of B_s, together with any zero stratum enveloped by one of these H_i. Note that the set of almost invariant subgraphs {X_1,…,X_S} partitions the set of the nonfixed strata. 









  §.§.§ Admissible S-tuples; quasi-exceptional families
 





<cit.>   For each S-tuple =(a_1,…, a_S) of non-negative integers define f^ : G → G on each edge E ⊂ G as follows:[In  the notation f_ was used for what we here denote f^.] if E ⊂ X_s for some s then f^(E) = f^a_s_#(E); otherwise E is fixed by f and then f^(E) = E. Each such f^ is a homotopy equivalence representing an outer automorphism denoted ϕ^. By construction f^ preserves the given filtration G_1 ⊂⋯⊂ G_u=G.

One would like to arrange (among other things) that (f^)^k_#(E) = f_#^a_s k(E) for all k ≥ 1, all s=1,…,S, and all edges E ⊂ X_s. This need not hold for general S-tuples, but it does hold when a certain relation amongst linear edges and QE-splitting terms is satisfied. 

Given an almost invariant subgraph X_r and linear edges E_i, E_j, we say that the triple (X_r,E_i,E_j) is quasi-twist related if there exists a stratum H_k ⊂ X_r, a short path μ for H_k, a twist path w, and an integer p ≥ 1 such that E_i, E_j ∈_w, and such that E_i w^p E_j is a term in the QE-splitting of f_#(μ).


We say that an S-tuple  is admissible if for all quasi-twist related triples (X_r,E_i,E_j), letting d_i,d_j be the twist coefficients of E_i,E_j respectively, and letting X_s,X_t be the almost invariant subgraphs containing E_i,E_j respectively, the following “twisting relation” holds:

    a_r(d_i - d_j) = a_s d_i - a_t d_j


<cit.>   Consider an almost invariant graph X_r. For each pair of linear edges E_i,E_j such that the triple (X_r,E_i,E_j) is quasi-twist related, the associated quasi-exceptional family is defined to be the set of all paths of the form E_i w^* E_j. We let _r denote the set of all quasi-exceptional families associated to quasi-twist related triples (X_r,E_i,E_j). 

QE-equivalence of linear edges. We say that a pair of linear edges E_i,E_j is QE-related if there exists an almost invariant subgraph X_r such that (X_r,E_i,E_j) is quasi-twist related. Equivalently E_i,E_j are QE-related if and only if they are in the same linear family and, letting w be the unique twist path for which E_i, E_j ∈_w, there exists r such that the family E_i w^* E_j is in the set _r. 

The equivalence relation on linear edges generated by being QE-related is called QE-equivalence and is written ∼_QE. Note that the QE-equivalence relation amongst linear edges is a refinement of the LIN-equivalence relation. 

Note that if an exceptional path E_i w^p E_j occurs as a term in the complete splitting of some iterate of some short path in X_r then the quasi-exceptional family E_i w^* E_j is an element of _r and the linear edges E_i and E_j are QE-related.



  §.§.§ X_s paths.
 

<cit.>   For each almost invariant subgraph X_s, we use the terminology X_s-paths to refer to the subpaths of G which form the elements of the set 𝒫_s that is defined in <cit.> — namely, the completely split paths γ such that each term of the QE-splitting of γ is one of the following: 


  * a Nielsen path; or

  * a short path in a stratum contained in X_s, for example any edge in H^z_i for any  stratum H_i ⊂ X_s; or

  * a quasi-exceptional path in the family _s. 




Furthermore, for any admissible S-tuple , any almost invariant subgraph X_s, and any X_s path γ, we have the following:




  * 
Each iterate f^k_#(γ) is also an X_s path;

  * 
f^_#(γ) = f^a_s_#(γ).







  §.§.§ The disintegration subgroup (f)


  


Here we recall the definition of the disintegration subgroup (f) (F_n). We also recall the “admissible semigroup” (f), 

which we will use as an aid to understanding properties of (f).




  
The subgroup (f). 
<cit.>   This is the subgroup of (F_n) generated by the set of elements 

    {ϕ^  is an admissible S-tuple}

The subgroup (f) is abelian and rotationless. 


The dependence of (f) on f rather than just ϕ was suppressed in <cit.> where the notation (ϕ) was used; see <cit.>.





  
The admissible semigroup (f). <cit.> Let (f) ⊂^S denote the set of admissible S-tuples, which forms a sub-semigroup of ^S. Let Ł(f) ^S be the subgroup generated by (f). The map (f) ↦(f) defined by ↦ϕ^ is an injective semigroup homomorphism, and it extends to an isomorphism Ł(f) ↦(f). Every element of Ł(f) can be written as the difference - of two elements ,∈(f), and so every ψ∈(f) can be written in the form ψ = (ϕ^)^ϕ^ for some ,∈(f).

We record here a simple consequence of these definitions for later use:


The function which assigns to each ∈(f) the map f^ G → G satisfies the following “homomorphic” property: 

    f^_# (f^_#(E)) = f^+_#(E)   for each ,∈(f) and each edge E ⊂ G




If E is a fixed edge of f then both sides equal E. Otherwise E is an edge in some almost invariant graph X_s, hence E is an X_s-path, hence f^_#(E) = (f^b_s)_#(E) is an X_s path, hence both sides equal (f^a_s+b_s)_#(E).





We repeat here the theorem cited in Section <ref>:

[<cit.>]
For every rotationless abelian subgroup (F_n) there exists ϕ∈ such that for every  f  G → G representing ϕ, the intersection (f) has finite index in .


Remark. In this statement of the Disintegration Theorem we have made explicit the fact that <cit.> holds for any choice of  representing ϕ. This was already implicit in the notation (ϕ) used for disintegration groups in <cit.> where any representative  is allowed.



  §.§.§ The coordinate homomorphism of a disintegration group (f)
 

The disintegration group (f), being rotationless and abelian, has an injective coordinate homomorphism Ω(f) ↦^N as defined at the end of Section <ref>. The individual coordinate functions of Ω are the comparison homomorphisms  and the expansion factor homomorphisms of the abelian group (f). 

We review here how one may use the structure of the  f to sift through the coordinate functions of Ω — keeping a subset of the comparison homomorphisms, and keeping a normalized version of each expansion factor homomorphism — to obtain a homomorphism denoted Ω^f (f) →^ which is still injective. We will not normalize comparison homomorphisms, because it would screw up the “twisting relation” described in Section <ref>.

<cit.>.   Let = {i  H_i is either  linear or }. The homomorphism Ω^f (f) →^ has coordinate functions denoted ω_i (f) → for each i ∈. For each i ∈, the function ω_i is characterized in one of two ways, depending on whether H_i  is  linear or . Let X_s be the almost invariant subgraph containing H_i. 


   linear coordinate homomorphism (a component of Ω and of Ω^f):   If H_i = E_i is  linear, with twist path w and twist coefficient d_i satisfying f(E_i) = E_i w^d_i, then for each admissible S-tuple  we set ω_i(ϕ^) = a_s d_i.

Consider also the difference homomorphisms ω_i,j(f) →, one for each LIN-equivalent pair of linear edges E_i,E_j, defined by:

    ω_i,j = ω_i - ω_j

As shown in <cit.> just following Theorem 7.2, each of these difference homomorphisms is a comparison homomorphism on (f) (Section <ref>), and furthermore the set of comparison homomorphisms on (f) is exactly the set of functions consisting of the  linear coordinate homomorphisms ω_i, their additive inverses -ω_i, and their differences ω_i,j = ω_i - ω_j for LIN-equivalent linear edges E_i,E_j.




  Unnormalized  coordinate homomorphism (a component of Ω):   If H_i is  with associated attracting lamination Λ_i ∈Ł(ϕ) then for each admissible S-tuple  we set ω_i(ϕ^) = _Λ_i (ϕ^). 

As alluded to in Section <ref>, this definition makes sense because (f) (Λ_i).



In both the  linear case and the  case, for each admissible S-tuple  the following equation holds  <cit.>:

    ω_i(ϕ^) = a_s ω_i(ϕ)


Noting that the constant sequence = (1,1,…,1) is admissible and satisfies the equation ϕ^ = ϕ, it follows that the subgroup ⊷(ω_i) is generated by the integer ω_i(ϕ). If H_i is  then we normalize the function ω_i by dividing it by the integer ω_i(ϕ); the preceding equation still holds. After this normalization we have:


  Normalized  coordinate homomorphism (a component of Ω^f):   If H_i is  then, with notation as above, we have

    ω_i(ϕ^) = a_s

and in particular ω_i(ϕ) = 1. It follows that if λ_i is the Perron-Frobenius eigenvalue of the transition matrix of f on H_i, then the expansion factor of ϕ^ on the lamination Λ_i is equal to λ_i^ω_i(ϕ^)  = λ_i^a_s.






















For later reference we summarize the properties of Ω^f (f) →^ as follows:

∙ The homomorphism Ω^f (f) →^ is injective.

∙  For each i ∈ and each admissible S-tuple  we have ω_i(ϕ^) = a_s ω_i(ϕ).

∙ The coordinate function ω_i of Ω^f associated to an  stratum H_i is the normalized version of _Λ_i satisfying ω_i(ϕ)=1.







§ A TRAIN TRACK SEMIGROUP ACTION



Throughout this section we continue to adopt Notations <ref> regarding a rotationless ϕ∈(F_n) with a   representative f  G → G whose top stratum H_u is  with associated attracting lamination Λ. 

Consider the disintegration group =(f), reviewed in Section <ref>. Recall that each element of  stabilizes Λ. We shall focus on the subsemigroup _+ consisting of all ψ∈ for which the value of top coordinate homomorphism ω(ψ) is non-negative. The value of ω(ψ) is a logarithm of the asymptotic stretch factor of ψ on Λ, hence for ω(ψ) to be non-negative means that either ψ does not stretch Λ or Λ is an attracting lamination of ψ.

Using the theory of disintegration groups, for each ψ∈_+ we construct a topological representative f^ψ G → G whose action on the edges of H_u agrees with the action of the appropriate iterate of the  f  G → G, namely f^ω(ψ)_#. Then, by lifting these topological representatives to the universal cover G, projecting to the tree T, and extending to the coned off graph T^*, we obtain semigroup actions of _+ on T and on T^*, detailed properties of which are described in Section <ref>. An important feature of the action _+  T is that it is not an isometric action, in fact the action of each Ψ∈_+ will stretch lengths in T by a uniform amount depending on the appropriate value of the coordinate homomorphism ω (see Section <ref> Item_f_PsiEdges). On the other hand, by restricting the semigroup actions _+  T,T^* to the subgroup _0 we obtain true isometric actions _0  T,T^*, some properties of which are studied in Section <ref>. 

The actions constructed in this section will be the basis for the construction in Section <ref> of an isometric group action of  on the hyperbolic suspension complex .



For use throughout Sections <ref> and Section <ref> we establish some further notations.






 

Given a rotationless ϕ∈(F_n) with  representative f  G → G having associated filtration G_1 ⊂⋯⊂ G_u with its top stratum H_u being , and with all associated objects and notations as described in Notations <ref>, we have the following additional objects and notations:


  * 
[Section <ref>, “Properties of zero strata”]  
H^z_u is the union of H_u with all zero strata enveloped by H_u.
For some t ≤ u-1 this set of zero strata has the form {H_i  t+1 ≤ i ≤ u-1} (which is empty if t=u-1), where 


  * H_t is the highest irreducible stratum below H_u, 

  * G_t is the union of noncontractible components of G_u-1, the contractible components being the zero strata H_i, t+1 ≤ i ≤ u-1. 


  * 
[Definitions <ref>, <ref>]   
F_n  T denotes the free splitting corresponding to the marked graph pair (G,G_u-1), with associated Nielsen set {N_j}_j ∈ J, each N_j having basepoint set Z_j ⊂ N_j. Also, F_n  T^* denotes the action obtained from F_n  T by coning off each basepoint set Z_j to a cone point P_j with edges P_n Q (Q ∈ Z_j).

  * 
[Definition <ref>]  
Associated to each automorphism Φ∈(F_n) representing ϕ are unique Φ-twisted equivariant maps as follows: 


  * f^Φ G → G, a lift of f; 

  *  f^Φ_T  T → T, induced by f^Φ with respect to the collapse map G ↦ T.

  * f_T^* T^* → T^*, an extension of f^Φ_T that permutes cone points and cone edges.

The maps f^Φ_T, f^Φ_T^* satisfy the “Stretch properties” recorded in Section <ref>. 

  * 
The disintegration group of f is denoted = (f). Its full pre-image in (F_n) is denoted  and is called the extended disintegration group. 

  * Associated to the top  stratum H_u we have the following objects: 


  * The almost invariant subgraph X_s ⊂ G containing H^z_u; 

  * 
 (Section <ref>) The coordinate homomorphism ω→ associated to X_s, a scaled copy of the expansion factor homomorphism _Λ that is normalized so that

    ω(ϕ)=1

In particular, ω is surjective.

  * The lifted coordinate homomorphism ω→, obtained by pre-composing ω with the projection map ↦.

  * The kernels of these homomorphisms denoted

    _0 = (ω)     _0 = (ω)


We thus have a commutative diagram with short exact rows:

    1 [r]    (ω) = _0 [r] 
          [r]^ω@=[d] 
          [r]     1 
    
    1 [r]     F_n ≈(F_n) [r] @=[d] [u]_⊂   [r] [d]^⊂   [r] [d]^⊂[u]_ω   1 
    
    1 [r]     F_n ≈(F_n) [r]    (F_n) [r]    (F_n) [r]     1


  * 
Letting [0,∞) = {0,1,2,…} we have subsemigroups D_+ and _+ and inclusions as follows:

    _0 = ω^(0)     _+ = ω^[0,∞)


    _0 = ω^(0)     _+ = ω^[0,∞)









 §.§ A “homotopy semigroup action” of _+ on G.
 


To prepare for the construction of the semigroup action _+  T, in this section we work downstairs in G and construct a “homotopy semigroup action” of _+ on G. What this means will be clear from the construction, but the intuitive idea is we associate a topological representative f^ψ G → G to each ψ∈_+ so that although the action equation f^ψ∘ f^ψ' = f^ψψ' does not hold exactly, it does hold “up to homotopy relative to G_t”. The values of f^ψ on edges of H^z_n are determined by appropriate iterates of f itself, and the values on G_t are determined, up to homotopy, by the “graph homotopy principle” of Section <ref>.



Letting ‖(G) denote the set of vertices of G, we define subsets

    P ⊂ Fix ⊂ V ⊂‖(G)

as follows. First, V = ‖(G)  H_u = ‖(G)  H^z_u, the latter equation holding because each edge E ⊂ H^z_u ∖ H_u has both endpoints in H_u <cit.>. Next, P = H^z_u  G_t = H_u  G_t; note that P ⊂‖(G) because H^z_u and G_t are subgraphs having no edges in common; also P is the frontier both of H^z_u and of G_t in G = H^z_u  G_t. Finally, Fix denotes the set of points in V fixed by f, and the inclusion P ⊂ Fix follows from <cit.>. 



Here is a summary of the construction of the “homotopy semigroup action”: 







  * 
For each ψ∈_+ we define a topological representative f^ψ G → G such that the following hold:


  * 
We have f^__+ = _G (where __+∈_+ denotes the identity group element, and _G  G → G the identity map).

  *  f^ψ(G_u-1) ⊂ G_u-1 and f^ψ(G_t) ⊂ G_t.

  * 
For each edge E ⊂ H^z_u we have f^ψ_(E) = f^ω(ψ)_#(E). In particular, if ψ∈_0 then f^ψ H^z_u is the identity.

  *  f^ψ V = f^ω(ψ) V. In particular f^ψ(V) ⊂ V and f^ψ fixes each point of Fix and of its subset P.


  *  If a height u indivisible Nielsen path ρ exists then f^ψ fixes each endpoint of ρ and f^ψ_#(ρ)=ρ.







  * 
For each ψ,ψ' ∈_+ we define a homotopy h^ψ,ψ' G × [0,1] → G between f^ψ f^ψ' and f^ψψ' such that the following hold:


  * 
For each edge E ⊂ H_u the homotopy h^ψ,ψ' straightens the edge path f^ψ f^ψ' E relative to its endpoints to form the path f^ψψ' E by cancelling only edges of G_u-1, without ever cancelling any edges of H_u.

  * 
h^ψ,ψ' is stationary on V. 

  * 
h^ψ,ψ' preserves G_t.


Recall that for a homotopy h  A × [0,1] → A to “preserve” a subset B ⊂ A means that h(B × [0,1]) ⊂ B, and for h to be “stationary” on B means that h(x × [0,1]) is constant for each x ∈ B.

First we will construct the maps f^ψ and then the homotopies h^ψ,ψ', along the way proving the various requirements of ItemOnHzN and ItemSemiUnique.

Constructing f^ψ. We being with the construction of f^ψ_t = f^ψ G_t  G_t → G_t by applying the graph homotopy principle Lemma <ref>.

Recall from Section <ref> the abelian semigroup of admissible S-tuples (f). Recall also the injection (f) ≈Ł(f) defined by ↦ϕ^ with topological representative f^ G → G; this injection is a semigroup homomorphism whose image Ł(f) generates . 

Since the restricted map f^_t = f^ G_t  G_t → G_t is a homotopy equivalence that fixes each point of P, by Lemma <ref> ItemGPHomInvExists the map of pairs f^_t  (G_t,P) → (G_t,P) is a homotopy equivalence in the category of pairs, and hence its homotopy class rel P is an element [f^_t] of ^_0(G_t,P), the pure automorphism group of (G_t,P) in the graph point category (Lemma <ref> ItemGPAut). By Fact <ref>, for each ,∈(f) the maps f^_t ∘ f^_t and f^+_t  (G_t,P) → (G_t,P) are homotopic rel P, and so the map Ł(f) ↦^_0(G_t,P) defined by (ϕ^) = [f^_t] is a semigroup homomorphism. Since the commutative group  is generated by its subsemigroup Ł(f), a simple semigroup argument shows that  extends uniquely to a group homomorphism →^_0(G_t,P). For each ψ∈ choose f^ψ_t  (G_t,P) → (G_t,P) to be a representative of (ψ); if ψ = ϕ^ is already in Ł(f) then we choose f^ψ_t=f^_t, and so if ψ is the identity then f^ψ_t is the identity. Notice that no straightening is carried out when f^ψ_t is applied, and so there is no need for the “#” operator in the definition of f^ψ_t.

For later use, for each ∈(f) we denote f̅^_t = f^(ϕ^)^_t  (G_t,P) → (G_t,P), which is the homotopy inverse of f^_t that represents ((ϕ^)^) = (ϕ^)^∈^_0(G_t,P).

For each ψ∈_+ we may now define f^ψ G → G as follows: 

    f^ψ(E)    = f^ω(ψ)_#(E)   for each E ⊂ H^z_u, and
    
    f^ψ G_t     = f^ψ_t

If ψ∈ is the identity then clearly f^ψ G → G is the identity, verifying ItemOnIdentity. By construction f^ψ satisfies ItemOnGNMinusOne and ItemOnE. For item ItemOnFrontier, note first that for each x ∈ V there exists an oriented edge E ⊂ H_u with initial vertex x, and for each such E the initial vertex of the path f^ψ(E) = f^ω(ψ)_#(E) equals f^ω(ψ)(x); and if in addition x ∈ P then both of f^ω(ψ) and f^ψ_t fix x. This shows that f^ψ is well-defined on V, and that it restricts to f^ω(ψ) on V, completing the proof of item ItemOnFrontier; it also follows that f^ψ is continuous. 

The proof that f^ψ is a homotopy equivalence and is a topological representative of ψ will be delayed to the very end.

X_s-paths under f^ψ. Item ItemOnNielsen is encompassed in the following generalization of ItemOnE, which will also be used for item ItemSemiUnique:


  *  For each ψ∈_+ and each X_s-path γ with endpoints in V we have f^ψ_#(γ) = f^ω(ψ)_#(γ).

To see why this holds, the general X_s-path γ with endpoints in V is a concatenation of three special types, and it suffices to check Item_f_psiOnXsPath only for these types: 


  Type (a): edges in H^z_u; 

  Type (b): X_s-paths in G_t having endpoints in the set P. 

  Type (c): a height u indivisible Nielsen path of f;

Type (a) is the special case handled already in item ItemOnE. 

If γ is of type (b), first note that for ∈(f) and ψ = ϕ^∈Ł(f) we have 

    f^ψ_#(γ) = (f^ψ_t)_#(γ)=(f^_t)_#(γ)=f^_#(γ)=f^a_s_#(γ) = f^ω(ψ)_#(γ)

where the second-to-last equation follows from Section <ref> ItemXsPathImages. For more general ψ∈_+, choose ,∈(f) so that ψ = (ϕ^b)^ϕ^, and note that a_s - b_s = ω(ϕ^) - ω(ϕ^) = ω(ψ) ≥ 0. In the group ^(G_t,P) we have the equation [f^ψ_t] = [f̅^_t]   [f^_t] and so we may calculate





    f^ψ_#(γ)  = (f^ψ_t)_#(γ)     =  (f̅^_t  f^_t)_#(γ)  =  (f̅^_t)_#((f^_t)_#(γ)) 
        =  (f̅^_t)_#(f^_#(γ))  =  (f̅^_t)_#(f^a_s_#(γ)) 
        =  (f̅^_t)_#(f^b_s_#(f^a_s-b_s_#(γ)))  =  (f̅^_t)_#(f^_#(f^ω(ψ)_#(γ))) 
        =  (f̅^_t)_#((f^_t)_#(f^ω(ψ)_#(γ)))  =  (f̅^_t  f^_t)_#(f^ω(ψ)_#(γ)) 
        =  f^ω(ψ)_#(γ)

where the second equations of the second and third lines follow from Section <ref> ItemXsPathIterates and ItemXsPathImages.

For type (c), let γ be a height u indivisible Nielsen path of f. We may write γ as an alternating concatenation of nontrivial paths of the form

    γ = η_0  μ_1  η_1  ⋯ η_K-1 μ_K  η_K

where each η_k is a path in H^z_u and each μ_k is a Nielsen path of f in G_t with endpoints in P <cit.>. By definition of f^ψ we have

    f^ψ_#(γ) = [f^ω(ψ)_#(E_1)    (f^ψ_t)_#(μ_1)   f^ω(ψ)_#(η_1)   ⋯   f^ω(ψ)_#(η_K-1)    (f^ψ_t)_#(μ_K)    f^ω(ψ)_#(η_K)]

We claim that each μ_k is a Nielsen path of f^ψ_t for each ψ∈ D, and so (f^ψ_t)_#(μ_k)=μ_k. To prove this claim, it holds if ψ = ϕ^∈Ł(f) for some ∈(f), because in that case the left hand side equals (f^)_#(μ_k) = μ_k. Using that [f̅^ψ_t] = [f^ψ_t]^∈^(G,P), we have μ_k = (f̅^ψ_t)_#(f^ψ_t(μ_k)) = (f̅^ψ_t)_#(μ_k), and so the claim holds if ψ = (ϕ^)^ for some ∈(f). The general case holds because ψ = (ϕ^)^ϕ^ for some ,∈(f). Applying the claim we have:

    f^ψ_#(γ)    = [f^ω(ψ)_#(E_1)   μ_1   f^ω(ψ)_#(η_1)   ⋯   f^ω(ψ)_#(η_K-1)   μ_K    f^ω(ψ)_#(η_K)] 
       = [f^ω(ψ)_#(E_1)    f^ω(ψ)_#(μ_1)   f^ω(ψ)_#(η_1)   ⋯   f^ω(ψ)_#(η_K-1)    f^ω(ψ)_#(μ_K)    f^ω(ψ)_#(η_K)] 
       = f^ω(ψ)_#(γ)

This complete the proof of Item_f_psiOnXsPath. 

This also completes the construction of f^ψ G → G and the proof of ItemOnHzN for each ψ∈_+, except that we will delay further the proof that f^ψ is a homotopy equivalence and a topological representative of ψ.


Constructing h^ψ,ψ'. Given ψ,ψ' ∈_+ we turn to the construction of the homotopy h^ψ,ψ' G × [0,1] → G from f^ψ∘ f^ψ' to f^ψψ'. Let θ = ψψ' ∈_+. First, using the homomorphism →^_0(G,P) we have (ψψ')=(ψ)(ψ') which translates to [f^ψψ'_t] = [f^ψ_t]   [f^ψ'_t] and so there exists a homotopy rel P from f^ψ_t ∘ f^ψ'_t to f^ψψ'_t denoted h^ψ,ψ'_t  G_t × [0,1] → G_t. Second, for each E ⊂ H^z_u we have 

    f^ψ_# (f^ψ'_#(E)) = f^ψ_#(f^ω(ψ')_#(E)) = f^ω(ψ)_#(f^ω(ψ')_#(E)) = f^ω(ψψ')_#(E) = f^ψψ'_#(E)

where the second equation holds by applying Item_f_psiOnXsPath together with the fact that f^ω(ψ')_#(E) is an X_s-path (Section <ref> ItemXsPathIterates). Putting these together, we may define h^ψ,ψ' so that for each edge E ⊂ H^z_u its restriction to E × [0,1] is a homotopy rel endpoints from f^ψ∘ f^ψ' E to f^ψψ' E, and its restriction to G_t ×[0,1] is equal to h^ψ,ψ'_t. Items ItemVNStationary and ItemHomotopyPreservesGNMinusOne are evident from the construction. Item ItemPsiPsiPrimeEEquation follows the definition of a relative train track map, which tells us that for each edge E ⊂ H_u and each integer i ≥ 0 the path f^i_#(E) is u-legal, and that for each u-legal path γ the homotopy that straightenes f^i(γ) to produce f^i_#(γ) cancels no edges of H_u. 

Topological representatives. For each ψ∈_+, since Ł(f) generates  we may choose ,∈(f) so that ψ = (ϕ^)^ϕ^, and hence ϕ^ψ = ϕ^. Since all three of ϕ^, ψ, ϕ^ are in _+, we have a homotopy h^ϕ^,ψ from f^∘ f^ψ to f^. Since f^,f^ are homotopy equivalences it follows that f^ψ is also, and since f^,f^ are topological representatives of ϕ^,ϕ^ respectively it follows that f^ψ is a topological representative of (ϕ^)^ϕ^=ψ.



 §.§ Semigroup actions of _+ on T and T^*


We turn now to the construction of the action _+  T, deriving various properties of the construction, and then we extend the action to _+  T^*. 

Associated to each Ψ∈_+ we define a map f^Ψ T → T by carrying out the following steps. First let ψ∈_+ be the image of Ψ under the homomorphism (F_n) ↦(F_n), and consider the map f^ψ (G,G_u-1) → (G,G_u-1), part of the “homotopy semigroup action” constructed in Section <ref>. Let f^Ψ ( G, G_u-1) → ( G, G_u-1) be the unique Ψ-twisted equivariant lift of f^ψ to G. Let f^Ψ T → T be the Ψ-twisted equivariant map induced from f^Ψ by collapsing each component of G_u-1 to a point and then straightening on each edge E ⊂ T so that f^Ψ E stretches length by a constant factor. 

We record several properties of this semigroup action:




  * Twisted equivariance:

The map f^Ψ is Ψ-twisted equivariant. 
  * Semigroup action property:

f^Ψ f^Ψ' = f^ΨΨ' for all Ψ,Ψ' ∈_+. 
 


Property Item_f_PsiComp follows from the fact that f^Ψ(f^Ψ'(E)) = f^ΨΨ'(E) for each edge E ⊂ T, which is a consequence of Section <ref> item ItemPsiPsiPrimeEEquation  applied to edges of H_u. 



  * Vertex Action Property:


f^Ψ takes vertices to vertices, and it restricts to a bijection of the set of vertices having nontrivial stabilizer.



For the proof, denote vertex sets by V(G) ⊂ G and V(T) ⊂ T, and let V^(T) denote the subset of all v ∈ V(T) such that _F_n(v) is nontrivial. By Section <ref> items ItemOnFrontier and ItemOnGNMinusOne, the map f^ψ takes the set G_u-1 V(G) to itself, and since f^ψ is a topological representative it follows that f^ψ restricts further to a homotopy equivalence amongst the noncontractible components of G_u-1 V(G). Letting G_u-1⊂ G and V( G) ⊂ G be the total lifts of G_u-1 and V(G), it follows that f^Ψ G → G restricts to a self-map of the components of G_u-1 V( G), and it restricts further to a bijection amongst the components having nontrivial stabilizer. Under the quotient map G → T, the set of components of G_u-1 V( G)  corresponds bijectively and F_n-equivariantly to the vertex set V(T). It follows that f^Ψ T → T restricts to a self map of V(T), and that it restricts further to a bijection of V^(T), proving property Item_f_PsiVertices. 



  * Stretch Property:

f^Ψ maps each edge E ⊂ T to an edge path f^Ψ(E) ⊂ T, stretching length by a uniform factor of λ^ω(Ψ). 



This follows from the definition of the piecewise Riemannian metric on T in Section <ref>, the eigenlength equation in T in Section <ref>, and Section <ref> item ItemOnE. 



  * Train Track Property:

For each Ψ,Ψ' ∈ and each edge E of T, the restriction of f^Ψ to the edge path f^Ψ'(E) is injective. 



This follows from property Item_f_PsiComp together with property Item_f_PsiEdges as applied to ΨΨ'. 



For the statement of property Item_f_KernelActs, recall from Section <ref> the subgroup [T] (F_n) and its pre-image [T] (F_n). Recall particularly Lemma <ref> of that section, which says that each subgroup of [T] containing (F_n) has a unique action on T extending the given action of the free group F_n ≈(F_n) such that each element of the action satisfies a twisted equivariance property.




  * Restricted action of _0:

We have _0 [T] and hence _0 [T]. Furthermore, the restriction to _0 of the semi-action _+  T is identical to the action _0  T given by Lemma <ref>: the unique isometric action of _0 on T such that each Ψ∈_0 satisfies Ψ-twisted equivariance.



For the proof, consider Ψ∈_0 with projected image ψ∈_0. The map f^ψ G → G restricts to the identity on each edge E ⊂ H_u, by Section <ref> item ItemOnE. The map f^Ψ G → G therefore permutes the edges of H_u, mapping each isometrically to its image. It follows that the map f^Ψ T → T is an isometry. Since f^Ψ satisfies Ψ-twisted equivariance (property Item_f_Twisted), it follows that ψ∈[T] (Lemma <ref>). Since ψ∈_0 is arbitrary, we have proved _0 [T] and that _0 [T]. Applying, Lemma <ref> we have also proved that the map Ψ↦ f^Ψ is the same as the restriction to _0 of the action [T]  T given in that lemma, namely the unique action assigning to Ψ the unique Ψ-twisted equivariant isometry of T. 



  * Invariance of Nielsen data:
 
For each Ψ∈ and each j ∈ J there exists j' ∈ J such that N_j' = (f^Ψ)_#(N_j), and f^Ψ restricts to an order preserving bijection of basepoint sets Z_j  Z_j'. In particular f^Ψ_# induces a bijection of the Nielsen paths in T (Definition <ref>).


This follows immediately from Section <ref> item ItemOnNielsen. 



Finally, as an immediate consequence of property Item_f_PsiNielsen we have:


  * 
Extension to _+  T^*: The semigroup action _+  T extends uniquely to a semigroup action _+  T^*, denoted f^Ψ_* = f^Ψ_T^* T^* → T^* for each Ψ∈_+, such that f^Ψ_* permutes the cone points P_j, and f^Ψ_* permutes the cone edges P_j Q by isometries. In particular, for each j ∈ J and Q ∈ Z_j, letting N_j'=f^Ψ_#(N_j) and Q' = f^Ψ(Q), we have f^Ψ_*(P_j)=P_j'  and  Ψ( P_jQ) = P'_jQ'. 


























































 §.§ Dynamics of the group action _0  T^*



Isometries of a simplicial tree equipped with a geodesic metric satisfy a dichotomy: each isometry is either elliptic or loxodromic. This dichotomy does not hold for all isometries of Gromov hyperbolic spaces — in general, there is a third category of isometries, namely the parabolics. In this section we prove Lemma <ref> which says in part that the dichotomy does hold for the action _0  T^*. 



Under the action _0  T^*, each element Δ∈_0 acts either loxodromically or elliptically on T^*. More precisely, _0 acts loxodromically on T^* if and only if _0 acts loxodromically on T and its axis A ⊂ T is not a Nielsen line (Definition <ref>), in which case A is a quasi-axis for Δ in T^*. Furthermore, the loxodromic elements of _0 have “uniform uniformly positive stable translation length” in the following sense: there exist constants η,κ>0 such that for each Δ∈_0 acting loxodromically on T^*, and for each x ∈ T^* and each integer k ≥ 0, we have

    d_*(Δ^k(x),x) ≥ k  η - κ





    Remarks.

In this lemma, the terminology “uniform uniformly positive stable translation length” refers to the corollary that the stable translation length 

    lim_k →∞1/k d_*(Δ^k(x),x))

has a uniform positive lower bound η>0 independent of the choice of a loxodromic element Δ∈_0, and the rate at which this positive lower bound is approached is uniform. This property will be applied in Section <ref>, for purposes of characterizing the loxodromic elements of the action  that is described in Section <ref>.



Lemma <ref> and proof could already have been presented almost word-for-word back in Section <ref> for the restricted action (F_n) ≈ F_n  T^*. Other than the methods available in Section <ref>, the additional things needed to prove the lemma are that the larger action _0  T^*, like its restriction, satisfies twisted equivariance and preserves Nielsen paths and associated objects (Section <ref> Item_f_PsiNielsen, Item_f_StarExtension).

 Throughout the proof we will apply Section <ref> Item_f_PsiNielsen, Item_f_StarExtension regarding the action of _0 on Nielsen paths, elements of the Nielsen set {N_i}, the basepoint sets Z_i ⊂ N_i, the cone points P_i, and the cone edges P_i Q, Q ∈ Z_i. In particular, because the action of each Δ∈_0 on T preserves Nielsen paths, it takes maximal ρ^*-subpaths of any path α to maximal ρ^*-subpaths of the path Δ(α).

Consider Δ∈_0. If Δ acts elliptically on T then it fixes a vertex of T; and if in the geometric case Δ acts loxodromically on T and its axis is a Nielsen line N_j, then Δ fixes the corresponding cone point P_j; in either case Δ acts elliptically on T^*.

Suppose that Δ acts loxodromically on T and its axis A ⊂ T is not a Nielsen line (this always happens in the nongeometric case, but the terminology of our proof applies to all cases). For each vertex x ∈ T and each integer k ≥ 1 consider the Proposition <ref> decomposition of the path [x,Δ^k(x)] into “ν-subpaths” meaning maximal ρ^* subpaths, alternating with “μ-subpaths” each having no ρ^* subpath. Since the intersection [x,Δ^k(x)]  A contains a concatenation of k fundamental domains for the action of Δ on A, it follows that if [x,Δ^k(x)]  A contains a ν-subpath of [x,Δ^k(x)] then it must contain k-1 distinct ν-subpaths of [x,Δ^k(x)], between which there are k-2 distinct μ-subpaths of [x,Δ^k(x)]; here we are applying the fact that Δ takes maximal ρ^* paths to maximal ρ^* paths. As a consequence, the collection of μ-subpaths of [x,Δ^k(x)] contains at least k-2 distinct edges of T. Applying Proposition <ref> ItemDPFFormula we obtain D_(x,Δ^k(x)) ≥ (k-2) η' where η' = min{l_(E) E is an edge of T}. Applying Proposition <ref>, in T^* we have

    d_*(x,Δ^k(x)) ≥ k η'/K_=  η - (2η'/K + C_=  κ')

It immediately follows that Δ acts loxodromically on T^*. This estimate is for vertices of T, but a similar estimate holds for arbitrary points x ∈ T^*, replacing κ' by κ = κ' + 2 δ where δ is an upper bound for the distance in T^* from an arbitrary point of T^* to the vertex set of T.





§ THE SUSPENSION ACTION.
 


In this section we complete the proof of the multi-edge case of the Hyperbolic Action Theorem. 

Throughout this section we adopt Notations <ref> and <ref> concerning a rotationless ϕ∈(F_n) and a  representative f  G → G with top  stratum H_u, with disintegration group =(f) (F_n), and with free splitting F_n  T associated to the pair (G,G_u). In particular, in Section <ref> we used the coned off free splitting F_n  T^* to construct the suspension space , proving that  is Gromov hyperbolic by using flaring properties of the action F_n  T^*. In Section <ref>  we studied the extended disintegration group (F_n) and its subsemigroup _+, and we constructed the train track semigroup action _+  T^*.

In Section <ref> we shall show how to suspend the semigroup action _+  T^* to an isometric action  (this is a completely general construction that applies to the disintegration group of any  having top  stratum).

In Section <ref> we put the pieces together to complete the proof. 






 §.§ The suspension action .
 


Recall from Notations <ref> the short exact sequence

    1 ↦_0 ↦→ 1

Choose an automorphism Φ∈ representing ϕ∈, and so ω_u(Φ)=ω(ϕ)=1. It follows that Φ determines a splitting of the above short exact sequence, and hence a semidirect product structure ≈_0 _Φ expressed in terms of the inner automorphism on  determined by Φ, namely I(Ψ) = I_Φ(Ψ) = ΦΨΦ^.  






Thus each element of  may be written in the form ΔΦ^m for unique Δ∈_0 and m ∈, and the group operation in this notation is defined by

    (Δ_1 Φ^m_1)(Δ_2 Φ^m_2) = (Δ_1 I^m_1(Δ_2)) Φ^m_1+m_2







Recall from Definition <ref> the construction of : using the chosen automorphism Φ representing ϕ, the corresponding Φ-twisted equivariant map of T^* is denoted in abbreviated form as f_* = f^Φ_T^*, and  is the suspension space of f_*, namely the quotient of T ×× [0,1] with identifications [x,k,1] ∼ [f_*(x),k+1,0]. Recall also various other notations associated to  that are introduced in Definitions <ref> and <ref>.

To define the action on  of each element Ψ = ΔΦ^k ∈ it suffices to carry out the following steps: first we define the group action _0; next we define the action of the element Φ; then we define the action of each element ΔΦ^k by composition; and finally we verify that the two sides of the semidirect product relation ΦΔ = I(Δ) Φ act identically. 

Let each Δ∈_0 act on [x,k,t] ∈ by the equation

    Δ· [x,k,t] = [f^I^k(Δ)_*(x), k, t] = [f^Φ^k ΔΦ^-k_*(x),k,t]

and note that this formula is well defined because, using the properties of the semigroup action _+  T from Section <ref>, we have

    Δ· [x,k,1]    = [f^Φ^k ΔΦ^-k_*(x),k,1] 
     = [f^Φ_* ∘ f^Φ^k ΔΦ^-k_*(x) ,k+1,0] 
       = [f^Φ^k+1ΔΦ^-(k+1)_* ∘ f^Φ_*(x),k+1,0] 
       = Δ· [f^Φ_*(x),k+1,0] 
       = Δ· [f_*(x),k+1,0]

Each Δ∈_0 evidently acts by an isometry. Again using the semigroup action properties, the action equation is satisfied for each Δ',Δ∈_0:

    Δ' · (Δ· [x,k,t])    = [f^I^k(Δ')_*∘ f^I^k(Δ)_*(x),k,t] 
       = [f^I^k(Δ') ∘ I^k(Δ)_*(x),k,t] 
       = [f^I^k(Δ'Δ)_*(x),k,t] 
       = Δ' Δ· [x,k,t]

This completes the definition of the isometric action _0.
















We note that the restriction to F_n ≈(F_n) of the action _0 agrees with the action F_n given in Definition <ref>, defined in general by γ· [x,k,r] = [Φ^k(γ) · x, k, r] for each γ∈ F_n. In the special case k=0, r=0, the equation γ· x = f^i_γ_*(x) holds by Lemma <ref> ItemLTUInner. The extension to the general case follows easily.

Next, let Φ act by shifting downward,

    Φ· [x,k,t] = [x,k-1,t]

which is evidently a well-defined isometry. 





As required, for each Δ∈_0 the two sides of the semidirect product equation act in the same way:

    Φ·Δ· [x,k,t]    = Φ· [f^I^k(Δ)_*(x),k,t] 
       = [f^I^k(Δ)_*(x),k-1,t] 
    
    I(Δ) ·Φ· [x,k,t]    = I(Δ) · [x,k-1,t] 
       = [f^I^k-1(I(Δ))_*(x),k-1,t] 
       = [f^I^k(Δ)_*(x),k-1,t]

Notice that since the action of each Δ∈_0 preserves each “horizontal level set” _s, and since the action of Φ has “vertical translation” equal to -1=-ω(Φ) meaning that Φ(_s) = _s-ω(Φ), it follows that for each Ψ∈ the integer -ω(Ψ) is the “vertical translation length” for the action of Ψ in . We record this for later use as:


For each Ψ∈ and each fiber _s (s ∈), we have 

    Ψ(_s) = _s - ω(Ψ)  for any s ∈






 §.§ Hyperbolic Action Theorem, Multi-edge case: Proof.



First we set up the notation, based on the formulation of the multi-edge case in Section <ref> and the outline of the proof in Section <ref>, and we apply the results of Section <ref> to obtain a hyperbolic action . After that, Conclusions ItemThmF_EllOrLox, ItemThmF_Nonelem and ItemThmF_WWPD of the Hyperbolic Action Theorem are proved in order. 



  
Setup of the proof.   We have a subgroup (F_n), having image (F_n) under the quotient homomorphism (F_n) ↦(F_n), and having kernel J = (↦), which satisfies the hypotheses of the Hyperbolic Action Theorem:   is abelian;   is finitely generated and not virtually abelian;  and no proper, nontrivial free factor of F_n is -invariant. 

The conclusion of the Hyperbolic Action Theorem is about the existence of a certain kind of hyperbolic action of a finite index normal subgroup of , and so we are free to replace  with any finite index subgroup ', because once the conclusion is proved using some hyperbolic action ' of some finite index normal subgroup ' ', the same conclusion follows for the restriction of ' to the action  where  is the intersection of all of its  conjugates of '; one need only observe that the conclusions of the Hyperbolic Action Theorem for the action '  S imply the same conclusions when restricted to any finite index subgroup of '.

We may therefore assume that  is a rotationless abelian subgroup, using the existence of an integer constant k such that the k^th power of each element of (F_n) is rotationless, and replace the abelian group  by its finite index subgroup of k^th powers.

We have a maximal, proper, -invariant free factor system  of co-edge number ≥ 2 in F_n. Applying the Disintegration Theorem, we obtain ϕ∈ such that for any  representative f  G → G of ϕ with disintegration group =(f), the subgroup  has finite index in . We choose f so that the penultimate filtration element G_u-1 represents the free factor system . Since the co-edge number of  is ≥ 2, the top stratum H_u is . Replacing  with , we may thus assume . 

We now adopt Notations <ref> and <ref>, regarding the free splitting F_n  T associated to the marked graph pair (G,G_u-1), the action F_n  T^* obtained by coning off the elements of the Nielsen set, the disintegration group =(f), and its associated extended disintegration group . Using that  it follows that . Setting J = (↦) and K = () we may augment the commutative diagram of Notations <ref> ItemDisintGroup to obtain the commutative diagram with short exact rows shown in Figure <ref>.


From Notations <ref> ItemSubsemigroups we also have the subgroups and subsemigroups _0  _+ and _0 _+. 
Let _+  T^* be the semigroup action described in Section <ref>, associating to each Ψ∈_+ a map f^Ψ_T^* T^* → T^*. Pick Φ∈ to be any pre-image of ϕ∈, and so ω_u(Φ) = ω_u(ϕ)=1 (see Notations <ref> ItemCoordHomDf). Let  be the suspension space of f^Φ_T^* T^* → T^* as constructed in Definition <ref>. Let  be the isometric action constructed in Section <ref>. We will make heavy use of the integer sections _j (j ∈) and of the _0-equivariant identification _0 ↔ T^*.



We shall abuse notation freely by identifying F_n ≈(F_n) given by γ↔ i_γ, where i_γ(δ)=γδγ^. For example, using this identification we often think of J  K as subgroups of F_n. We also note the equation

    i_Ψ(γ) = Ψ∘ i_γ∘Ψ^,   Ψ∈(F_n), γ∈ F_n

which expresses the fact that the isomorphism F_n ≈(F_n) is equivariant with respect to two actions of (F_n): its standard action on F_n; and its action by inner automorphisms on its normal subgroup (F_n).  Combining this equation with Item_f_Twisted Twisted equivariance of Section <ref>, it follows that under the isomorphism F_n ≈(F_n), the action F_n  T agrees with the action (F_n)  T obtained by restricting the action _0  T. 



We turn to the proof of the three conclusions of the Hyperbolic Action Theorem for the action of = on .



  
Proof of Conclusion ItemThmF_EllOrLox: Every element of  acts either elliptically or loxodromically on . We show more generally that every element of  acts either elliptically or loxodromically on . From Section <ref> the general element of  has the form Ψ = ΔΦ^m for some Δ∈_0, and

Ψ(_j) = _j-m for any j ∈. If m  0 then, by Lemma <ref>, for each k ∈ and each x ∈ we have d(x,Ψ^k(x)) ≥km, and so Ψ is loxodromic. Suppose then that m=0 and so Ψ=Δ preserves _j for each j ∈. Consider in particular the restriction Δ_0 ≈ T^* and the further restriction Δ T. If Δ is elliptic on T then it fixes a point of T, hence Δ fixes a point of T^* ≈_0 ⊂, and so Δ is elliptic on T^* and on . If Δ is loxodromic on T with axis L_Δ⊂ T, and if L_Δ is a Nielsen line N_j then Δ fixes the corresponding cone point P_j ∈ T^* ∈_0 ⊂, and so Δ is elliptic on T^* and on . 

It remains to consider those Δ∈_0 which act loxodromically on T and whose axes in T are not Nielsen lines. Applying Lemma <ref>, each such Δ acts loxodromically on T^* and for each x ∈ T^* and each integer k ≥ 1 we have

    (*)          d_*(x,Δ^k(x)) ≥ k  η - κ

where the constants η,κ > 0 are independent of Δ and x.

Consider the function which assigns to each integer k the minimum translation distance of vertices v ∈ under the action of Δ^k:

    σ(k) = inf_v ∈ d_(v,Δ^k(v))

To prove that Δ acts loxodromically on  we apply the classification of isometries of Gromov hyperbolic spaces <cit.>, which says that every isometry is elliptic, parabolic, or loxodromic. But if Δ is elliptic or parabolic then the function σ(k) is bounded. Thus it suffices to prove that lim_k →∞σ(k) = ∞. 

For each integer i, consider Δ'_i = Φ^i ΔΦ^-i∈_0. Consider also the Φ^-i-twisted equivariant map j_i  T^* = _0 → S_i given by j_i[x,0,0] = [x,i,0], which is an isometry from the metric d_*=d_0 to the metric d_i. This map j_i conjugates the action of Δ^k on _i to the action of (Δ'_i)^k = Φ^i Δ^k Φ^-i∈_0 on _0, because

    Δ^k j_i[x,0,0]    = Δ^k[x,i,0] = [f_*^Φ^i Δ^k Φ^-i(x),i,0] 
       = [f_*^Δ'_i(x),i,0]  = j_i [f_*^Δ'_i(x),0,0] 
       = j_i Δ'_i [x,0,0]

Applying the inequality (*) to Δ'_i, and then applying the twisted equivariant isometric conjugacy j_i between Δ^k and (Δ'_i)^k, for each vertex p ∈_i we have 

    (**)          d_i(p,Δ^k(p)) ≥ k  η - κ

As seen earlier, the uniformly proper maps _i have a uniform gauge function δ [0,∞) → [0,∞) (see (*) in Section <ref> just before Lemma <ref>). If σ(k) does not diverge to ∞ then there is a constant M and arbitrarily large values of k such that d_(p,Δ^k(p)) ≤ M holds for some i and some vertex p ∈_i, implying by that d_i(p,Δ^k(p)) ≤δ(M), contradicting (**) for sufficiently large k. 

This completes the proof of Conclusion ItemThmF_EllOrLox. 



Furthermore, we have proved the following which will be useful later:


  * For each Δ∈_0 the following are equivalent: Δ acts loxodromically on ; Δ acts loxodromically on T^*;   Δ acts loxodromically on T and its axis is not a Nielsen line.




  
Proof of Conclusion ItemThmF_Nonelem: The action  is nonelementary. We shall focus on the restricted actions of the subgroup J (often abusing notation, as warned earlier, by identifying J (F_n) with the corresponding subgroup of F_n). We prove first that J has nonelementary action on T, then on T^*, and then on ; from the latter it follows that the whole action  is nonelementary.

We shall apply Lemma <ref> and so we must check the hypotheses of that lemma. Let V^ be the set of vertices v ∈ T having nontrivial stabilizer under the action of F_n. As shown in Section <ref> Item_f_PsiVertices, the semigroup action _+  T restricts to a semigroup action _+  V^ having the property that each element of Ψ∈_+ acts by a Ψ-twisted equivariant bijection of V^, and it follows immediately that this semigroup action extends uniquely to a group action V^ having the same property. Restricting to  we obtain an action V^ satisfying the hypotheses in the first paragraph of Lemma <ref>. By hypothesis of the Hyperbolic Action Theorem, no proper nontrivial free factor of F_n is fixed by , and in particular no subgroup _F_n(v) (v ∈ V^) is fixed by . Finally, the free group J has rank ≥ 2 because otherwise, since  is abelian, it would follow that  is a solvable subgroup of (F_n) (F_n+1) and hence is virtually abelian by , contradicting the hypothesis that  is not virtually abelian. Having verified all the hypotheses of Lemma <ref>, from its conclusion we have that the action J  T is nonelementary.

Since the action J  T is nonelementary, its minimal subtree T^J is not a line, and so T^J contains a finite path α which is not contained in any Nielsen line. Furthermore, α is contained in the axis of some loxodromic element γ∈ J whose axis L_γ is therefore not a Nielsen line. Applying Lemma <ref>, the action of γ on T^* is loxodromic and L_γ is a quasi-axis for γ. Choosing δ∈ J - _F_n(L_γ) and letting γ' = δγδ^, it follows that γ' also acts loxodromically on T and on T^*, and that its axis L_γ' in T is also a quasi-axis in T^*. By choice of δ the intersection L_γ L_γ' is either empty or finite. Since neither of the lines L_γ, L_γ' is a Nielsen axis, each ray in each line has infinite D_ diameter and so goes infinitely far away from the other line in T^*-distance. It follows that γ,γ' are independent loxodromic elements on T^*, proving that J  T^* is nonelementary. 

Finally, using the same γ,γ' as in the previous paragraph whose axes L_γ,L_γ' in T are not Nielsen lines, we showed in the proof of Conclusion ItemThmF_EllOrLox that γ,γ' act loxodromically on . Furthermore, since the lines L_γ,L_γ' have infinite Hausdorff distance in T^*, it follows by they also have infinite Hausdorff distance in , as shown in item (*) of Section <ref>. This proves that γ,γ' are independent loxodromics on  and hence the action J is nonelementary. 



  
Proof of Conclusion ItemThmF_WWPD: Each element of J that acts loxodromically on  is a strongly axial, WWPD element of .   

Given γ∈ J = (F_n)  K, as was shown earlier under the proof of Condition ItemThmF_EllOrLox, we know that the action of γ on  is loxodromic if and only if its action on T^* is loxodromic, if and only if its action on T is loxodromic with an axis L_γ⊂ T that is not a Nielsen line. It follows that L_γ is a quasi-axis for the actions of γ on T^* and on .

We shall prove that each such element γ is a WWPD element with respect to three actions:

Step 1: The action K  T, with respect to the metric d_u (Definition <ref>); 

Step 2: The action K  T^*, with respect to the metric d^* (Definition <ref>); 

Step 3: The action , with respect to the metric d_ (Definition <ref>). 

The proofs in Steps 2 and 3 are bootstrapping arguments, deriving the WWPD property for the current step from the WWPD property of the previous step. 

Once the WWPD arguments are complete, we will verify that γ is strongly axial with respect to the action .

Step 1: The action K  T. 

As shown in Section <ref> Item_f_KernelActs, the action J  T is the restriction of the action K  T. 

Since J is normal in K, and since J  T is the restriction to J of the free splitting action (F_n) ≈ F_n  T, we may apply Lemma <ref> to conclude that γ is a WWPD element of the action K  T with respect to the edge counting metric d_u.




Step 2: The action K  T^*.  The underlying intuition of this bootstrapping proof is that WWPD behaves well with respect to coning operations, for elements whose loxodromicity survives the coning process. We shall use the WWPD property of γ with respect to the action K  T and metric d_u to derive the WWPD property of γ with respect to the action K  T^* and the “coned off” metric d^*. For this purpose we shall use the original version of WWPD from <cit.>, referred to in <cit.> as “WWPD (3)”:


  WWPD (3): Given a hyperbolic action Γ X, a loxodromic element h ∈Γ satisfies WWPD if and only if for any quasi-axis ℓ of h and for any closest point map π X →ℓ there exists B ≥ 0 such that for any g ∈Γ - ( h), the set g(ℓ) has diameter ≤ B.  

Remark: This statement is equivalent to any alternate version in which either of the universal quantifiers on ℓ or on π is replaced with an existential quantifier, because any two quasi-axes of h have finite Hausdorff distance, and any two closest point maps X →ℓ have finite distance in the sup metric. We shall use these equivalences silently in what follows.


Let π T → L_γ be the retraction which maps each component C of T-L_γ to the point where the closure of C intersects L_γ. This map π is the unique closest point map from T to L_γ with respect to the metric d_u, and so in the context of Step 1 we can apply WWPD (3) to conclude that there is a constant B ≥ 0 such that for each g ∈ K - (γ), the set π(g(L_γ)) has d_u-diameter ≤ B, and hence the set π(g(L_γ)) has D^u-diameter ≤ B (Corollary <ref>). 

Recall two facts: the coarse metrics D_u and D_ are quasicomparable on T (Definition <ref>); and the inclusion map T  T^* is a quasi-isometry with respect to D_ on T and d^* on T^* (Proposition <ref>). It follows from these two facts that the sets π(g(L_γ)) have uniformly bounded d^*-diameter for all g ∈ K - (γ). Property WWPD (3) for h with respect to the action K  T^* and the metric d^* will therefore be proved if we can demonstrate the following: for any closest point map π^*  T^* → L_γ with respect to the metric d^*, the distance d^*(π(x),π^*(x)) is uniformly bounded as x varies over T. For this purpose, using the same two facts above, it suffices to prove that the quasidistance D_u(π(x),π^*(x)) is uniformly bounded as x varies over T. Applying the Coarse Triangle Inequality for the path function L_u (Lemma <ref>), and using that π^*(x) minimizes D^u distance from x to L_γ, we have

    D_u(π(x),π^*(x))    ≤ D_u(π(x),x) + D_u(x,π^*(x)) +  C_<ref>
       ≤ D_u(π(x),x) + D_u(x,π(x)) + C_<ref>
       ≤ 2B + C_<ref>

This completes the proof that γ is a WWPD element of the action K  T^*.











































































































    Step 3: The action .
 Arguing by contradiction, suppose that γ fails to be a WWPD element of the action  with respect to the metric d_. Fix a closest point map π→ L_γ. Applying WWPD (3) (stated above, and see the following remark), we obtain an infinite sequence Ψ_i ∈ - _(γ) (i=1,2,…) such that the diameter of π(Ψ_i(L_γ)) goes to +∞ as i → +∞. Denote L_i = Ψ_i(L_γ) which is a (k,c)-quasiaxis in  for Ψ_i γΨ_i^, where k ≥ 1, c ≥ 0 are independent of i. 



Using that the coordinate homomorphism ω_u → is surjective with kernel K, and that ω_u(Φ)=1 (see Figure <ref>), for each i we have Ψ_i = Δ_i Φ^m_i for a unique Δ_i ∈ K and a unique integer m_i. Choose points w_i,x_i ∈ L_γ and y_i,z_i ∈ L_i such that π(y_i)=w_i, π(z_i)=x_i, the point w_i precedes the point x_i in the oriented axis L_γ, and d_(w_i,x_i) → +∞. By replacing Ψ_i with Δ_i Φ^m_iγ^k for an appropriate exponent k that depends on i, after passing to a subsequence we may assume that the subpaths [w_i,x_i] of L_γ are nested:

    (*)          [w_1,x_1] ⊂⋯⊂ [w_i,x_i] ⊂ [w_i+1,x_i+1] ⊂⋯


We next prove that the sequence of integers (m_i) takes only finitely many values. Consider the (k,c)-quasigeodesic quadrilateral Q_i in  having the (k,c)-quasigeodesics [w_i,x_i] ⊂ L_γ and [y_i,z_i] ⊂ L_i as one pair of opposite sides, and having geodesics w_i y_i, x_i z_i as the other pair of opposite sides. By a basic result in Gromov hyperbolic geometry <cit.>, there exists a finite, connected graph G_i equipped with a path metric, and having exactly four valence 1 vertices denoted w̅_i, x̅_i, y̅_i, z̅_i, and there exists a quasi-isometry f_i  (Q_i,w_i,x_i,y_i,z_i) → (G_i,w̅_i, x̅_i, y̅_i, z̅_i) with uniform quasi-isometry constants independent of i (depending only on k, c, and the hyperbolicity constant of ). We consider two cases, depending on properties of the graph G_i. In the first case, the paths [w̅_i, x̅_i] and [y̅_i, z̅_i] are disjoint in G_i, and so the points y̅_i, z̅_i have the same image under the closest point projection G_i ↦ [w̅_i, x̅_i]; it follows that π(y_i)=w_i and π(z_i)=x_i have uniformly bounded distance, which happens for only a finite number of values of i. In the second case the paths [w̅_i, x̅_i] and [y̅_i, z̅_i] are not disjoint in G_i, and so the minimum distance between the paths [w_i,x_i] and [y_i,z_i] in  is uniformly bounded, implying that the minimum distance in  between _0 and Φ^m_i(_0) is uniformly bounded, but that distance equals m_i by Lemma <ref>. In each case it follows that m_i takes on only finitely many values. 

We may therefore pass to a subsequence so that m_i=m is constant, hence Ψ_i = Δ_i Φ^m and L_i ⊂_m. By restriction of the action  we obtain an isometric action K _m. Reverting to formal action notation, let _0  K _0 and _m  K _m denote the two restrictions of the action K. We have a commutative diagram 

    K [rrr]^_0[d]_i_Φ^m         (_0) [d]^Ad_Φ^m
    
    K [rrr]^_m         (_m)

in which the left vertical arrow is the restriction to the normal subgroup K of the inner automorphism i_Φ^m→, and the right vertical arrow is the “adjoint” isomorphism given by the formula 

    Ad_Φ^m(Θ)(y) = Φ^m ·Θ(Φ^-m· y)   for each y ∈_m and each Θ∈(_0)
 
Using the above commutative diagram, observe that i_Φ^m takes loxodromic WWPD elements of the action _0  K _0 to loxodromic WWPD elements of the action _m  K _m. Combining this with Step 2, it follows that γ' = i_Φ^m(γ) is a loxodromic WWPD element of the action K _m. 

Noting that the line L_γ' = Φ^m(L_γ) is a quasi-axis for γ' in _m, we may apply the property WWPD (3) to L_γ', with the conclusion that for each Δ∈ K - (γ') the image of an _m-closest point map Δ· L_γ'↦γ' has uniformly bounded diameter. By careful choice of Δ, namely the members of the sequence Δ_i^Δ^_i-1∈ K - (γ'), we shall use the nesting property (*) to obtain a contradiction. 

Knowing that [y_i-1,z_i-1] ⊂ L_i-1 is uniformly Hausdorff close in  to [w_i-1,x_i-1], and using that the inclusion _m → is uniformly property (Lemma <ref>), it follows that the diameter of [y_i-1,z_i-1] in _m goes to +∞ as i → +∞. Also, knowing that [w_i-1,x_i-1] is a subpath of [w_i,x_i], and that [w_i,x_i] is uniformly Hausdorff close in  to [y_i,z_i] ⊂ L_i, it follows that [y_i-1,z_i-1] is uniformly Hausdorff close in  to a subpath of [y_i,z_i]. Again using that _m → is uniformly proper (Lemma <ref>), it follows that [y_i-1,z_i-1] is uniformly Hausdorff close in _m to a subpath of [y_i,z_i]. It follows that the diameter of the image of the _m-closest point map L_i-1↦ L_i goes to +∞, because it is greater than __m[y_i,z_i] - C for some constant C independent of i. Applying the isometric action of the group element Δ_i^, it follows that diameter of the image of the _m-closest point map from Δ_i^Δ_i-1 L_γ' = Δ_i^ L_i-1 to Δ_i^ L_i = L_γ' goes to +∞ as i → +∞. This gives the desired contradiction, completing Step 3.







    The strong axial property.
 
For each γ∈ J that acts loxodromically on the tree T, we regard its axis L_γ as an oriented line with positive/negative ideal endpoints equal to the attracting/repelling points _+ γ, _-γ, respectively. Assuming also L_γ is not a Nielsen line, we shall prove that L_γ is a strong axis with respect to the action . It suffices to prove this for the set Z ⊂ J consisting of all root free γ∈ J such that L_γ is not a Nielsen line. 

We have indexed sets of oriented lines Ł = {L_γγ∈ Z} and of ideal points points = {_+ γγ∈ Z}, with bijective indexing maps Z ↦ L_Z and Z ↦. Having shown already that each γ∈ Z acts loxodromically on T^* and on , with attracting/repelling pairs denoted _±γ∈ T^* and _±γ∈, we obtain indexed sets ^* = {_+γγ∈ Z} and ^ = {_+γγ∈ Z}.


We next show that the indexing map Z ↦^ is a bijection. The oriented line L_γ is also a quasi-axis for γ in , and hence the positive end of L_γ limits on _+γ in . Consider γδ∈ Z. The line L_γδ⊂ T with ideal endpoints _+γ,_+δ in T is a concatenation of the form L_γδ = R̅_γ A R_δ where R_γ is a positive ray in L_γ and R_δ is a positive ray in L_δ, hence the two ends of L_γδ limit on _+γ,_-δ in . To show that _+γ_+δ it therefore suffices to prove that L_γδ is a quasigeodesic line in . Since L_γδ is a reparameterized quasigeodesic in T^* (Proposition <ref>), and since its disjoint subrays R_γ, R_δ each have infinite D_ diameter in T and hence infinite d^* diameter in T^* (Proposition <ref>), it follows that L_γδ is a quasigeodesic line in T^*; this shows that _+γ_+δ. Arguing by contradiction, suppose that _+γ = _+δ∈, and so both ends of L_γδ limit on that point of . Since the rays R_γ, R_δ are quasigeodesics in , it follows that there are sequences (x_i) in R_γ and (y_i) in R_δ such that the distances d^(x_i,y_i) are bounded. Since x_i,y_i approach _+γ_+δ in T^*, respectively, it follows that d^*(x_i,y_i) approaches +∞. Since the inclusion T^* is uniformly proper (Lemma <ref>), it also follows that the distances d^(x_i,y_i) approach +∞, a contradiction.

We claim that there exist actions of the group  on the sets Z,  and ^ satisfying the following properties:


  *  The indexing bijections Z ↦, ^ are -equivariant.

  *  No element of - K fixes any element of Z. 

Once this claim is proved, from ItemHEqBij and ItemHMinusJ it follows that for each γ∈ Z we have _K(_±γ) = _K(^_±γ) = _ (^_±γ). In the place where Lemma <ref> is applied in Step 1 we may further conclude that each γ∈ Z is strongly axial with respect to the action K  T, and hence the action of _K(_±γ) on T preserves L_γ. The action of _ (^_±γ) on  therefore preserves L_γ, proving that γ is strongly axial with respect to the action , and we are done. 

For proving the claim, we will for the most part restrict our attention to the semigroup _+ = _+ and its semigroup action _+  T which is obtained by restricting _+  T (Section <ref>).

The action Z is defined by restricting to Z the natural inner action of  on its own normal subgroup J = (F_n), but we must verify that Z is invariant under that action. Given γ∈ J, and Ψ∈_+ = _+, we note three things.

First, applying Property Item_f_Twisted Twisted Equivariance of Section <ref>, the action of γ on T is loxodromic with axis L_γ if and only if the action of Ψ(γ) on T is loxodromic with axis f^Ψ_#(L_γ)=L_Ψ(γ). Second, γ is root free if and only if Ψ(γ) is root free. Third, applying Property Item_f_PsiNielsen Invariance of Nielsen data of Section <ref>, the axis L_γ is a Nielsen line if and only if L_Ψ(γ) is a Nielsen line. It follows that γ∈ Z if and only if Ψ(γ) ∈ Z, hence Z is -invariant.

The argument in the previous paragraph yields a little more: the bijection Z ↔Ł is equivariant with respect to the action Z and the action Ł, the latter of which is defined by Ψ· L_γ = f^Ψ_#(L_γ) = L_Ψ(γ). Existence of actions of  on  and ^ satisfying item ItemHEqBij is an immediate consequence: for each Ψ∈_+, the ideal point in  or in ^ represented by the positive end of the oriented line L_γ is taken by Ψ∈_+ to the ideal point represented by the positive end of L_Ψ(γ).

Finally, item ItemHMinusJ is a special case of the following: for each Ψ∈ - _0 and each γ∈ F_n, if γ is loxodromic and if its axis L_γ is not a Nielsen line then Ψ(γ) γ. Suppose to the contrary that Ψ(γ)=γ, let Ψ = ΔΦ^m where Δ∈(F_n) and m  0, and let c be the conjugacy class of γ in F_n, and so ϕ^m(c)=c. Let σ be the circuit in G represented by c (Notations <ref>). We may assume m > 0, and so f^m_#(σ)=σ. It follows that σ is a concatenation of fixed edges and indivisible Nielsen paths of f (, Fact 1.39). However, no edge of H_u is fixed, and the only indivisible Nielsen path that crosses an edge of H_u is ρ_u^± 1 which has at least one endpoint disjoint from G_u-1 (Notations <ref> ItemCTiNP). One of two cases must therefore hold: either σ is contained in G_u-1, implying that γ is not loxodromic (Definition <ref>); or ρ exists and is closed, and σ is an iterate of ρ or ρ^, implying that L_γ is a Nielsen line (Definition <ref>). In either case, we are done verifying item ItemHMinusJ.
































































































































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