Abstract
Abstract. We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space.
We apply this to characterize hyperbolically embedded subgroups in terms of nice actions on (relatively) hyperbolic spaces. We also study the divergence of (properly) relatively hyperbolic groups, in particular showing that it is at least exponential.
Our main tool is the generalization of a result proved by Bowditch for hyperbolic spaces: if a family of paths in a space satisfies a list of properties specific to geodesics in a relatively hyperbolic space then the space is relatively hyperbolic and the paths are close to geodesics.
Introduction
The notion of (strong) relatively hyperbolicity first appeared in [Gro87] , see also [Bow99, Far98] , and has been studied only in the context of groups until Druţu and Sapir [DS05a] gave a metric characterization and discovered several properties of the geometry of Cayley graphs of relatively hyperbolic groups. Many examples of relatively hyperbolic spaces that are not Cayley graphs of relatively hyperbolic groups occur "in nature", as we will show in the last section of the present paper. It has to be noted that the term "asymptotically tree-graded space" in use in, e.g., [DS05a] and [Dru09] , is replaced everywhere in this paper by the term "relatively hyperbolic space". We use the latter term as the terminology in the group case is well-established and asymptotically tree-graded/relatively hyperbolic spaces are generalizations of (Cayley graphs of) relatively hyperbolic groups.
We generalize the characterizations of relative hyperbolicity given in the context of groups in [Gro87, Bow99, GM08] in terms of what we will call Bowditch space and in [Far98] in terms of the coned-off graph, under a mild (necessary) hypothesis on the collection of candidate peripheral sets that is always satisfied for groups. These characterizations will be referred to as (RH1) and (RH2), and their precise statements are given in Section 3. The characterization (RH0) is similar to one that can be found in [Dru09] (which we will take as our main definition of relative hyperbolicity), except that it requires control of geodesic triangles only, as opposed to almost-geodesic triangles.
Theorem 1.1. Let X be a geodesic metric space and let P be a collection of uniformly coarsely connected subsets of X. Then X is hyperbolic relative to P if and only if it satisfies (RH0), (RH1), (RH2) or (RH3).
(Coarse connectedness is not required for (RH0),(RH3).) Moreover, we obtain some information regarding quasi-geodesics in X in terms of geodesics in the Bowditch and coned-off spaces, see Corollaries 4.10 and 4.12.
The last characterization (RH3) is possibly, among those found so far, the characterization of relative hyperbolicity that is nearest to the standard definition of hyperbolicity. It relies on a relative Rips condition which prescribes that every transient point on a side of a geodesic triangle is close to a transient point on one of the other sides. A point on a geodesic is transient, roughly speaking, if it is not the midpoint of a long subgeodesic with endpoints in a suitable neighborhood of some peripheral set.
An informal statement of our main tool, Theorem 4.2, is the following.
Guessing Geodesics Lemma: Let X be a geodesic metric space and P a collection of subsets of X. Suppose that for each pair of points x, y a path η(x, y) connecting them and a subset trans(x, y) ⊆ η(x, y) have been assigned. Suppose that the pairs trans(x, y)/η(x, y) behave like the pairs transient points/geodesics in a relatively hyperbolic space (e.g. they satisfy the relative Rips condition). Then X is hyperbolic relative to P and the set trans(x, y) coarsely coincides with the set of transient points on a geodesic from x to y.
There are a few variations of the Guessing Geodesics Lemma for hyperbolic spaces, the first of which appeared in [Bow06] , where it is used to show hyperbolicity of curve complexes. Our formulation is closer to that given in [Ham07] . Here is an example of how we will use the Guessing Geodesics Lemma. When P is a collection of (say connected) subsets of the metric space X, the Bowditch space associated to the pair (X, P) is obtained gluing combinatorial horoballs to each P ∈ P (the Bowditch space also depends on other choices, this is ignored here). Suppose that the Bowditch space for the pair (X, P) is hyperbolic. Then for each pair of points x, y in X we can define a path connecting them by substituting each subpath of a geodesic γ from x to y contained in a horoball with any path in the corresponding P ∈ P, and define trans(x, y) = γ ∩ X. Due to the hyperbolicity of the Bowditch space, the relative Rips condition is easily checked (and the other properties required by the Guessing Geodesics Lemma are easily checked as well). Hence, we will easily obtain that if the Bowditch space for the pair (X, P) is hyperbolic then X is hyperbolic relative to P (for P coarsely connected).
The last section is dedicated to applications. We show that a construction due to Bestvina, Bromberg and Fujiwara [BBF10] gives rise to relatively hyperbolic spaces. This is weaker than a very recent result by Hume [Hum12] , but it will suffice for our further applications. We also provide alternative characterizations of the notion of hyperbolically embedded subgroup, as defined in [DGO11] . Examples of hyperbolically embedded subgroups include peripheral subgroups of relatively hyperbolic groups and maximal virtually cyclic subgroups containing a pseudo-Anosov (resp. a rank one element, iwip, element acting hyperbolically) in a mapping class group (resp. a CAT (0) group, Out(F n ), a group acting acylindrically on a tree), and therefore it is a quite general notion. We show that, when H is a subgroup of G, H is hyperbolically embedded in G, as defined in [DGO11] , if and only if G acts with certain properties on a relatively hyperbolic space, and also if and only if G acts with certain properties on a hyperbolic space with H acting parabolically (in both cases, the extra condition of properness for the actions gives back a definition of relative hyperbolicity). Here is a simplified version of one of the results. Theorem 1.2. Let {H λ } λ∈Λ be a finite collection of subgroups of the group G and let X be a (possibly infinite) generating system for G. Then {H λ } is hyperbolically embedded in (G, X) if and only if the Cayley graph Γ = Cay(G, X) is hyperbolic relative to the left cosets of the H λ 's and d Γ | H λ is quasi-isometric to a word metric.
This result implies in particular that an infinite order element is weakly contracting as defined in [Sis11] if and only if it is contained in a virtually cyclic hyperbolically embedded subgroup, see Corollary 6.9. We will use it to show, with an additional hypothesis, that hyperbolically embedded subgroups are preserved in subgroups of the ambient group, i.e. that if H is hyperbolically embedded in G and K < G then H ∩ K is hyperbolically embedded in K (Corollary 6.10). We will also study the divergence function of relatively hyperbolic groups. Roughly speaking, the divergence of a metric space measures the length of detours avoiding a specified ball as a function of the radius of the ball. It has been first studied in [Gro93] and [Ger94] , and in recent years in [Beh06, OOS09, DR09, DMS10, BC12, BD11] .
The divergence of one-ended hyperbolic groups is known to be exponential; the exponential upper bound is folklore, even though there seems to be no published reference for it. In Subsection 6.3 we show the following. Theorem 1.3. Suppose that the one-ended group G is hyperbolic relative to the (possibly empty) collection of proper subgroups H 1 , . . . , H k . Then
We make the observation that the divergence of a one-ended finitely presented group (e.g., a hyperbolic group) is at most exponential (Lemma 6.15), a fact that does not seem to be recorded in the literature.
It is well-known that finite configurations of points in a hyperbolic space can be approximated by trees, and this is very helpful to reduce computations in hyperbolic spaces to computations in trees. In Subsection 6.4 we will show a similar result for relatively hyperbolic spaces.
Finally, in Subsection 6.5 we give an example of how the Guessing Geodesics Lemma can be used to show a known combination result for relative hyperbolicity.
Further applications to certain notions of boundary at infinity of a space will be presented in [Sis12] .
Acknowledgement. The author would like to thank Cornelia Druţu, Vincent Guirardel and Denis Osin for very helpful comments and suggestions.
Main definition of relative hyperbolicity
The following definitions are taken from [DS05a, Dru09] (except that of being weakly ( * )−asymptotically tree-graded, which is relevant to (RH0)). We will use the notation N K (A) = {x ∈ X|d(x, A) ≤ K} for the K−neighborhood of the subset A of the metric space X.
Definition 2.1. The collection P of subsets of the geodesic metric space X is said to satisfy (α 1 ) if for each K there exists B so that diam(
Definition 2.2. The collection P of subsets of the geodesic metric space X is said to satisfy (α 2 ) if there exists ∈ (0, 1/2) and M ≥ 0 so that for each P ∈ P and x, y ∈ X with x, y ∈ N d(x,y) (P ) any geodesic from x to y intersects N M (P ).
Continuous quasi-geodesics will be referred to as almost-geodesics. Notice that in a geodesic metric space each quasi-geodesic is within bounded Hausdorff distance from an almost-geodesic (with related constants).
Definition 2.3. The geodesic metric space X is said to be ( * )−asymptotically tree-graded (resp. weakly ( * )−asymptotically tree-graded) with respect to the collection of subsets P if the following holds. For every C ≥ 0 there exist σ, δ so that every triangle with (1, C)−almost geodesic edges (resp. every geodesic triangle) γ 1 , γ 2 , γ 3 satisfies either (C) there exists a ball of radius σ intersecting all sides of the triangle, or (P ) there exists P ∈ P with N σ (P ) intersecting all sides of the triangle and the entrance (resp. exit) points x i (resp. y i ) of the sides γ i in (from)
If a group is ( * )−asymptotically tree-graded with respect to the left cosets of a collection of subgroups H 1 , . . . , H n then it is said to be ( * )−hyperbolic relative to H 1 , . . . , H n . This property is relevant to the Rapid Decay property, important in K-theory: it is shown in [DS05b] that if G is ( * )−hyperbolic relative to H 1 , . . . , H n then G satisfies the Rapid Decay property if and only if H 1 , . . . , H n do.
We take the following as our main definition of relative hyperbolicity.
Definition 2.4.
[Dru09] The geodesic metric space X is hyperbolic relative to the collection of subsets P, called peripheral sets, if P satisfies (α 1 ) and (α 2 ) and X is ( * )−asymptotically tree-graded with respect to P.
The definition above is known to be equivalent to the definition given in [DS05a] in terms of asymptotic cones as well as the definition given in [Sis10] in terms of (almost) closest-points projections on the peripheral sets, so that we can use results from the said papers.
The following a priori less restrictive definition will turn out to be equivalent to the one above, see Proposition 5.1.
Definition 2.5. The pair (X, P), where X is a geodesic metric space and P is a collection of subsets, is said to satisfy (RH0) if P satisfies (α 1 ) and (α 2 ) and X is weakly ( * )−asymptotically tree-graded with respect to P.
We record the following result that will often be tacitly used.
Lemma 2.6. [DS05a] Let P, Q be collections of subsets of the geodesic metric space X. Suppose that there exists K ≥ 0 and a bijection between P and Q so that corresponding elements are at Hausdorff distance bounded by K. Then X is hyperbolic relative to P if and only if it is hyperbolic relative to Q.
We will say that a (finitely generated) group is hyperbolic relative to the collection of its (finitely generated) subgroups H 1 , . . . , H n if some (any) Cayley graph of G is hyperbolic relative to the collection of the left cosets of the H i 's.
Convention 2.7. All groups we consider in the present paper will be tacitly assumed to be finitely generated.
The characterizations
When Γ is a graph, we will denote by Γ 0 and Γ 1 the vertex set and edge set of Γ.
Definition 3.1. Suppose Γ is a connected metric graph. The horoball H(Γ) is defined to be the graph with vertex set Γ 0 × N and edges ((v, n), (v, n + 1)) of length 1, for all v ∈ Γ 0 , n ∈ N, and edges ((v, n), (v , n)) of length e −n l((v, v )), for all (v, v ) ∈ Γ 1 .
We will call vertical ray a geodesic ray in H(Γ) obtained by concatenating all edges of type ((v, n), (v, n + 1)), for some v ∈ Γ 0 . It is readily seen that it actually is a geodesic ray. It can be checked that H(Γ) is δ−hyperbolic for some δ depending on the supremum of the lengths of the edges of Γ (if finite). In fact, the horoballs as defined above are quasi-isometric to the combinatorial horoballs defined in [GM08] . (One can also derive this directly from a description of the geodesics in H(Γ) as the concatenation of two vertical and one short horizontal segment, see [GM08, Lemma 3 .10] or [MS11, Lemma 2.2].) Also, for each n the level Γ 0 ×{n} lies within uniformly bounded Hausdorff distance from a horosphere defined in the usual sense, as a level set of a Busemann function. Definition 3.2. A maximal k−net in a metric space X is a maximal collection of distinct points at reciprocal distance at least k. An approximation graph Γ with constants k, R for a metric space X is a connected metric graph whose vertex set is a maximal k−net in X and so that two distinct vertices are connected by an edge of length R if and only if their distance (as points of X) is at most some fixed R.
We emphasize that approximation graphs are required to be connected, so that they can be endowed with the path metric. Such metric is clearly quasi-isometric to that of X when X is, say, a length space.
Definition 3.3. Let X be a geodesic metric space and let P be a collection of subsets of X. Fix approximation graphs Γ P for each P ∈ P. The Bowditch space Bow(X, {Γ P } P ∈P ) is the metric space obtained identifying Γ 0 P ⊆ H(Γ P ) with the corresponding maximal net in P for each P ∈ P. We will often write Bow(X) instead of Bow(X, {Γ P }). Notice that the metric d X of X and the restriction to X of the metric of Bow(X) are coarsely equivalent. The following is the metric analogue of the characterization of relatively hyperbolic groups that can be found in [Bow99, GM08] .
Definition 3.4. The pair (X, P) satisfies (RH1) if each (some) Bowditch space Bow(X) is hyperbolic.
The equivalence of each/some, under a mild hypothesis on P, will be shown later, in Proposition 4.9.
Definition 3.5. Let X be a geodesic metric space and let P be a collection of subsets of X. Fix maximal k−nets N P for each P ∈ P. The coned-off spaceX {N P } is the metric space obtained adding segments e x,y of length 1, called P -components, connecting all pairs of distinct points x, y ∈ N P , with P ∈ P.
We will often writeX instead ofX {N P } . The paths defined below will play the role that combinatorial paths play in coned-off graphs of groups. Definition 3.6. We will say that two P −components (for the same P ∈ P) are tied. A standard path in a coned-off spaceX {N P } is a concatenation of geodesics in X and P-components. A P−component of a standard path γ is isolated if it is not tied to any other P−components of γ. A standard path is said to be without backtracking if all its P−components are isolated. The length of a P−component e x,y is simply d X (x, y).
Definition 3.7. The coned-off spaceX {N P } is said to satisfy the BCP property if the following holds. For each L there exists K so that for each L−quasi-geodesic standard paths γ 1 , γ 2 without backtracking and with corresponding endpoints at d X -distance at most 1:
(1) If γ 1 contains a P−component of length at least K then γ 2 contains a P−component tied to it. (2) If γ 1 , γ 2 contain, respectively, the P −components e x,y and e x ,y , for some
The following is the metric analogue of the characterization of relatively hyperbolic groups that can be found in [Far98] .
Definition 3.8. The pair (X, P) satisfies (RH2) if each (some) coned-off graphX is hyperbolic and the BCP property holds forX.
Again, the equivalence of each/some, under a mild hypothesis on P, will be shown later, in Proposition 4.11.
There is (at least) one more characterization of relatively hyperbolic groups that would be interesting to turn into a metric characterization, i.e. the one in terms of relative isoperimetric inequalities given in [Osi06b] .
The following definition is taken from [Hru10] .
Definition 3.9. Let (X, d) be a geodesic metric space and P a collection of subsets. When α is a path in X we will denote by deep µ,c (α) the subset of α defined in the following way. If x ∈ α then x ∈ deep µ,c (α) if and only if there exists a subpath β of α with endpoints x 1 , x 2 in N µ (P ) for some P ∈ P so that x ∈ β and d(x, x i ) > c. The set of transient points trans µ,c (α) of α is defined as α\deep µ,c (α).
Remark 3.10. We will later show that, roughly speaking, in a relatively hyperbolic space quasi-geodesics with the same endpoints have the same transient set, up to finite Hausdorff distance (Proposition 5.7).
Recall that the collection P of subsets of the geodesic metric space X is said to satisfy (α 1 ) if for each K there exists B so that diam(N K (P ) ∩ N K (Q)) ≤ B for each distinct P, Q ∈ P (Definition 2.1).
Definition 3.11. The pair (X, P) satisfies (RH3) if there exist µ, R 0 so that (1) (α 1 ) holds; (2) for each k and R ≥ R 0 there exists
The content of (2) is that when x, y are close to a peripheral set then any transient point between x, y is close to either x or y, and also there is a transient point in a specified neighborhood of the peripheral set.
In Proposition 4.6 we will show that (RH3) is another characterization of relative hyperbolicity.
Guessing geodesics
We wish to generalise the following result, which is similar to a an earlier result due to Bowditch [Bow06] . For a path α and p, q ∈ α we denote by α| [p,q] the subpath of α joining p to q.
Proposition 4.1. [Ham07, Proposition 3.5] Let X be a geodesic metric space. Suppose that for all x, y ∈ X there is an arc η(x, y) connecting them so that the following hold for some D.
(
for all x, y, z. Then X is hyperbolic and there exists κ so that
The second part of the conclusion does not appear in the statement of [Ham07, Proposition 3.5], but the proof strategy, as explained in the first paragraph of the proof, is to show the second part and then deduce the first one.
Theorem 4.2. Let (X, d) be a geodesic metric space and P a collection of subsets. Suppose that for each pair of points x, y ∈ X a path η(x, y) connecting them and a closed subset trans(x, y) ⊆ η(x, y) have been assigned so that the following conditions hold for some large enough D.
(
(2) for all x , y ∈ η(x, y), we have
where y) ) for all x, y, z ∈ X; (4) if x , y ∈ η(x, y) do not both lie on P ∈ P for any P , then there exists z ∈ trans(x, y) between x , y ; (5) (α 1 ); (6) for every k there exists K so that if d(x, P ), d(y, P ) ≤ k and d(x, y) ≥ K then trans(x, y) ⊆ B K (x)∪B K (y) and there exists z ∈ trans(x, y)∩ N D (P ). Then (a) X is hyperbolic relative to P; (b) for every C there exist µ, c 0 , L with the following property. For any c ≥ c 0 and (1, C)−almost-geodesic β with endpoints x, y the Hausdorff distance between trans(x, y) and trans µ,c (β) is at most some L + c.
The fact that transient sets in relatively hyperbolic spaces satisfy the properties listed above is the content of Lemma 4.7.
We will first show part (b). Given part (b), we will be able to show a new characterization of relative hyperbolicity. Using this characterization, part (a) follows immediately.
Proof of part (b). Let us start with some general remarks. First of all, for each n−gon η(x 1 , x 2 ) ∪ · · · ∪ η(x n , x 1 ), with n ≥ 3, we have
In particular, subdividing a geodesic from x to y and using 1, we get
Notice that by 2) with x = x, y = y for each x, y there is p ∈ trans(x, y) with d(p, x) ≤ D. For convenience we will always assume x, y ∈ trans(x, y).
Lemma 4.3. Up to increasing D, we can substitute 6) with the following:
Proof. Fix the notation of 6 ), where M is large enough to carry out the following argument. Consider x , y ∈ P so that d(x, x ) ≤ d(x, P )+1, d(y, y ) ≤ d(y, P ) + 1. Fix K as in 6) for k = 4D + 1. Then by 6) with k = 0 we have that trans(x , y ) is contained in the K−neighborhood of {x , y }, so that keeping into account diam(trans(x, x )) ≤ Dd(x, x ) + D and the same estimate for y, y we have
and we get trans(x, y) ⊆ N M k ({x, y}) as trans(x, y) ⊆ N 2D (A). The second part is obvious for d(x, P ) ≤ 4D + 1 or d(y, P ) ≤ 4D + 1 (as we allow an increase of D), so assume that this is not the case. Consider the first points x , y in η(x, x ), η(y, y ) at distance 4D + 1 from P . By 6) there exists z ∈ trans(x , y ) ∩ N D (P ) ∩ B K (x ). Such z cannot be 2D−close to trans(x, x ) (keeping 2) into account) and trans(y, y ) (as we can bound d(x, z) linearly in d(x, P )). Hence, it is 2D−close to trans(x, y), so that trans(x, y) intersects N 3D (P ). We are done up to increasing D to 4D+1.
Lemma 4.4. Let α be any rectifiable path connecting, say, x to y. Then
Proof. Indeed, this holds if α has length at most 2 by condition 1). Suppose l(α) ≥ 2. Let α 1 , α 2 be subpaths of α of equal length and let z be the common endpoint. Then, by 3), trans(x, y) ⊆ N D (trans(x, z)∪trans(z, y)).
On the other hand, we can assume by induction that trans(x, z), trans(z, y) are both contained in the (D log 2 (l(α)/2) + D)−neighborhood of α. So, trans(x, y) is contained in the (D log 2 (l(α)/2) + 2D)−neighborhood of α, and we are done as
We will also need "logarithmic quasi-convexity", i.e. the following estimate.
Lemma 4.5. Suppose that α is a (1, C)-almost geodesic joining x to y. Denote
Notice that ξ(α) is bounded logarithmically in l(α) by Lemma 4.4.
We are now ready for the core of the proof and we will proceed as follows. We first show (I) that any point in trans(x, y) is close to any (1, C)-almostgeodesic β connecting x to y. Then we show (II) that points in trans µ,c (β), for suitable µ, c , are close to trans(x, y), and in order to do so we will actually show that if p ∈ β is far from trans(x, y) then p ∈ deep µ,c (β). Finally, we show (III) that points in trans(x, y) are close to trans µ,c 0 (β) for some c 0 ≥ c , but we will actually show that points in deep µ,c 0 (β) are far from trans(x, y). Furthermore, µ, c will be chosen so that each deep µ,c (β) is a disjoint union of subpaths each contained in a large subpath with endpoints in N µ (P ) for some P ∈ P. It is easily deduced then that
Let now β be a (1, C)−almost-geodesic connecting x to y.
Step (I). For the purposes of this step, we can assume that β is C−Lipschitz, up to increasing the constant we find with the following argument. In fact, any (1, C)−quasi-geodesic lies within Hausdorff distance C from a C −Lipschitz (1, C )−quasi-geodesic, where C = C (C). Let p ∈ trans(x, y) be the point at maximal distance from β. We want to give a bound on d(p, β). Let q be a closest point to p in β and set d(p, q) = ξ. We can assume ξ ≥ 4D + 1 and l(β) ≥ 1, for otherwise we are done. Also, up to substituting β with a suitable subpath, we can assume ξ ≥ ξ(β) − 1, as defined in Lemma 4.5.
Consider the closest x ∈ trans(x, y) before p so that d(x , p) ≥ min{2ξ, d(p, x)}, and define y similarly. By 2) we have d(p, p 0 ) ≤ D for some p 0 ∈ trans(x , y ). The proof in the hyperbolic setting (in the case x = x, y = y) relies on d(x , p), d(y , p) = 2ξ, but a linear upper bound in ξ would make the proof work as well. So, we would like to reduce to the case when x is either x or is not too far from p 0 (and similarly for y ) by substituting x with some w if necessary. Set w = x and
is large enough to allow us to carry out the following argument. If d(x , p) is large, by 4) we have that x , p lie in N 2ξ (P ) for some P ∈ P. Let x ∈ β be so that
Up to running the same argument on a final subpath of η(w , y ) in case d(y , p) is large, we get a point p 2 (possibly
and similarly for z . Consider now a path α obtained concatenating in the suitable order a geodesic of length at most L + 1 from w to a ∈ β, a geodesic of length at most L + 1 from b ∈ β to y and β| [a,b] (choose a = x and/or b = y if x = x and/or y = y). The length of α is at most
Step (Q-C). We now show that (1, C)−almost-geodesics with endpoints in N µ (P ) for some P ∈ P are contained in N µ (P ), where µ depends on C (but not on P ), a fact that will be needed later. As a consequence of this and 5), for each c large enough deep µ,c (β) is a disjoint union of subpaths each contained in a large subpath with endpoints in N µ (P ) for some P ∈ P, as subpaths of β with endpoints in N µ (P ) and N µ (P ) for P, P ∈ P with P = P have controlled intersection. Suppose that an almost-geodesic α as above contains a subpath α outside N µ+D+1 (P ) with endpoints x, y. Then, if by contradiction d(x, y) is larger than some suitable R, by 6) there exists z ∈ trans(x, y) ∩ N D (P ). But such point cannot be µ−close to α. So, d(x, y) ≤ R and hence α ⊆ N µ =µ+D+R+C+1 (P ), which in turn gives α ⊆ N µ (P ).
Step (II). Fix c as determined in (Q-C). Let now p ∈ β be at distance at least max{c + µ, 2µ + C} + 1 from trans(x, y). We wish to show that p ∈ deep µ,c (β). Let p 1 , p 2 be the closest points on the sides of p in β at distance at most µ from trans(x, y). Clearly, d(p 1 , p 2 ) ≥ max{2c , 2µ + 2C} + 1. Let q 1 be the last point on trans(x, y) so that d(q 1 , β 1 ) ≤ µ, where β 1 is the sub-quasi-geodesic of β with final point p 1 . Define q 2 and β 2 similarly. Notice that there are no points in trans(x, y) between q 1 and q 2 , as any point in trans(x, y) is µ−close to either a point in β 1 or a point in β 2 and
In particular, q 1 , q 2 both lie on some P ∈ P by 4), which implies (considering
Step (III). Finally, we have to show that if p ∈ deep µ,B (β), where B will be determined later, then d(p, trans(x, y)) > µ. Let p 1 , p 2 be the endpoints of the subpath of β contained in the neighborhood of some P ∈ P as in the definition of deep µ,B (β) (in particular,
) for any P, P ∈ P with P = P and for k as in (Q-C). We have, by Step (II), d(p i , q i ) ≤ max{c + µ, 2µ + C} + 1 = A for some q i ∈ trans(x, y). In particular, d(q i , P ) ≤ A + µ so that for B large enough we have, by 6) and 2),
for K as in 6) with k = A + µ. This easily implies the claim, up to further increasing B, because for B large enough
as β is a (1, C)−quasi-geodesic. This concludes the proof of part (b).
As mentioned earlier, we now use (b) to give the characterization of relative hyperbolicity (RH3) below. In the setting of Theorem 4.2, the conditions stated in (RH3) are readily checked in view of (b), as they basically are some of the condition in Theorem 4.2 with trans µ,R substituting trans. (Substituting trans with trans µ,R allows us to drop some of the "coherence" conditions we had to require in Theorem 4.2.) In particular, the implication ⇐ below concludes the proof of Theorem 4.2.
Proposition 4.6 (Characterization (RH3)). The geodesic metric space X is hyperbolic relative to P ⇐⇒ there exist µ, R 0 so that
Proof. ⇐: First of all, let us show that any geodesic γ with endpoints y 1 , y 2 in N µ (P ), for some P ∈ P, is contained in N µ (P ) for a suitable µ . Indeed, suppose that x ∈ γ lies outside N µ (P ) and let [x 1 , x 2 ] be the minimal subgeodesic of γ containing x with endpoints in N µ (P ). Consider the points
was sufficiently large, we would get a contradiction with 2) as [
As a consequence, we see that, given R ≥ R 0 large enough, deep µ,R (γ) is a disjoint union of subgeodesics each with both endpoints in N µ (P ) for some P ∈ P, for any geodesic γ. In fact, if γ 1 , γ 2 are subgeodesics of γ so that the endpoints of γ i are in N µ (P i ) and P 1 = P 2 then we can assume by 1) that diam(γ 1 ∩ γ 2 ) ≤ R. Fix such R from now on and let D be the constant given by 3).
Another consequence, in view of 2), is that there exists E so that for all w, z ∈ [x, y] we have
(Just consider the cases when w, z are/are not in trans µ,R ([x, y]).)
In particular, it is now readily checked that all conditions of Theorem 4.2 are satisfied for η(x, y) = [x, y] any geodesic and trans(x, y) = trans µ,R (x, y) (with P substituted by {N µ (P )} P ∈P in order to ensure 4)). In view of part (b), the argument to show ( * )−asymptotic tree-gradedness below work, with suitable constants, for (1, C)−almost-geodesics as well as geodesics. For simplicity, we will spell out the proof for geodesics only.
Let us check ( * )−asymptotic tree-gradedness. We will use the readily checked property that for each geodesic n−gons γ 1 ∪ · · · ∪ γ n , with n ≥ 3, we have
Let us also show the following preliminary fact.
Claim 1:
If Q is large enough we can argue as follows. By 2) and the quasiconvexity (which can be shown with the same argument for almost-geodesics as well) of each N µ (P ) there exists L so that for all w ∈ [y 0 , z 0 ] we have
Consider a pentagon with vertices w 0 , z 0 , y 1 , z 1 , y 2 . By 2) there exists p 0 ∈ trans([w 0 , y 2 ])∩N µ (P 2 ). It is readily checked that p 0 cannot be 3D−close to
Consider a geodesic triangle with sides γ 0 , γ 1 , γ 2 with γ i = [x i , x i+1 ]. Suppose that there exists i and y ∈ trans µ,R (γ i ) so that d(y, trans µ,R (γ i±1 ) ≤ Q + D, for Q as in Claim 1 with K = D + 2R. Then the triangle clearly falls into case (C). Hence, suppose that this is not the case. Then there exist deep components [y i , z i ] ⊆ γ i with d(z i , y i+1 ) ≤ D + 2R. Let P i be so that y i , z i ∈ N µ (P i ). Claim 1 implies P 0 = P 1 = P 2 , and we are done.
Let us now verify the following stronger version of (α 2 ). Claim 2: If x 1 , x 2 ∈ X and P ∈ P have the property that d(
Otherwise, a similar argument can be carried out with w i = x i and considering a triangle instead of a quadrangle. Consider a geodesic quadrangle with vertices x 1 , x 2 , w 2 , w 1 . By 2), there exists
⇒: We will show (RH3) starting from the weaker hypothesis (RH0) in Proposition 5.1.
In retrospect, i.e. once the proposition is established, we see that in the first part of the proof of the proposition we have shown the following lemma.
Lemma 4.7. Let X be hyperbolic relative to P. All hypotheses of Theorem 4.2 are satisfied substituting P with {N µ (P )} P ∈P and for η(x, y) = [x, y] a geodesic, trans(x, y) = trans µ,R ([x, y]), for some suitably chosen µ, µ , R. Definition 4.8. A subset P of a metric space X is K−coarsely connected if for all x, y ∈ P there exists a chain x = x 0 , . . . , x n = y of points x i ∈ P satisfying d(x i , x i+1 ) ≤ K. A collection P of uniformly coarsely connected subsets is a collection of K−coarsely connected subsets for some K.
Recall that we defined Bowditch spaces in Definition 3.3.
Proposition 4.9 (Characterization (RH1)). Let X be a geodesic metric space and let P be a collection of uniformly coarsely connected subsets of X. Then (X, P) is relatively hyperbolic ⇐⇒ each (some) Bowditch space Bow(X) is hyperbolic.
Moreover, given a model of Bow(X) there exist K, µ, R 0 so that for each geodesic γ in Bow(X) connecting x, y ∈ X we have that γ ∩ X is within
The following is an easy consequence of the proposition.
Corollary 4.10. Suppose that (X, P) is relatively hyperbolic and fix a model for Bow(X). Then there exists K so that if γ is a geodesic in Bow(X) then substituting each subpath of γ lying outside X with a geodesic in X we get a (K, K)−quasi-geodesic.
Proof of Proposition 4.9. ⇐: First of all, let us show that all combinatorial horoballs H = H(P ) in Bow(X) are within bounded Hausdorff distance from actual horoballs. As this is true in the path metric of H, we just need to show that any geodesic with endpoints in H is contained in H except in uniformly bounded balls around its endpoints. This is easy to see: consider x, y ∈ P , a geodesic γ in Bow(X) connecting them and the vertical rays γ x , γ y at x, y. Such vertical rays are geodesic rays in Bow(X), and their Hausdorff distance is finite. The ideal triangle γ, γ x , γ y is thin, and this easily implies that γ is contained in H, except in balls around the endpoints of radius bounded in terms of the hyperbolicity constant.
For convenience, we will assume that each P ∈ P is path-connected, which we can guarantee by adding suitable paths to X in view of uniform coarse connectedness. We can use Theorem 4.2 applied to η(x, y) constructed substituting each subpath of [x, y] ⊆ Bow(X) contained in some combinatorial horoball with a path in P with the same endpoints, with trans(x, y) being [x, y] ∩ X. In view of the previous paragraph and the fact that the metric d Bow(X) is coarsely equivalent to d X on X, all conditions are readily checked. More precisely, 1) and 4) are clear, 5) and 6) follow from the corresponding statements in Bow(X) and 2) can be proven in a similar way to 3), which we are about to show. Consider a geodesic triangle in Bow(X) with vertices x, y, z ∈ X. Consider a point w ∈ [x, y] ∩ X and assume without loss of generality that there exists p ∈ [y, z] with d Bow(X) (p, w) ≤ δ, the hyperbolicity constant. As y, z ∈ X and we can regard the levels of H as horospheres, there is on [y, z]∩X a point q not too far away from p, meaning that d Bow(X) (p, q) is bounded by some constant depending on Bow(X) only. Given this, 3) follows from the coarse equivalence of d Bow(X) and d X on X.
⇒: As earlier, assume that P is path-connected. Consider any Bowditch space Bow(X). Let φ : Bow(X) → X be the map restricting to the identity on X and mapping in the natural way each combinatorial horoball to the corresponding P ∈ P. Define η(x, y) to be the concatenation of a geodesic in the horoball from x to φ(x), any geodesic γ x,y in X from φ(x) to φ(y) and a geodesic in the horoball from φ(y) to y. Also, let trans(x, y) = trans µ,R (γ x,y ), where µ, R are given by Lemma 4.7. The hypotheses of Theorem 4.2 are satisfied in the metric of Bow(X) as well, because the restriction of such metric on X is coarsely equivalent to the metric of X. In particular, Bow(X) is hyperbolic relative to the collection of combinatorial horoballs, which are themselves hyperbolic. Hence, Bow(X) is hyperbolic.
Proposition 4.11 (Characterization (RH2)). Let X be a geodesic metric space and let P be a collection of uniformly coarsely connected subsets of X.
Then (X, P) is relatively hyperbolic ⇐⇒ each (some) coned-off graphX is hyperbolic and the BCP property holds forX.
Moreover, given a model ofX there exist K, µ, R 0 so that geodesics in X are within HausdorffX−distance K from geodesics inX and, conversely, if γ is a geodesic inX connecting x, y ∈ X then γ ∩ X is within Hausdorff X−distance K + R from trans µ,R ([x, y]), for each R ≥ R 0 .
Corollary 4.12. Suppose that (X, P) is relatively hyperbolic and fix a model forX. Then there exists K so that if γ is a geodesic inX then substituting each component of γ with a geodesic in X we get a (K, K)−quasi-geodesic.
Proof of Proposition 4.11. ⇒: Hyperbolicity ofX and the fact that geodesics in X are close in the dX metric toX−geodesics follow from Proposition 4.1 using Proposition 4.6-(3) (the requirement for the paths η(x, y) to be arcs is inessential, as one can see taking a product of X with [0, 1], for example). We have to show the BCP property. For each P ∈ P denote by π P : X → P be a map satisfying d(x, π P (x)) ≤ d(x, P ) + 1. We will use the following lemmas.
Lemma 4.13. Let (X, P) be relatively hyperbolic. There exists K 0 with the following property. Consider a standard path α : I →X with endpoints x, y ∈ X that does not contain a P −component, for some P ∈ P. Then
Proof. π P is coarsely Lipschitz in X and also for each P ∈ P with P = P we have a bound on diam(π P (P )) [Sis10] . Hence, in order to prove the lemma we just need to consider a subdivision of α.
Lemma 4.14. Let (X, P) be relatively hyperbolic. Then there exists K 1 with the following property. For each L there exists K 2 so that if the L−quasigeodesic α inX with endpoints x, y ∈ X does not intersect BX K 2 (P ), for some P ∈ P, then
Proof. AsX is hyperbolic and geodesics in X are close to geodesics inX, by taking K 2 large enough we can assume that a geodesic in X from x to y stays as far as we wish, inX and hence in X, from P . The uniform bound then follows from (AP 2) in [Sis10] .
Lemma 4.15. Let (X, P) be relatively hyperbolic. Then for each L there exists K 3 with the following property. Let α be an L−quasi-geodesic standard path without backtracking with endpoints x, y. Suppose that α contains a P −component e p,q , for some P ∈ P.
Proof. Let α be the subpath of α from x to p. Subdivide α in subpaths α 1 , α 2 so that α 1 ∩ BX K 2 (P ) = ∅ and l(I) ≤ LK 2 + L 2 + 1. We get the desired bound applying the first lemma to α 2 and the second one to α 1 .
This lemma readily implies the second part of the BCP property. Let us show the first part. Consider L−quasi-geodesic standard paths γ 1 , γ 2 without backtracking with corresponding endpoints (in X and) at d X −distance at most 1 and suppose that γ 1 has a P −component e p,q . Also, suppose that γ 2 does not have a P −component. We have to provide a bound for d(p, q). Let K = max{K i }. We can split γ 2 into three subpaths α 1 , α 2 , α 3 so that α 1 , α 3 ∩ BX K (P ) = ∅ and l(I) ≤ 2LK + L 2 + 1, where α 2 : I →X. As α 2 does not contain a P −component, by the previous lemmas we easily get a bound on d X (p, q). For completeness, we remark that we just showed the following.
Lemma 4.16. Let (X, P) be relatively hyperbolic. Then for each L there exists K 4 with the following property. Suppose that d(π P (x), π P (y)) ≥ K 4 , for some x, y ∈ X and P ∈ P. Then any L−quasi-geodesic standard path without backtracking with endpoints x, y contains a P −component.
⇐: For convenience, we will assume that each P ∈ P is path-connected, which we can guarantee by adding suitable paths to X in view of uniform coarse connectedness. Let x, y ∈ X and consider a geodesic [x, y] inX connecting them. Define η(x, y) by substituting each P−components of [x, y] with paths in the corresponding P ∈ P. Also, let trans(x, y) be [x, y]\ e p,q , where the union is taken over all P−components of [x, y]. We want to show that Theorem 4.2 applies.
We note the following easy result.
Lemma 4.17. For each L there exists K 0 with the following property. If γ is an L−quasi-geodesic standard path so that all its P−components have length bounded by L, then for all x, y ∈ γ ∩ X we have d X (x, y) ≤ K 0 dX (x, y) + K 0 .
Condition 1) follows from the BCP property applied to a geodesic from x to y and a trivial path, together with the lemma. Condition 2) in the present setting follows from condition 3). In fact, it is enough to consider a polygon inX consisting of [x , y ], a suitable subpath of [x, y], possibly trivial geodesics in X of length at most K, where K is so that all P ∈ P are K−coarsely connected, and possibly trivial P−components.
Let us now show 3) (the containment is obvious in the dX −metric, not so in the d X −metric). The proof will also show that given L-quasi-geodesic standard paths without backtracking γ 0 , γ 1 with common endpoints we have a bound depending on L only on d X Haus (γ 0 ∩ X, γ 1 ∩ X). Consider a geodesic triangle γ 0 , γ 1 , γ 2 . We can form an L−quasi-geodesic standard path without backtracking α with the same endpoints as γ 0 by concatenating subpaths γ 1 , γ 2 of γ 2 and a suitable geodesic γ so that d(γ 0 , γ) > 1, where L depends on the hyperbolicity constant ofX only. If γ contains a sufficiently long P −component e p,q , then p, q are close to endpoints of a P −component of α, and hence of either a P −component of γ 1 or a P −component of γ 2 , as γ 0 and γ do not have tied P−components.
The proof of 3) is then readily completed given the following lemma.
Lemma 4.18. For each L there exists K with following property. Let α 0 , α 1 be L−quasi-geodesic standard paths without backtracking and with corresponding endpoints at distance at most L. Also, suppose that all P−components of α 0 and α 1 have length at most L. Then for each x ∈ α 0 ∩ X we have
Proof. There exists K 0 depending on L and the hyperbolicity constant ofX with the following property. First, if x, y ∈ γ 0 then d X (x, y) ≤ K 0 dX (x, y) + K 0 . Also, for x as in the statement, we can find a geodesic γ inX of length at most K 0 connecting x ∈ γ 0 to γ 1 , where d X (x , x) ≤ K 0 , and so that the concatenation α of an initial subpath of α 0 , γ and a final subpath of α 1 is a K 0 −quasi-geodesic standard path with no backtracking. By the BCP-property we see that there is a bound K 1 on the lengths of the P−components of α, which can in turn be used, together with l(γ) ≤ K 0 , to give a bound of d X (x, γ 1 ∩ X).
Condition 4) is clear. Property (α 1 ) can be shown as follows. If x, y ∈ N R (P 0 ) ∩ N R (P 1 ) for P 0 , P 1 ∈ P then we can construct an L−quasi-geodesic standard path γ i without backtracking from x to y, where L depends on R only, concatenating a geodesic in X from x to P i , a P i −component and a geodesic in X from P i to y. The BCP property implies that such components have bounded length, which then easily gives (α 1 ).
In the setting of 6), we can construct an L−quasi-geodesic standard path without backtracking γ from x to y, where L depends on k only, by concatenating a geodesic in X from x to P , a P −component and a geodesic in X from P to y. Recall that the proof of 3) gives that quasi-geodesic standard paths without backtracking γ 0 , γ 1 with common endpoints have the property that γ 0 ∩ X and γ 1 ∩ X are at bounded Hausdorff distance. In order to conclude the proof of 6) we just need to use this fact with γ 0 a geodesic from x to y and γ 1 = γ.
Weak ( * )−ATG suffices
In this section we show that in our main definition of relative hyperbolicity we can just require control on geodesics rather than (1, C)−almost-geodesics.
Proposition 5.1 (Characterization (RH0)). Let X be a geodesic metric space and let P be a subsets of X. Then (X, P) is relatively hyperbolic ⇐⇒ P satisfies (α 1 ) and (α 2 ) and X is weakly ( * )−asymptotically treegraded with respect to P.
Proof. It suffices to show that (RH0) implies (RH3). We will use the following lemmas, that are proven in [DS05a] and [Sis10] for relatively hyperbolic spaces. In the following lemmas, (X, P) is assumed to satisfy (RH3) and for P ∈ P and x ∈ X we denote by π P (x) a point so that d(x, π P (x)) ≤ d(x, P ) + 1. Let M be as in (α 2 ).
Lemma 5.2. There exists t so that for each L ≥ 1 any geodesic connecting points in N L (P ), for some P ∈ P, is contained in N tL (P ).
Proof. Fix and L ≥ 1 as in (α 2 ). We can assume L > M . Let γ be a geodesic as in the statement and consider a subgeodesic [x, y] of γ so that γ ∩ N L (P ) = {x, y}. We clearly cannot have l(γ ) ≥ L/ , which easily implies the lemma.
Lemma 5.3. There exists R such that each geodesic from x to N M (P ) intersects N R (π P (x)), for each x ∈ X and P ∈ P.
Proof. For convenience, let us assume M ≥ σ, for σ as in (P ). Let t be as in the previous lemma and B be a uniform bound on the diameters of N tM (P 1 ) ∩ N tM (P 2 ) for P 1 , P 2 ∈ P, P 1 = P 2 . Set R = max{tM + 4σ + 1, B + tM + 3δ + 1, M + δ + 1}.
Set p = π P (x) and consider some y ∈ N M (P ) and a geodesic γ from x to y. Suppose that a geodesic triangle with vertices x, y, p and such that γ is one of its sides falls under case (C). Then we have points a ∈ [x, p], b ∈ [p, y] and c ∈ γ with reciprocal distances at most 2σ. We have that d(a, p) ≤ tM + 2σ + 1, for otherwise
Let us then consider a geodesic triangle with vertices x, y, p such that γ is one of its sides which falls under case (P ). Consider A ∈ P and x 1 , x 2 , y 1 , y 2 , p 1 , p 2 as in case (P ), where, e.g., p 2 , y 1 ∈ [p, y] and p 2 is closer to p. Notice that d(p, p 1 ) ≤ tM + δ + 1, for otherwise we would have
Lemma 5.4. There exists L such that for each x, y ∈ X, P ∈ P, if d(π P (x), π P (y)) ≥ L, then any geodesic from x to y intersects B L (π P (x)) and B L (π P (x)).
Proof. Once again, let B be a uniform bound on the diameters of N tM (P 1 ) ∩ N tM (P 2 ) for P 1 , P 2 ∈ P, P 1 = P 2 . Set L = max{tR + 4σ + 1, 2R + 2σ + 2, B + tR + 3δ + 1, R, 2R + δ + 2}. Setx = π P (x) andŷ = π P (y). Consider a geodesic γ from x to y. Suppose that we have a case (C) geodesic triangle with vertices x,ŷ, y containing γ. Therefore, we have points a ∈ [x,ŷ], b ∈ [ŷ, y], c ∈ γ with reciprocal distances at most 2σ. Use the previous lemma to find p ∈ [x,ŷ] such that
which implies L ≤ 2R + 2σ + 1, a contradiction.
Suppose we have a case (P ) geodesic triangle as above. Consider x 1 , x 2 ,ŷ 1 ,ŷ 2 , y 1 , y 2 and A ∈ P as in case (P ) (where, e.g., [x 1 ,ŷ 2 ] ∈ [x,ŷ] and x 1 is closer to x). Let p be as above, and suppose first thatŷ 2 ∈ [p,ŷ]. Proceeding as above we
Otherwise A = P . In this case, by Lemma 5.3, there is a point on z ∈ [y,
We are only left to prove that the caseŷ 2 ∈ [x, p] is impossible. In fact, doing the estimates as above, we obtain
Hence L ≤ 2R + δ + 1, a contradiction.
Corollary 5.5. For each µ ≥ L there exists R with the following property. If γ is a geodesic starting from x and p ∈ γ is the first point in N µ (P ) for some P ∈ P, then d(p, π P (x)) ≤ R .
Proof. We have d(p, π P (p)) ≤ µ + 1. Suppose µ > L, and otherwise use the following argument to a point just before p on γ.
As this is not the case, we have
Lemma 5.6. Let (X, P) be relatively hyperbolic. Then there exists K so that the following holds. Suppose x, y ∈ X, P ∈ P are so that d(x, P )+d(y, P ) ≤ d(x, y)−K. Then there exists z ∈ [x, y]∩N K (P ) with |d(x, z)−d(x, P )| ≤ K.
Proof. It follows from the previous lemma that if d(π P (x), π P (y)) is large enough then any geodesic from x to y intersects balls of uniformly bounded radius around the projection points. Up to increasing K, we can make sure that the said condition holds.
Property 2) follows directly from Lemma 5.6. Let us now show 3), with µ = max{L, σ} and R 0 large enough so that the following conditions are satisfied. First, we want that, for each geodesic γ, deep µ,R 0 is a disjoint union of subpaths each contained in N tµ (P ) for some P ∈ P. Also, we require that, for each geodesic γ and P ∈ P, the entrance point of γ in N µ (P ) is in trans µ,R 0 (γ) if diam(γ ∩ N µ (P )) ≥ 2R 0 . Both conditions can be arranged in view of (α 1 ) and Lemma 5.2. Fix R ≥ R 0 . An easy argument based on weak ( * )−asymptotic tree-gradedness shows that it is enough to prove that for each K there exists D 1 = D 1 (K) so that whenever The following fact will be used in [Sis12] and its proof is very similar to the previous one. Roughly speaking, the content is that quasi-geodesics with the same endpoints have the same transient set, up to finite Hausdorff distance. Furthermore there exists t ≥ 1 so that (1) for every continuous (L, C)−quasi-geodesic γ, deep µ,R (δ) is a disjoint union of subpaths each contained in N tµ (P ) for some P ∈ P, (2) for every P ∈ P the entrance point of γ in
Proof. Similarly to the proof above, we need the following facts. Fix L, C.
Lemma 5.8. [DS05a, Lemma 4.15] There exists t so that for every d ≥ 1 all (L, C)−quasi-geodesics with endpoints in N d (P ), for some P ∈ P, are contained in N td (P ).
As usual, denote by π P a (coarse) closest point projection on P ∈ P.
Lemma 5.9. [Sis10, Lemma 1.17] There exists µ so that if d(π P (x), π P (y)) ≥ µ for some x, y ∈ X and P ∈ P then all (L, C)−quasi-geodesics from x to y intersect B µ (π P (x)) and B µ (π P (x)).
We also have the following, that can be obtained just as in Corollary 5.5.
Corollary 5.10. There exists A with the following property. Let δ be a continuous (L, C)−quasi-geodesic starting at x so that {y} = δ ∩ N µ (P ) is the final point of δ, for µ as in Lemma 5.9 and some P ∈ P. Then d(y, π P (x)) ≤ A.
We are now ready for the proof. We will only show that trans µ,R (δ) is contained in a suitable neighborhood of trans µ,R (γ), the other containment is symmetric. Let t, µ be as in the lemmas above. As usual, we use (α 1 ) and Lemma 5.8 to choose R large enough so that, for every continuous (L, C)−quasi-geodesic γ, deep µ,R (δ) is a disjoint union of subpaths each contained in N tµ (P ) for some P ∈ P. Also, we require that, for each continuous (L, C)−quasi-geodesic γ and P ∈ P, the entrance point of γ in N µ (P ) is in
By Lemma 5.9 and Corollary 5.10, we have that if p ∈ γ is the entrance point of γ in N µ (P ) (the exit point behaves similarly) and diam(N µ (P )∩γ) ≥ Q, for some suitable Q, then diam(δ ∩ N µ (P )) ≥ 2R and the entrance point of δ in N µ (P ) is within bounded distance from p.
Hence, by passing to subpaths of γ, δ connecting exit and entrance points in neighborhoods of peripheral sets, we see that it suffices to show the following statement:
For each K there exists D = D(K) so that whenever γ, δ are continuous (L, C)−quasi-geodesics connecting x, y and x , y respectively, where d(x, x ), d(y, y ) ≤ K, and for each P ∈ P we have diam(
). Also, we can just show δ ⊆ N D (γ), as with our hypothesis trans µ,R (γ) is within bounded Hausdorff distance from γ. Pick p ∈ δ. Let δ 1 , δ 2 be the sub-quasi-geodesics of δ with endpoints x , p and y , p respectively. Consider the quasi-geodesic triangle with sides γ, δ 1 , δ 2 (as we regard it as a quasi-geodesic triangle it does not matter that the endpoints do not coincide as they are within distance K). In view of our hypothesis and [DS05a, Lemma 8.17,Proposition 8.16-(1)] (which essentially give ( * )−asymptotictree-gradedness for quasi-geodesic triangles), all sides of the said triangle intersect a ball of radius, say, σ. Consider x 1 , y 1 in, respectively, δ 1 and δ 2 with d(x 1 , y 1 ) ≤ 2σ and d(y 1 , γ) ≤ 2σ. As x 1 , y 1 are on a quasi-geodesic on opposite sides of p and their distance is bounded, we can also bound d(y 1 , p), and hence we get a bound on d(p, γ). 
, whose exact definition we do not need. Using the enumeration in [BBF10] , consider the following Axioms: 
The following is proven in [BBF10] .
We can improve this result to the following.
This result probably follows from the proof of Theorem 6.2. Instead, we will use a trick to reduce this theorem to Theorem 6.2. As mentioned in the introduction, [Hum12] contains a stronger result than the theorem above. We included this result as the proof is very short and as it suffices for our next application.
6.2. Hyperbolically embedded subgroups. We refer the reader to [DGO11] for the definition of hyperbolically embedded subgroup. In the following theorem, the technical statement 4) is essentially just a more general set of hypotheses for [DGO11, Theorem 4.42] that allows to carry out essentially the same proof, and statement 3) is just a simplified version of 4). The motivating setting for statement 4) is when H λ acts parabolically on a hyperbolic space and its orbit is, up to finite Hausdorff distance, a horosphere contained in the horoball X λ 1 , as in Corollary 6.8. Theorem 6.4. Let {H λ } λ∈Λ be a finite collection of subgroups of the group G. Then the following are equivalent.
(1) There exists a generating set X so that {H λ } → h (G, X).
(2) There exists a generating set X (the same as in (1)) so that Γ = Cay(G, X) is hyperbolic relative to the left cosets of the H λ 's and d Γ | H λ is quasi-isometric to a word metric. (3) G acts by isometries on a geodesic space S that is hyperbolic relative to the orbits (of a given point) of the cosets of the H λ 's, and each H λ acts properly. (4) G acts by isometries on a geodesic space S so that S is hyperbolic relative to a G-invariant collection {X λ
for some fixed s ∈ S, (c) H λ stabilises X λ 1 and acts coboundedly on N 1 (S\X λ 1 ) ∩ X λ 1 . The implication 1) ⇒ 2) can probably also be proven as in the appendix of [DS05a] .
Proof. 1) ⇒ 2) : We set S = Cay(G, X). For H = H λ \{1}, we have that Cay(G, X H) is a coned-off graph for S with P the collection of the left cosets of the H λ 's. By definition of being hyperbolically embedded, Cay(G, X H) is hyperbolic. We have to check the BCP property. It is easily seen that we can restrict to combinatorial paths instead of standard paths. We will use [DGO11, Proposition 4.14], which gives a quantity D = D(L, n) with the following property. Consider an n−gon each of whose sides is either an L−quasi-geodesic (combinatorial path) in Cay(G, X H) or an isolated component (meaning not tied to any component of the other sides). Then each of the isolated components has length bounded by D. (The proposition actually gives a bound on the sum of such lengths but we will not need this.) Here we also used [DGO11, Lemma 4.11-(b)] which shows the compatibility of the notion of length of components used in [DGO11, Proposition 4.14] with the one we use.
It is immediate to see that the first part of the BCP property holds. Also, the second part follows considering a quadrangle with two sides being subpaths of γ 0 , γ 1 , one side being a short geodesic connecting endpoints of γ 0 , γ 1 and the last side being a P −component.
4) ⇒ 1) : [DGO11, Theorem 4.42] is a similar statement for actions on a hyperbolic space. We will adapt its proof to our setting.
We now start with an application of a result from [BBF10] . Let Y be the collection of all X λ g i,λ Lemma 6.5. Given a relatively hyperbolic space, there exists C so that whenever Q 1 = Q 2 are distinct peripheral sets we have is actually a quasi-tree) .
We now wish to define a generating system X of G so that, first of all, the Cayley graph Cay(G, X H), where H = H λ \{1}, is quasi-isometric to P K (Y). Denote s λ = X λ 1 ⊇ H λ s. Recall that for each h ∈ H λ we have hs λ = s λ . For every edge e in P K (Y) going from some s λ to some gs µ choose x e ∈ H λ gH µ so that d(s, x e s) ≤ inf{d(s, ys)|y ∈ H λ gH µ } + 1.
In order to make the generating system symmetric, we can require x f = x −1 e when f , in the notation above, is the edge g −1 e. Remark 6.6. (cfr. [DGO11, Remark 4.48] ) For x e as above there is an edge in P K (Y) from s λ to x e s µ . In fact, as x e = h λ gh µ for some h λ ∈ H λ , h µ ∈ H µ , we have
The proof of [DGO11, Lemma 4.49] applies verbatim to show the desired fact that Cay(G, X H) is quasi-isometric to P K (Y) and hence hyperbolic.
We now have to show our version of [DGO11, Lemma 4.50], which requires a slightly different proof.
Lemma 6.7. There exists α so that if for some Y ∈ Y and x ∈ X ∪ H we have diam(π Y ({s, xs})) ≥ α then x ∈ H λ and Y = s λ for some λ ∈ Λ.
Proof. Suppose first x = x e ∈ X, for e connecting s λ to gs µ in P K (Y). There are 3 cases to consider, and in each one we will get a contradiction for α large enough. 
a contradiction for α > 2(R + C) + 1. Case 3: xs µ = Y = s λ . We can translate by x −1 to reduce to the previous case (in view of our choice x g −1 e = x −1 e ). Hence, in the hypothesis of the lemma, we must have x ∈ H λ , for some λ ∈ Λ. Notice that λ∈Λ π s λ µ =λ s µ has finite diameter by Lemma 6.5 (and s belongs to the set), so that for α large enough we can exclude that s λ = Y . Therefore, s λ = Y as required. Corollary 6.8. Let {H λ } λ∈Λ be a finite collection of subgroups of the group G. Then there exists a generating set X so that {H λ } → h (G, X) ⇐⇒ G acts on a geodesic space S so that each H λ acts properly, is a maximal parabolic subgroup and there is an invariant system of disjoint horoballs U in 1-1 correspondence with the cosets of the H λ 's in G so that each H λ acts coboundedly on the corresponding horosphere.
Proof. The if part follows from the theorem since S is hyperbolic relative to U. The only if part follows adding combinatorial horoballs to S as in 2) in view of (RH1).
The notion of weakly contracting element is defined in [Sis11] in terms of the existence of an action so that the closest point projection on an orbit satisfies essentially the same properties as a closest point projection on a peripheral set of a relatively hyperbolic space.
Corollary 6.9. Let G be a group and g ∈ G be an infinite order element. Then g is weakly contracting if and only if it is contained in a virtually cyclic hyperbolically embedded subgroup.
Proof. The only if part is proven in [Sis11] . If g is contained in the virtually cyclic subgroup E(g) hyperbolically embedded in (G, X) then the action of G on Cay(G, X) satisfies the requirements of the definition of weakly contracting in view of the properties of closest point projections on peripheral sets [Sis10] .
We say that H < G is a virtual retract if there exists a subgroup G < G and a retraction φ H : G → H , where H = G ∩ H. For example, all subgroups of finitely generated abelian groups are virtual retracts.
Corollary 6.10. Suppose {H 1 , . . . , H λ } is hyperbolically embedded in G, and let K < G. Suppose further that, for each λ, H λ ∩ K is a virtual retract of K. Then {H 1 ∩ K, . . . , H λ ∩ K} is hyperbolically embedded in K.
It is quite possible that the corollary holds in greater generality.
Question 6.11. Can the virtual retraction condition be substituted by a more general condition? Is the corollary true without additional hypotheses?
Proof. Let φ H λ : H λ → K λ be as in the definition of virtual retract (with K λ being the finite index subgroup of H λ ∩ K). Denote by S the metric graph obtained adding edges (gx, gφ H λ (x)) to Cay(G, X), for each x ∈ H λ and g ∈ G. Notice that G, and hence K, acts on S.
We claim that S is hyperbolic relative to the collection P of the left cosets of the H λ 's it contains. The easiest definition to check is (RH2). In fact, the natural coned-off graphĈ of Cay(G, X) (the one obtained adding edges connecting each pair of vertices contained in a left coset of some H λ ) is a coned-off graph for S as well, as S is obtained adding to Cay(G, X) some of the edges that have to be added to getĈ. The BCP property for S follows from the BCP property for Cay(G, X) and the observation that
Notice that each P ∈ P is within bounded Hausdorff distance in S from a left coset of H λ ∩ K. Hence, in order to apply Theorem 6.4-(3), we now only have to show that each H λ ∩ K acts properly on S. We will actually show that the restriction of the metric of S to H λ ∩ K is quasi-isometric to a word metric. Denote by π λ : Cay(G, X) → H λ a closest point projection. Choose now any quasi-isometry t : H λ → H λ (both endowed with word metrics) that restricts to the identity on K λ .
We claim that
Lipschitz (notice that the closest point projection is defined in terms of the metric of Cay(G, X), and now we are regarding G as endowed with the restriction of the metric of S). This will prove that K λ , and hence H λ ∩ K λ , is quasiisometrically embedded in S, as f λ restricts to the identity on K λ . It suffices to show that f λ maps the endpoints of any edge of S to a set of uniformly bounded diameter. Let e = (p, q) be an edge of S.
Case 1: e ⊆ Cay(G, X). Closest points projections in relatively hyperbolic spaces are coarsely Lipschitz [Sis10] (it is a consequence of Lemma 5.4), so that d Cay(G,X) (π λ (p), π λ (q)) can be uniformly bounded. But this implies that d H λ (π λ (p), π λ (q)) can be uniformly bounded as well as H λ is quasi-isometrically embedded in Cay(G, X). The conclusion follows from the fact that t and φ H λ are Lipschitz (when restricted to the vertices of their domain).
Case 2: e = (gx, gφ H λ (x)) and g ∈ H λ . The edge is contained in H λ , so that π λ acts as the identity, and we can conclude as t and φ H λ are Lipschitz.
Case 3: e = (gx, gφ H λ (x)) and g / ∈ H λ . The projection of a peripheral set on another peripheral set has uniformly bounded diameter by Lemma 6.5, so that d Cay(G,X) (π λ (p), π λ (q)) can be bounded. We can conclude as in Case 1.
6.3. Divergence. We now recall the definition of divergence of a metric space X, following [DMS10] . Choose constants 0 < δ < 1 and γ ≥ 0. For functions f, g : R + → R + write f g if there exists C so that f (n) ≤ Cg(Cn + C) + Cn + C, and define , similarly. By [DMS10, Corollary 3.12], there exist δ 0 , γ 0 so that when X is a Cayley graph we have Div X γ (n, δ) Div X γ 0 (n, δ 0 ) whenever 0 < δ ≤ δ 0 and γ ≥ γ 0 . Also, the -equivalence class of Div X γ (n, δ) is a quasi-isometry invariant (of Cayley graphs). We will write Div X (n) for (the -equivalence class of) Div X γ 0 (n, δ 0 ). We now show the following.
Theorem 6.13. Suppose that the one-ended group G is hyperbolic relative to the (possibly empty) collection of proper subgroups H 1 , . . . , H k . Then
(We write max{e n , e n Div H i (n)} instead of max{e n Div H i (n)} because we allow the collection of subgroups to be empty.)
Not all super-exponential functions f (n) satisfy f (n) e n f (n), hence the following questions arise.
Question 6.14. Does there exist a group G of super-exponential divergence so that e n Div G (n) Div G (n)? If so, is it true that Div G (n) max{e n , Div H i (n)}? Do there exist one-ended properly relatively hyperbolic groups with isomorphic peripheral subgroups but different divergence?
In the case of finitely presented groups we have a cleaner statement, due to the following lemma.
Lemma 6.15. The divergence of a finitely presented one-ended group is at most exponential.
Proof. All pairs of points x, y, say with r x = d(x, 1) ≤ d(y, 1) = r y , can be joined by a path in B(1, 3r y )\B(1, r x /3), see e.g. [MS11, Lemma 5.6]. The intersection of such subset and G has e ry points, and the conclusion easily follows.
A relatively hyperbolic group is finitely presented if and only if its peripheral subgroups are, see [DG10] and [Osi06b] . Hence we have the following.
Corollary 6.16. Suppose that the one-ended group G is hyperbolic relative to its proper subgroups
We split the proof of the theorem in two propositions.
Proposition 6.17. Let the group G be hyperbolic relative to its proper subgroups
Proof. Denote H = {H 1 , . . . , H n }. By [Osi06a] there exists an infinite virtually cyclic subgroup E(g) in G so that (G, H ∪ {E(g)}) is relatively hyperbolic. In particular, E(g) is within bounded Hausdorff distance from a bi-infinite geodesic α so that, for each µ, diam(α ∩ N µ (gH i )) is uniformly bounded for each g ∈ G and each H i (by (α 1 ) and quasi-convexity of peripheral sets). We would like to show that if the path β, which we assume to have length at least 1 for simplicity, connects points a, b ∈ α on opposite sides of some c ∈ α then d(c, β) ≤ C log 2 (l(β)) + C for some constant C. In order to do so, we modify a standard argument which can be found, e.g., in [BH99, Proposition III.H.1.6] (and which we already used in Lemma 4.4). For µ as in (RH3) and R large enough we have trans µ,R (α) = α, and similarly for all subgeodesics of α. Let q ∈ β be so that l(β| [a,q] ) = l(β| [q,b] ) and consider a geodesic triangle with vertices a, b, q. We have some D so that α| [a,b] 
We can assume by induction (starting with paths of length at most, say, 100), that trans µ,R ([a, q]) ⊆ N D log 2 (l(β)/2)+100 (β) and similarly for [q, b], and we get α [a,b] ⊆ N D log 2 (l(β))+100 (β) as required.
The author was not able to find a reference for the following fact in the case of hyperbolic groups.
Proposition 6.18. Let the one-ended group G be hyperbolic relative to H 1 , . . . , H k . Then Div G (n) max{e n , e n Div H i (n)}.
Proof. It is enough to find a bound on div γ (a, b, c; δ) when a, b, c are elements of G (as opposed to points in the interior of edges). Fix a, b, c ∈ G, and suppose without loss of generality r = d(a, c) ≤ d(b, c). We would like to find a "short" path connecting a to b outside B = B(c, r/δ − γ), for δ > 0 small enough and γ large enough. We can assume r ≤ 10n for n = d(a, b) (for otherwise a geodesic from a to b would avoid B). As G is one-ended, by [DMS10, Lemma 3.4] there exists a path β connecting a, b outside B. Consider a = x 0 , . . . , x k = b on β so that d(x i , x i+1 ) ≤ 1, where x i ∈ G. Define y i ∈ G to be the first point of [x i , c] in B = B (c, d(b, c) ). We then have a sequence of points a = y 0 , . . . , y k = b in (B ∩ G)\B, which contains e n elements.
If G was hyperbolic, we would have a uniform bound on d(y i , y i+1 ). Then it would be readily seen that we can choose 0 = j 0 < · · · < j l = k so that d(y j i , y j i+1 ) is bounded and l e n , so that the desired path can be obtained concatenating geodesics of the form [y j i , y j i+1 ].
In the relatively hyperbolic case, it is easily seen by the relative Rips condition (and the deep components being contained in neighborhoods of peripheral sets) that for an appropriate D we have either d(y i , y i+1 ) ≤ D or there exists a left coset P i of some H i so that y i , y i+1 ∈ N D (P i ). Also, all geodesics from c to P i pass D−close to some z i ∈ P i , see Lemma 5.3. Assume for convenience that we are considering a Cayley graph with respect to a generating set containing generating sets for the H i 's, so that P i is connected. As d(y i , y i+1 ) ≤ 2d(c, b) and peripheral subgroups are undistorted, there exists a path α in P i of length Div P i (2d(c, b)) connecting points y i , y i+1 at distance at most D from y i , y i+1 which avoids a P i −ball of radius δ 0 min{d(y i , z i ), d(y i+1 , z i )} − γ 0 around z i . Again as peripheral subgroups are undistorted, we have that α avoids B, if we chose δ, γ large enough. Now, we can choose 0 = j 0 < · · · < j l = k so that l e n and y j i can be connected to y j i+1 by a path avoiding B and of length max{Div H i (n)}. So, we can construct a path of length max{e n Div H i (n)} connecting a to b.
6.4. Approximation with tree-graded spaces. A standard result for hyperbolic spaces, see e.g. [Ghy90, Lemma 2.12], is that any finite configuration of points and geodesics can be approximated by some tree, the error in the approximation depending on the hyperbolicity constant and the number of points and geodesics involved only. This result is very useful to reduce computations in hyperbolic spaces to the tree case. We generalize this to relatively hyperbolic spaces. Fix from now on a relatively hyperbolic space (X, P) and µ, R as in (RH3).
Definition 6.19. Let A be a collection of points and peripheral sets. We denote by γ(A) the union of A and all trans µ,R (γ) for γ a geodesic connecting points each lying on some A ∈ A.
We will approximate finite configurations in X with tree-graded spaces, a notion that has been defined in [DS05a] .
Definition 6.20. A geodesic complete metric space F is tree-graded with respect to a collection of closed geodesic subsets of F (called pieces) if the following properties are satisfied: (T 1 ) distinct pieces intersect in at most one point, (T 2 ) each geodesic simple triangle is contained in one piece.
For the purposes of this section, a tree-graded space can be thought of as a tree-like arrangement of pieces. An example of tree-graded space is a the Cayley graph of A * B with respect to a generating set which is th union of generating sets of A and B.
Lemma 6.21. Let (X, P) be relatively hyperbolic and fix an approximating graph Γ P for each P ∈ P as well as µ, R as in Proposition 4.9. For each n ∈ N there exists C with the following property. Suppose that A is a collection of points and peripheral sets and A has cardinality at most n. Then there exists a tree-graded space T , each of whose pieces is isometric to some Γ P , and a (C, C)−quasi-isometric embedding f : γ(A) → T .
The idea is just to apply the statement for hyperbolic spaces in the Bowditch space and to use Proposition 4.9, "moreover" part included, to translate the information we get back to X.
Proof. Fix the Bowditch space Bow(X) with respect to the approximating graphs {Γ P } and let A be as in the statement. LetÂ be the collection of all points inÂ and quasi-geodesic lines (with uniform constants) which are asymptotic to vertical rays in a horoballs corresponding to some P ∈ A ∩ P in both directions.
Denoting by γ(Â) the collection of all geodesics with endpoint inÂ, we have that there exists a (1,Ĉ)−quasi-isometryĝ :T → γ(Â) with quasiinversef , for some treeT andĈ =Ĉ(n). We will denote the convex core of a set S inT again by γ(S). By substituting each combinatorial horoball H(Γ P ) with (the full subgraph with vertex set) Γ P × (k + N) for a suitable k = k(Ĉ, Bow(X)), we can make sure that S P = γ(f (H(Γ P ))) are all disjoint and the distance between two such sets is at least some > 0.
For each P ∈ P, removeS P fromT and construct T by gluing copies of Γ P so that each each point p ∈ ∂S P is glued to a point q ∈ Γ 0 P at minimal distance fromĝ(p). Notice that T is tree-graded.
We have that whenever a geodesic α in X and a geodesic β in Bow(X) have the same endpoints then trans µ,R (α) is within Hausdorff distance bounded by some D = D(k, Bow(X)) from β ∩ X. Also, moving the endpoint of a geodesic inside a peripheral set affects the transient set only up to finite Hausdorff distance. Using these two facts one can show that each x ∈ γ(A) lies within uniformly bounded distance from some y = y(x) ∈ γ(Â) ∪ P ∈P∩A Γ P . Define f (x) =f (y) if y lies in γ(Â) and f (x) = y if y ∈ P ∈P∩A Γ P (notice that such Γ P 's are actually subsets of T ).
Let x, x ∈ γ(A) and consider y = y(x), y = y(x ) as above. Letγ be a geodesic from y to y and notice thatγ ∩ X is contained in a suitable neighborhood of γ(Â) ∪ {y, y }. As we shrinked the horoballs, we can subdivide γ into subgeodesics either of length at least k > 0 and contained in N k (X) (type 1) or contained in a horoball (type 2). We know (similarly to Corollary 4.10) that we can obtain a quasi-geodesic in X from y to y by substituting the subgeodesics γ i of type 1 with geodesics in X whose endpoints are within distance k from the endpoints of the γ i 's and the geodesics of type 2 with geodesics each contained in a suitable neighborhood of a peripheral set. Also, each subgeodesic of type 1 has endpoints close to the image througĥ g of some geodesic α i in T (of length at least > 0), and the α i 's can be concatenated with geodesics each contained in some Γ P to get a geodesic from a point close to f (x) to a point close to f (y).
Let us write A ≈ B if the quantities A, B differ by some multiplicative and additive constants depending on the data in the statement of the lemma. From the discussion above we have, denoting T i the collection of subgeodesics as above of type i,
hence f is a quasi-isometric embedding.
6.5. Combination theorems. Theorem 4.2 can be used to give alternative proofs of some known combination theorems for relatively hyperbolic groups from [Dah03] . As the results are known, we consider only one example and provide a sketch of proof only.
Theorem 6.22 ( [Dah03] ). Let G 1 , G 2 be hyperbolic relative to a common subgroup H. Then G = G 1 * H G 2 is hyperbolic relative to H. Sketch of proof. Let x, y ∈ G. All paths from x to y have to cross a sequence x ∈ X 1 , . . . , X n y of left cosets of G 1 and G 2 corresponding to a geodesic in the Bass-Serre tree connecting vertices corresponding to x and y. Define η(x, y) by concatenating in the suitable order geodesics from x to X 1 ∩ X 2 , from X n−1 ∩ X n to y and from X i−1 ∩ X i to X i ∩ X i+1 , and let trans(x, y) be the set of all transient points for the said geodesics (with suitable constants). Let us for example check 3). Fix x, y, z ∈ G. Consider the corresponding tripod in the Bass-Serre tree we see that each point p ∈ trans(x, y) lies in some geodesic γ contained in a coset X of G 1 or G 2 satisfying the following property. There exists cosets Y 1 , Y 2 and possibly of Y 3 of H contained in X so that γ connects Y 1 to Y 2 and either
• η(x, z) or η(z, y) contains a subgeodesic with endpoints in Y 1 , Y 2 , or • η(x, z) contains a geodesic connecting Y 1 to Y 3 and η(y, z) contains a geodesic connecting Y 3 to Y 2 . In both cases, by the relative Rips condition for m−gons with m ≤ 6, we get the desired containment.

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