Abstract Abstract. We give a complete characterization of irreducible nonsurjective endomorphisms of Fn in terms of their topological representatives. Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an expanding irreducible immersion on a finite graph. We give a new proof of this and also show the converse holds when the immersion has connected Whitehead graphs. Consequently, irreducible nonsurjective endomorphisms are fully irreducible. The next result is a characterization of finitely generated subgroups of the free group that are invariant under some iterate of an irreducible nonsurjective endomorphism. These two characterizations imply that the mapping torus of an irreducible nonsurjective endomorphism is word-hyperbolic. Introduction Bestvina-Handel defined irreducible automorphisms of free groups in [3] as the base case of their proof of the Scott conjecture. Irreducible automorphisms are the dynamical free group analogues of pseudo-Anosov diffeomorphisms of surfaces and, as such, the study of their dynamics has been an active area of research since their introduction. For instance, a leaf of the attracting lamination of a pseudo-Anosov diffeomorphism cannot have compact support in an infinite cover of the surface, and analogously, Bestvina-Feighn-Handel constructed a lamination for fully irreducible automorphisms and showed that a leaf of the lamination cannot be supported by a finitely generated infinite index subgroup [2, Proposition 2.4]. Another analogy: pseudo-Anosov mapping classes act with north-south dynamics on the Thurston compactification of Teichmüller space [19] and Levitt-Lustig proved that fully irreducible automorphisms act with north-south dynamics on the compactification of Culler-Vogtmann outer space [12] . Introduction We would like to extend these results to irreducible endomorphisms. We begin by defining the (semi)action of an injective endomorphism on outer space and the first result is that irreducible nonsurjective endomorphisms act with sink dynamics on outer space. There is a correspondence between having a fixed point of the action and being able to represent the endomorphism with a graph immersion. Under this correspondence, we prove: Theorem 4.4. If φ : F → F is an irreducible nonsurjective endomorphism, then it can be represented by an irreducible immersion with connected Whitehead graphs. Moreover, the immersion is unique amongst irreducible immersions. Introduction All the terms in the theorem will be defined in Section 2. This is an unpublished result of Patrick Reynolds [16, Corollary 5.5] but the proof given here is original. We also note that Reynolds' proof assumes the endomorphism is fully irreducible but, as we shall see shortly, this is equivalent to being irreducible. Not much is currently known about the action; for instance, Problem 1.1. Does the action of an irreducible nonsurjective endomorphism on outer space always have bounded image? See Example 3.3. Introduction Using the sink dynamics on outer space, we prove the converse of Theorem 4.4: Theorem 4.5. If φ : F → F is represented by an irreducible immersion with connected Whitehead graphs, then φ is nonsurjective and fully irreducible. The analogous criteria for fully irreducible automorphisms due to Catherine Pfaff [15, Theorem 4 .1] and Ilya Kapovich [11, Theorem 1.2] require finding periodic Nielsen paths. So, in practice, the irreducibility of nonsurjective endomorphisms is easier to certify than that of automorphisms since it is straightforward to check that an immersion is irreducible and has connected Whitehead graphs. Furthermore, Theorem 4.5 can be combined with the action on the spine of outer space to get an algorithm that detects whether a nonsurjective endormorphism is irreducible. We sketch the algorithm at the end of Section 4. Introduction The proof of the next proposition determines which finitely generated subgroups support a leaf of the lamination of an irreducible endomorphism; roughly speaking, these are the iterated images of the endomorphism, up to finite index. Proposition 5.3. Suppose φ : F → F is injective and represented by an irreducible train track with connected Whitehead graphs. If H ≤ F is a finitely generated subgroup such that φ(H) ≤ H and H contains a φ-expanding conjugacy class, then [φ k (F ) : φ k (F ) ∩ H] < ∞ for some k ≥ 0. Introduction As a corollary of this technical result, we get a characterization for fully irreducible endomorphisms that applies to both automorphisms and nonsurjective endomorphisms. Introduction Corollary 5.5. Let φ : F → F be an injective endomorphism. Then φ is fully irreducible if and only if φ : F → F is primitively atoroidal, it is represented by an irreducible train track with connected Whitehead graphs, and its image φ(F ) is not contained in a proper free factor. Introduction Another consequence of Proposition 5.3 is the hyperbolicity of the mapping torus of an irreducible immersion with connected Whitehead graphs -this was our original motivation. Theorem 6.3. If φ : F → F is nonsurjective and irreducible, then F * φ is word-hyperbolic. Introduction By our previous work [14] , the proof boils down to showing that irreducible nonsurjective endomorphisms cannot have periodic laminations. As a quick application, we prove the hyperbolicity of the Sapir group: Example 1.2. Let F = F (a, b) be the free group on two generators and φ : F → F be the nonsurjective endomorphism given by φ(a) = ab and φ(b) = ba. The obvious map on the standard rose f : R 2 → R 2 will be an immersion that induces φ on the fundamental groups. It is easy to verify that f is an irreducible immersion with a connected Whitehead graph. By Theorem 4.5, φ is fully irreducible, and by Theorem 6.3, the Sapir group F * φ = a, b, t | t −1 at = ab, t −1 bt = ba is word-hyperbolic. Introduction In contrast, ψ : F → F given by (a, b) → (aba, bab) is not word-hyperbolic: ψ(ab) = (ab) 3 and so BS(1, 3) ∼ = ab, t ≤ F * ψ . As H 1 (F * φ ) = Z 2 has rank > 1, there are infinitely many isomorphisms F * ψ ∼ = F n * ψn [4] ; all such endomorphisms ψ n : F n → F n are nonsurjective, because BS(1, 3) ≤ F * ψ , and reducible by Theorem 6.3. Introduction In future work, we use Proposition 5.3 to show that the irreducibility of an endomorphism is a group-invariant of its mapping torus, i.e., suppose φ : F n → F n and ψ : F m → F m are endomorphisms (satisfying an additional necessary assumption for nonsurjective endomorphisms) and F n * φ ∼ = F m * ψ , then φ is irreducible if and only if ψ is irreducible [13] . This answers Question 1.4 posed by Dowdall-Kapovich-Leininger in [9] ; see also [8, Theorem C] . Introduction Outline: Section 2 contains the standard definitions and preliminary results that will be used throughout the paper. We define outer space and the action of injective endomorphisms on it in Section 3. Section 4 contains the proof of Theorem 4.4 and its converse Theorem 4.5 then ends with an algorithm for detecting irreducible nonsurjective endomorphisms. In Section 5, we prove the technical result classifying subgroups that support a leaf of a lamination, Proposition 5.3. While the two previous sections are independent, we combine their main results in Section 6 to prove Theorem 6.3. Introduction Acknowledgments: I would like to thank Ilya Kapovich for getting me started on the proof of Theorem 4.4. I also thank Derrick Wigglesworth and my advisor Matt Clay for the countless conversations on this material. Definitions and Preliminary Results In this paper, F is a finitely generated free group on at least two generators. Definition 2.1. An endomorphism φ : F → F is reducible if there exists a free factorization A 1 * · · · * A k * B of F , where B is possibly trivial if k ≥ 2, and a sequence of elements, Definitions and Preliminary Results where the indices are considered mod k. An endomorphism φ is irreducible if it is not reducible and it is fully irreducible if all its iterates are irreducible; equivalently, φ is fully irreducible if there does not exist a proper free factor A ≤ F , an element g ∈ F , and an integer n ≥ 1 such that φ n (A) ≤ gAg −1 . Definitions and Preliminary Results An application of Stallings folds implies the endomorphisms studied in this paper will be injective even though it may not be explicitly stated again. Definitions and Preliminary Results Proof. Stallings [17] showed that any endomorphism of F factors through folds whose nontrivial kernels corresponding to proper subgraphs, hence are normal closures of free factors. If φ is irreducible and not injective, then ker φ contains an invariant free factor; irreducibility implies the only invariant free factor is the group itself, but this implies φ is trivial, in which case, all proper free factors are also invariant -a contradiction. Definition 2.3. Fix an isomorphism F ∼ = π 1 (Γ) for some connected finite core graph Γ, i.e., a finite 1-dimensional CW-complex with no univalent vertices. For any subgroup H ≤ F , the Stallings subgroup graph S(H) is the core of the cover of Γ corresponding to H, i.e., it is the smallest subgraph of the cover containing all immersed loops of the cover. When H is nontrivial and finitely generated, S(H) is a finite core graph with rank ≥ 1. A Stallings subgroup graph comes with an immersion v : S(H) → Γ, which is a restriction of the covering map. We shall also assume that S(H) has been subdivided so that v is simplicial: v maps edges to edges. The immersion S(H) → Γ uniquely determines [H], the conjugacy class of H: if there is a subgroup H ≤ F with immersion v : S(H ) → Γ and homeomorphism h : Definitions and Preliminary Results The last statement makes Stallings subgroup graphs especially useful for studying a nonsurjective injective endomorphism φ : Definitions and Preliminary Results When φ is an automorphism, the maps v i : S i → Γ are all graph isomorphisms. Conversely, suppose there was a homeomorphism h : (F ) . But in free groups (subgroup separable), a finitely generated subgroup cannot be conjugate to a proper subgroup. Hence φ j (F ) = φ i (F ). But φ j−i φ i (F ) is conjugate to φ j−i as φ is injective. In particular, φ j−i is an automorphism and so is φ. Therefore, when φ is nonsurjective, the maps v i are all distinct and the number of edges in S i grows as i → ∞. We shall use this fact in Section 4 to show that, for an irreducible nonsurjective endomorphism, the sequence of maps S i → S i that induce φ on π 1 (S i ) ∼ = F converges to an immersion on Γ that induces φ on π 1 (Γ) ∼ = F . Definition 2.4. Let Γ be a finite graph. A map f : Γ → Γ is a train track if it maps vertices to vertices and all of its iterates are immersions when restricted to the interior of any edge. A train track is irreducible if for any pair of edges e i , e j in Γ, e i is in the image of some f -iterate of e j . Given a train track f : Γ → Γ, we fix an ordering of the edges of Γ and construct the transition matrix A(f ) as follows: it is a nonnegative square matrix whose size is given by the number of edges in Γ; the (i, j)-th entry of A(f ) is the number of times the edge e i appears in the image of e j . An irreducible train track is expanding if it is not a homeomorphism; equivalently, the transition matrix is irreducible and has a real eigenvalue λ f > 1 -this is the Perron-Frobenius eigenvalue. Definitions and Preliminary Results Let f : Γ → Γ be a train track map and v be a vertex of Γ. The Whitehead graph at v is a simple graph whose vertices are the half-edges of Γ attached to v, denoted by T v (Γ). A pair of elements of T v (Γ) is called a turn at v and it is nondegenerate if the pair consists of distinct elements. A nondegenerate turn is f -legal if it remains nondegenerate under iteration of f . A turn at v is an edge of the Whitehead graph if it appears in the image of an f -iterate of some edge of Γ. Note that the edges in the Whitehead graphs are f -legal. Definitions and Preliminary Results An irreducible train track f : Γ → Γ is weakly clean if the Whitehead graph at each vertex is connected. A weakly clean map is clean if there is an iterate such that every edge surjects onto the whole graph; equivalently, the transition matrix is primitive. It follows from the definition that if a map f is clean then so are all its iterates f i (i ≥ 1). Definitions and Preliminary Results The following is a proposition by Dowdall-Kapovich-Leininger [8, Proposition B.2] and it allows us to use clean and weakly clean interchangeably for the rest of the paper. We give a proof that does not assume the map f : Γ → Γ is a homotopy-equivalence. Definitions and Preliminary Results Proposition 2.5. Suppose Γ is a connected graph and f : Γ → Γ is a train track with an irreducible but not primitive transition matrix. Then f has a vertex with a disconnected Whitehead graph. In particular, if f is weakly clean, then f is clean. Definitions and Preliminary Results Proof. As A(f ) is irreducible but not primitive, it is permutation-similar to a transitive block permutation matrix [1, Theorem 1.8.3] . This block permutation form gives a partition of Definitions and Preliminary Results where indices are considered mod d. Thus, any vertex adjacent to two or more of these subgraphs has a disconnected Whitehead graph. Such a vertex exists since Γ is connected. Bestvina-Handel defined train tracks in [3] and one of the main results was the algorithmic construction of train track representatives for irreducible endomorphisms. We note that their result was stated for irreducible automorphisms but the proof itself never used nor needed the fact that the endomorphisms were surjective. See also Dicks-Ventura [7] . Notation. We denote by [p] (resp. [ρ]) the immersed path (resp. loop) homotopic rel. endpoints to a path p (resp. freely homotopic to a loop ρ). Definitions and Preliminary Results For any irreducible train track map f : Γ → Γ, there is the associated Perron-Frobenius eigenvalue of the transition matrix, λ f ≥ 1, and a metric on the graph Γ given by the Perron-Frobenius left eigenvector. With this metric, f is homotopic rel. vertices to a (unique) map whose restrictions to edges are local λ f -homotheties. For the rest of the paper, we shall assume an irreducible train track map has the latter linear property. The following lemma, known as the Bounded Cancellation Lemma, combined with the train track linear structure allow us to carefully study the dynamics of f . Definitions and Preliminary Results where the lengths of u and v are less than C and y is f -legal. With the assumed linear structure on train tracks, we have The Action of Injective Endomorphisms on Outer Space Culler-Vogtmann introduced outer space in [6] as a topological space with a nice Out(F )-action. We will give two equivalent definitions and then describe the Out(F )-action and, more generally, the semiaction of an injective endomorphism with respect to the two descriptions of outer space. The first description is given in terms of trees with free actions of F and the other in terms of marked graphs. See Karen Vogtmann's survey paper [20] . The Action of Injective Endomorphisms on Outer Space Definition 3.1. In general, trees refers to real trees (0-hyperbolic geodesic spaces) and a tree is simplicial if it is homeomorphic to a locally finite CW-complex. The CullerVogtmann outer space of F , denoted by CV (F ) , is the set of connected simplicial trees, T , with a free minimal (left) action of F , α : F → Isom(T ), up to the equivalence: The Action of Injective Endomorphisms on Outer Space Alternatively, CV (F ) is the set of connected finite metric core graphs, Γ, with a marking α : F → π 1 (Γ), up to the equivalence: The Action of Injective Endomorphisms on Outer Space . Identify π 1 (Γ) with the group of deck transformations ofΓ to get the correspondence between the two descriptions. The Action of Injective Endomorphisms on Outer Space We will now equip CV (F ) with a topology: for an equivalence class [Γ, α] ∈ CV (F ), choose a representative such that Γ has no bivalent vertices and vol(Γ) = 1; the volume of a metric graph vol(Γ) is the sum of the lengths of all edges in the graph. Let n be the number of edges in Γ and identify σ(Γ, α) with the (n − 1)-simplex one gets by varying the lengths of the edges of Γ to get homeomorphic Γ while still maintaining the equality vol(Γ ) = 1. In the tree description of CV (F ), this variation corresponds to equivariantly varying the metric on (Γ, α) to get (Γ , α ). The Action of Injective Endomorphisms on Outer Space This gives us a decomposition of CV (F ) into a disjoint union of open simplices σ(Γ, α). Attaching maps for these simplices are given by decreasing the volume of some forest of Γ to 0. This decomposition of CV (F ) and description of attaching maps makes CV (F ) a locally finite open simplicial complex, i.e., a simplicial complex with some missing faces corresponding to collapsing noncontractible subgraphs. The set of open simplices of CV (F ), denoted by SCV (F ), can be made into a locally finite simplicial complex known as the spine of outer space but, for the most part, we will treat it as a set with no added structure. The Action of Injective Endomorphisms on Outer Space For the rest of the paper, we will assume CV (F ) has this topology. It is a theorem of Culler-Vogtmann that CV (F ) is contractible [6, 21] . There is a natural right action of The Action of Injective Endomorphisms on Outer Space where T is the minimal tree of the twisted action αφ : F → Isom(T ), i.e., T is the minimal tree for φ (F ) . This induces an action on SCV (F ) by forgetting the metrics: σ(T, α)·[φ] = σ(T , αφ). For an F -tree (T, α) and a marked metric graph (Γ, α), we shall abuse notation and write only T and Γ when the actions and markings are clear. The goal of the next section is to describe dynamics of the action of irreducible nonsurjective endomorphisms on CV (F ). We will show that the action has a unique attracting point and no other fixed points. The Action of Injective Endomorphisms on Outer Space We now give the second description of the action of an injective endomorphism φ : F → F on CV (F ) and SCV (F ) . Let Γ be a marked metric graph, i.e., [Γ, α] ∈ CV (F ). Fix any topological representative f : Γ → Γ for φ. For any i ≥ 1, the map f i : Γ → Γ factors as f i = v i h i where h i : Γ → S i is a composition of folds and v i : S i → Γ is an immersion. The graph S i is subdivided so that the immersion is simplicial. If we let S i ⊂ S i be the core subgraph, then this is the Stallings subgroup graph for φ i (F ) and the graphs fits in the following commutative diagram: The Action of Injective Endomorphisms on Outer Space Since φ is injective, the composition of folds h i : Γ → S i is a homotopy equivalence which induces a marking h i * α : F → π 1 (S i ) = π 1 (S i ). Pullback the metric on Γ via the immersion v i : S i → Γ to get a metric on S i . By construction,ṽ i (S i ) ⊂Γ is the minimal tree for φ i (F ) The Action of Injective Endomorphisms on Outer Space The following lemma tells us precisely when φ fixes an element of SCV (F ) or CV (F ). Proof. The Action of Injective Endomorphisms on Outer Space . Then the composition of folds and core retraction h 1 : Γ → S 1 is homotopic to a homeomorphism. As f = v 1 h 1 and v 1 is an immersion, we get that f is homotopic to an immersion. The Action of Injective Endomorphisms on Outer Space By the previous sentence, we may assume f is an immersion, S 1 = S 1 , and h 1 : Γ → S 1 is a homeomorphism. Finally, for the point [Γ, α] to be fixed, the immersion v 1 : S 1 → Γ and homeomorphism h 1 : Γ → S 1 must induce the same projective metric on S 1 . Thus f is (homotopic to) a local homothety. The Action of Injective Endomorphisms on Outer Space We end the section with a description of the action of the endomorphism in Example 1.2. The Action of Injective Endomorphisms on Outer Space In rank 2, the spine of outer space has the structure of a regular trivalent tree with a spike at the midpoint of edges. Set F = π 1 (R 2 ), then the standard rose is the marked graph R * = (R 2 , id F ) and all other roses are given by (R 2 , ϕ) for some [ϕ] ∈ Out(F ). Let (B, β) be the marked barbell graph attached to R * and let (T, θ) be the theta graph between R * , (R 2 , ϕ a ), and (R 2 , ϕ b ) where ϕ a : (a, ab) and ϕ b : (ba, b) are the generators of Out (F ) . To compute σ(R 2 , ϕ) · [φ], first represent ϕφ on R * then fold. The composition of folds gives the resulting marked graph. For instance, Figure 1 below illustrates one of these computations. Along these lines, we can show that The Action of Injective Endomorphisms on Outer Space is the set of φ-periodic elements in SCV (F ); the first two are φ-fixed while the latter two have φ-period 2. By inducting on the roses of the spine, we can verify SCV (F ) · [φ] = S. Irreducible Nonsurjective Endomorphisms are Immersions Given an expanding irreducible train track map f : Γ → Γ representing an injective endomorphism φ : F → F , then φ has a right action on the F -treeΓ by taking minimal trees of the twisted actions: (Γ, α) · φ n = (Γ , αφ n ). We define the limit tree where the limit is taken in the space of length functions on F . Irreducible Nonsurjective Endomorphisms are Immersions ∈ CV (F ); here the limit is taken in the space of projective length functions. The class [T f ] need not be in CV (F ) and, in fact, it is not when φ is an automorphism. We will use [T φ ] = [T f ] and λ φ = λ f to emphasize the independence from the choice of train track representative f . The F -action on T φ is free if and only if φ is atoroidal, i.e., it has no periodic conjugacy class. When φ is represented by an irreducible immersion f : Γ → Γ where Γ has the train track linear structure, then f is a local λ φ -homothety, T φ =Γ, and [T φ ] ∈ CV (F ). In a certain sense, the immersion is unique since [Γ] is the unique φ-fixed point in CV (F ). Irreducible Nonsurjective Endomorphisms are Immersions In the general setting considered in the previous section, not much can be said about the graphs S i and immersions v i : S i → Γ. However, if f : Γ → Γ is assumed to be a clean representative, we gain more control; for example, we will show in this case that S i is a core graph, hence, S i = S i . To do this, we need to introduce a new structure: Definition 4.1. Let f : Γ → Γ be an irreducible train track map and, for some i ≥ 1, f i = hg where g : Γ → X and h : X → Γ are surjective graph maps and X is a finite core graph. A turn at a vertex v of X is legal if its image under h is an f -legal turn. The relative Whitehead graph of (X, g, h) at v is a simple graph whose vertex set is T v (X) and a turn at v is an edge of the relative Whitehead graph if it is in the edge-path g(f j (e)) for some j ≥ 0 and edge e in Γ. Note that edges of the relative Whitehead graphs are legal. Irreducible Nonsurjective Endomorphisms are Immersions The next lemma tells us how subdivision and folding affect the relative Whitehead graphs. Proof. Suppose h 1 is a subdivision. If v is a vertex of Y that was added by the subdivision, then T v (Y ) has two elements, v ∈ Int(e) for some edge e of X, and v ∈ h 1 (e ). In particular, the two elements of T v (Y ) are joined by an edge since g is surjective. Irreducible Nonsurjective Endomorphisms are Immersions If v is a vertex of both Y and X, then T v (Y ) = T v (X) and subdivision makes no changes to the relative Whitehead graph at v. So the relative Whitehead graph of (Y, h 1 g, h 2 ) is connected since we assumed (X, g, h) has connected relative Whitehead graphs. Irreducible Nonsurjective Endomorphisms are Immersions Suppose h 1 is a single fold. The only way to produce a univalent vertex in Y with a fold is by folding a bivalent vertex of X. But bivalent vertices only have one turn which is legal since the relative Whitehead graphs of (X, g, h) are connected. Irreducible Nonsurjective Endomorphisms are Immersions At the origin vertex, the fold identifies two distinct vertices of the relative Whitehead graphs and this preserves connectedness. At the two terminal vertices, the fold identifies two distinct vertices, one from each relative Whitehead graph, and this gluing of two connected graphs produces a connected graph. If a vertex in X is not part of the fold, then its image in Y will have the same connected relative Whitehead graph. Irreducible Nonsurjective Endomorphisms are Immersions Recall that to construct S i , we subdivide and fold the map f i : Γ → Γ to get the immersion v i : S i → Γ. Suppose f was a clean map. For the base case, the relative Whitehead graphs of (Γ, id Γ , f i ) are connected as they are isomorphic to the Whitehead graphs of f i . By Lemma 4.2 and induction on folds, S i is a core graph and the relative Whitehead graphs of (S i , h i , v i ) are connected. So the immersion v i maps edge-paths connecting branch points (natural edges) of S i to f -legal edge-paths in Γ. The commutative diagram in the previous section becomes: Irreducible Nonsurjective Endomorphisms are Immersions Aside from the construction of this diagram and the next lemma, this section and the next are independent of each other. This lemma will only be used in the proof of Proposition 5.3, the main result of the next section. Proof. For all n ≥ 1, ( Irreducible Nonsurjective Endomorphisms are Immersions Since f is a train track, v i is a simplicial immersion, and h i folds illegal turns only, it follows thatf i is a train track too. Since f is clean, there is an n ≥ 1 such that, for any edge e in Γ, f in (e) is surjective and contains all the edges of the Whitehead graphs of f i . As h i is surjective, (f i ) n+1 is surjective when restricted to any edge andf i has a primitive transition matrix. Since folding only identifies vertices of relative Whitehead graphs, h i f in (e) contains all the edges of the relative Whitehead graphs of (S i , h i , v i ). As v i maps edges to edges, we have that the Whitehead graphs off i contains the relative Whitehead graphs of (S i , h i , v i ) as subgraphs. Since v i is surjective, we have the reverse containment too. So the Whitehead graphs off i are connected and the map is clean. Irreducible Nonsurjective Endomorphisms are Immersions With this set-up, we are now ready to present a new proof of an unpublished theorem due to Patrick Reynolds [ Proof. Let f : Γ → Γ be a clean representative for φ : F → F given by Corollary 2.8. By the discussion following Lemma 4.2, we now construct core graphs S i , folds h i : Γ → S i , and immersions v i : S i → Γ such that f i = v i h i and the natural edges of S i are legal, i.e., they consist of legal turns. Nonsurjectivity implies that the number of edges in S i grows as i → ∞. So for some i 0, S i has a natural edge longer than the critical constant. By the bounded cancellation lemma, there is a nontrivial subpath s of the long natural edge such that, for all j > i, the folds in S i → S j are supported in the complement of s. Irreducible Nonsurjective Endomorphisms are Immersions Suppose i 0 is an index such that S i has the maximal number of natural edges longer than (F ) ) is the maximum number of natural edges of an embedded loop of a marked graph. Denote by K i the proper subgraph consisting of the remaining short natural edges of S i with length at most the critical constant. Let f : S i → S i be a map representing φ, i.e., [f * h i * ] = [h i * f * ]. Suppose K i has an embedded loop ρ, then by construction f v i (ρ) decomposes into at most L legal paths, each of length at most Irreducible Nonsurjective Endomorphisms are Immersions is anf i -invariant proper subgraph up to homotopy. Since φ is irreducible, K i must be a forest. Therefore, the loops in S i are growing exponentially, φ is atoroidal, and the action of F on T φ is free. Irreducible Nonsurjective Endomorphisms are Immersions We showed in the previous paragraph that folds in the map S i → S j (j > i) are supported in some forest K ⊂ S i . Since there are finitely many combinatorially distinct ways to fold a forest, there is a j > i and k ≥ 1 such that the composition of folds S j → S j+k is homotopic to a homeomorphism. This means Irreducible Nonsurjective Endomorphisms are Immersions So the projective limit tree for this sequence, Irreducible Nonsurjective Endomorphisms are Immersions is a simplicial tree, i.e., the F -quotient of T φ , call it Γ 0 , is a graph obtained by collapsing a (possibly empty) subgraph of S j . As T φ is free and simplicial, [T φ ] is a point in CV (F ) . By definition and Lemma 3.2, [T φ ] is fixed by φ and φ is represented by a local λ φ -homothety f 0 : Γ 0 → Γ 0 . Furthermore, the local homothety f 0 is expanding since λ φ > 1; so f 0 has no invariant forests and the irreducibility of φ implies f 0 is clean by Corollary 2.8. Uniqueness follows from Lemma 3.2 and uniqueness of the (attracting) fixed point [T φ ]. Irreducible Nonsurjective Endomorphisms are Immersions This theorem tells us that the action of an irreducible nonsurjective endomorphism on CV (F ) has a unique fixed point that is also a global attracting point. Reynolds studied this action further; for instance, the action converges to the fixed point uniformly on compact sets and, for admissible endomorphisms, the action can be extended to the compactification of outer space CV (F ) . Irreducible Nonsurjective Endomorphisms are Immersions We now use the action on outer space again to prove the converse to Theorem 4.4, which can also be thought of as a criterion for fully irreducible nonsurjective endomorphisms. Proof. Since φ is represented by a clean immersion f : Γ → Γ, it is nonsurjective. Suppose φ n (A) ≤ gAg −1 where A ≤ F is a proper free factor, g ∈ F , and n ≥ 1. Let S(A) be the Stallings subgroup graph corresponding to A with respect to Γ. Set ψ = i g • φ n so that ψ(A) ≤ A and the immersion f n lifts to an immersion g : S(A) → S(A) representing ψ| A . Complete ∆ 0 = S(A) to a graph Γ with a marking π 1 (Γ ) ∼ = F and extend g to the rest of Γ such that g : Γ → Γ is a topological representative for ψ and ∆ 0 is a noncontractible g-invariant proper subgraph corresponding to A. Irreducible Nonsurjective Endomorphisms are Immersions Recall from Section 3, the Stallings subgroup graph S i = S(ψ i (F )) with respect to Γ along with the marking given by folding g i : Γ → Γ determine the i-th iterate of Irreducible Nonsurjective Endomorphisms are Immersions , as discussed at the start of this section, and the spine of outer space is locally finite, we have that the sequence σ(S i ) (i ≥ 1) in SCV (F ) is eventually periodic. So for some i, j ≥ 1, we have σ(S i ) = σ(S i+j ) and, by Lemma 3.2, ψ j can be represented by an immersion on S i . Irreducible Nonsurjective Endomorphisms are Immersions Let h i : Γ → S i be the composition of folds given by the construction of S i . Since g| ∆ 0 is an immersion, the corresponding restriction h i | ∆ 0 is either a homeomorphism or an identification of vertices of ∆ 0 . Set ∆ i = h i (∆ 0 ) and let g j i : S i → S i be the induced map representing ψ j . Then ∆ i is a noncontractible g i -invariant subgraph and g i | ∆ i is an immersion. By the previous paragraph, g j i is homotopic to an immersion γ : S i → S i . The homotopy will preserve the invariance of ∆ i so that ∆ i is a γ-invariant subgraph. As σ(S i ) is fixed by ψ j , the sequence [S i+jm ] is constructed by pulling back the metric of S i via γ m . Irreducible Nonsurjective Endomorphisms are Immersions Iteratively pulling back the metric via γ has the effect of collapsing γ-invariant forests and normalizing the metric so the induced map is a local homothety. By uniqueness of the limit [S i+jm ] → [Γ], the induced map must be f nj and ∆ ∞ , the image of ∆ i under the collapse map, is an f nj -invariant subgraph. But f nj is a clean map, so ∆ ∞ = Γ. Irreducible Nonsurjective Endomorphisms are Immersions Let h ∞ : ∆ 0 → ∆ ∞ = Γ be the induced map. By construction, Irreducible Nonsurjective Endomorphisms are Immersions and, as h ∞ is an identification of some vertices and possibly a collapse of a forest, the Whitehead graphs of f nj are determined by where g j maps the edges of ∆ 0 . So f nj will have disconnected Whitehead graphs at the identified vertices -a contradiction. Irreducible Nonsurjective Endomorphisms are Immersions A result due to Dowdall-Kapovich-Leininger is that atoroidal irreducible automorphisms are fully irreducible [8, Corollary B.4] . Combining Theorem 4.4 and Theorem 4.5, we get that this holds for irreducible nonsurjective endomorphisms. Corollary 4.6. Let φ : F → F be an endomorphism. Then φ is nonsurjective and (fully) irreducible if and only if it is represented by a clean immersion. Irreducible Nonsurjective Endomorphisms are Immersions In the next section, we shall give a uniform characterization for fully irreducible endomorphism that holds for both surjective and nonsurjective endomorphisms. We now sketch an algorithm for checking whether a nonsurjective endomorphism is irreducible. Irreducible Nonsurjective Endomorphisms are Immersions Let φ : F → F be a given nonsurjective endomorphism: Irreducible Nonsurjective Endomorphisms are Immersions (1) Check if the Stallings subgroup graph S(φ(F )) with respect to the standard rose has the same rank as F . If not, then φ is not injective and we are done by Lemma 2.2. (2) Run Bestvina-Handel's algorithm to construct an irreducible train track representative [3, Theorem 1.7] . This algorithm either succeeds or produces a reduction for φ. Irreducible Nonsurjective Endomorphisms are Immersions In the latter case, we are done and φ is reducible. Otherwise, assume f : Γ → Γ is an irreducible train track representative for φ. (3) By Corollary 2.8, either the Whitehead graphs of f are connected or there is a reduction for φ. In the latter case, we are done. Otherwise, assume f is clean. (4) Construct graphs S i = S(φ i (F )) with respect to Γ by folding iterates of f . Let f i : Irreducible Nonsurjective Endomorphisms are Immersions S i → S i be the induced maps representing φ. Following the proof of Theorem 4.4, for i 0, either f i : S i → S i contains contains a noncontractible proper subgraph K i that is f i -invariant up to homotopy or all folds in S i → S j for j > i are supported in some forest of S i . In the former case, we get a reduction for φ and we are done. In the latter case, we can find a φ-periodic simplex σ(S j ) in SCV (F ) for some Irreducible Nonsurjective Endomorphisms are Immersions Collapse any maximal g-invariant forest if necessary and assume the immersion g has no invariant forest. If g is not irreducible, then we have a reduction for φ k and so φ is not irreducible by Corollary 4.6. Assume the immersion g is irreducible. Irreducible Nonsurjective Endomorphisms are Immersions (6) By Corollary 2.8 again, either some Whitehead graph of g is disconnected so φ k is reducible or g is clean. Irreducible Nonsurjective Endomorphisms are Immersions In the latter case, we get φ k is irreducible by Theorem 4.5. A careful analysis of Irreducible Nonsurjective Endomorphisms are Immersions Step (4) is required to understand the effectiveness of the algorithm. Subgroups Invariant Under Irreducible Endomorphisms In this section, we generalize a result by Bestvina-Feighn-Handel [2, Proposition 2.4] and I. Kapovich [10, Proposition 4.2] that characterizes the finitely generated subgroups of F that support a leaf of the lamination of an irreducible automorphism. Subgroups Invariant Under Irreducible Endomorphisms Definition 5.1. Given an expanding irreducible train track map f : Γ → Γ, the lamination of f , denoted by Λ(f ), is defined by iterated neighbourhoods of f -periodic non-vertex points. Precisely, suppose x ∈ Int(e) for some e ∈ E(Γ) and k ≥ 1 are such that f k (x) = x. Then a leaf of the lamination of f , is the isometric immersion γ x : R → Γ such that γ x (0) = x and f k (γ x (r)) = γ x (λ k f · r) for all r ∈ R, unique up to orientation. Λ(f ) = {γ x : x is an f -periodic non-vertex point.} For any integer m ≥ 1, f m -periodic points are f -periodic and vice-versa. So it follows that Λ(f ) = Λ(f m ) for any m ≥ 1. Subgroups Invariant Under Irreducible Endomorphisms We now address the difficulty that arises when generalizing Bestvina-Feighn-Handel and I. Kapovich's results. Any automorphism φ : F → F permutes the finite index subgroups with the same index; so given any finite index subgroup H , there exists i ≥ 1 such that φ i (H ) = H . This fact is used to lift a clean map representing φ to a clean map on the Stallings subgroup graph S(H ). However, this fails when dealing with nonsurjective injective endomorphisms, i.e., there is no reason why some φ-iterate of H must be a subgroup of H . The next lemma gives us a way of getting around this failure. The key observation is to look at backward iteration: when φ is injective, then pre-images of finite index subgroups are finite index subgroups with the same index or less. Subgroups Invariant Under Irreducible Endomorphisms Lemma 5.2. Let φ : F → F be an injective endomorphism of a free group F . If H ≤ F is a finite index subgroup, then there exist j > k ≥ 0 such that φ −k (H ) = φ −j (H ). Subgroups Invariant Under Irreducible Endomorphisms Furthermore, for K = φ −k (H ), there is an induced set bijection ϕ : F/K → F/K such that the following diagram commutes: Subgroups Invariant Under Irreducible Endomorphisms , and the induced function ϕ : F/K → F/K is an injection if and only if φ k−j (K) ≤ K. By construction, φ k−j (K) = K, i.e., both conditions are satisfied, so π • φ j−k = ϕ • π where ϕ is a bijection since it is an injection of the finite set F/K into itself. Subgroups Invariant Under Irreducible Endomorphisms The main result of the section follows: Proposition 5.3. Suppose φ : F → F is injective and represented by a clean map. If H ≤ F is a finitely generated subgroup such that φ(H) ≤ H and H contains a φ-expanding conjugacy class, then Subgroups Invariant Under Irreducible Endomorphisms Proof. Suppose φ(H) ≤ H for some finitely generated group H ≤ F . Let f : Γ → Γ be the given clean map and ρ be the f -expanding immersed loop in Γ that lifts to a loop in S(H). The invariance φ(H) ≤ H implies the loops [f k (ρ)] lift to loops in S(H) for all k ≥ 1. As ρ is f -expanding and f is a clean map, length of [f k (ρ)] grows arbitrarily with k while the number of f -illegal turns in [f k (ρ)] remains bounded. Thus, for some k ≥ 1, the loop [f k (ρ)] will contain an f -legal subpath longer than the critical constant. Therefore, there is a nontrivial subpath s of ρ such that f k (s) is a subpath of [f k (ρ)] for all k ≥ 1. As f is clean, some f -iterate of s maps onto Γ. Let x be an f -periodic non-vertex point of ρ and γ x : R → Γ be the corresponding leaf of Λ(f ). For all real r > 0, the path γ x ([−r, r]) is a subpath of [f k (ρ)] for some k ≥ 1 since x is contained in some f -iterate of s. Thus γ x | [−r,r] has a lift γ x,r : [−r, r] → S(H). There are only finitely many preimages of x in S(H), so after passing to an unbounded increasing subsequence r m , we can assume the sequence (γ x,rm (0)) ∞ m=1 is constant. By uniqueness of lifts, γ x,r m+1 is an extension of γ x,rm for m ≥ 1. The limit immersion γ x,∞ : R → S(H) is a lift of γ x and we say S(H) supports a leaf of Λ(f ). Subgroups Invariant Under Irreducible Endomorphisms As H is finitely generated, S(H) is a finite graph and we can add edges to S(H) if necessary to extend the immersion S(H) → Γ to a finite cover S(H ) → Γ corresponding to H ≤ H ≤ F . Thus [F : H ] < ∞ and we can apply Lemma 5.2 to get j > k ≥ 0 such that (F ) . Recalling the diagram preceding Lemma 4.3, the graph S(φ k (F )) = S k and the cover corresponding to φ k (K) ≤ φ k (F ) lie in the following commutative diagram: Subgroups Invariant Under Irreducible Endomorphisms is a finite cover. The map h k maps f k -periodic points to g-periodic points and v k maps g-periodic points to f k -periodic points. This allows us to identify Λ(f ) with Λ(g). Subgroups Invariant Under Irreducible Endomorphisms Let Subgroups Invariant Under Irreducible Endomorphisms . This is a noncontractible graph since φ(H) ≤ H. Note that S(φ k (K)) is the pullback of S(φ k (F )) → Γ and S(H ) → Γ and ∆ H is the pullback of S(φ k (F )) → Γ and S(H) → Γ. As S(H) is a subgraph of S(H ), we have that ∆ H is a subgraph of S(φ k (K)). The graphs fit in this commutative diagram: Subgroups Invariant Under Irreducible Endomorphisms Since S(φ k (F )) and S(H) support a leaf of Λ(f ), it follows that their pullback ∆ H supports a leaf of Λ(f ). Following the identification Λ(f ) ∼ = Λ(g), we have that ∆ H supports a leaf of Λ(g). Lemma 5.2 says φ j−k induces a permutation of Subgroups Invariant Under Irreducible Endomorphisms with the property: if g j−k (x) = x, thenĝ permutes the elements of p −1 (x). Subgroups Invariant Under Irreducible Endomorphisms The rest of the argument follows that of Bestvina-Feighn-Handel [2, Lemma 2.1]. We include the details for completeness; see also [10, Proposition 4.2]. Subgroups Invariant Under Irreducible Endomorphisms Proof. Let {a , b } be a turn at a vertex v ∈ S(φ k (K)) such that its projection under p, {a, b}, is an edge of the Whitehead graph of g j−k at p(v). Since g j−k is a clean map, we can replace it with an iterate and assume g j−k (a) = . . . ab . . .. So a contains a g j−k -fixed point x and, consequently,ĝ permutes the lifts p −1 (x). Replace g j−k with an iterate if necessary and assumeĝ fixes p −1 (x) and let x ∈ p −1 (x) be the lift of x in a . Then x is â g-fixed point andĝ(a ) = . . . a b . . .. An identical argument shows thatĝ(b ) = . . . a b . . . of φ l−1 (F ) . As φ l−1 is injective, we have φ| φ l−1 (F ) is conjugate to φ. Therefore, φ l (F ) is not contained in a proper free factor of φ l−1 (F ) and K ∩ φ l−1 (F ) = φ l−1 (F ) . Therefore φ l−1 (F ) ≤ K. The induction hypothesis is that φ l−1 (F ) is not contained in a proper free factor of F , hence K = F . So φ l (F ) is not contained in a proper free factor of F . Subgroups Invariant Under Irreducible Endomorphisms By induction, φ n (F ) is not contained in a proper free factor of F . We also have i g •φ n (F ) is not contained in a proper free factor of F since i g is an automorphism. By the same induction argument, (i g • φ n ) k (F ) is not contained in a proper free factor of F . Therefore, (i g • φ n ) k (F ) ≤ A implies A =
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