GEODESICS IN TREES OF HYPERBOLIC AND RELATIVELY HYPERBOLIC GROUPS FRANC ¸ OIS GAUTERO Abstract. We present a careful approximation of the geodesics in trees of hyperbolic or relatively hyperbolic groups. As an application we prove a combination theorem for finite graphs of relatively hyperbolic groups, with both Farb’s and Gromov’s definitions, so providing an answer to one of the problems raised in Bestvina’s list.

1. Introduction The main part of this paper is devoted to give a precise description of the geodesics in trees of hyperbolic (and thereafter relatively hyperbolic - see below) groups. Such a work might appear not very appealing, and somehow quite technic. In order to show that this however might be worthy, let us give an application: a combination theorem for hyperbolic and relatively hyperbolic groups. That is, a theorem giving a condition for the fundamental group of a graph of relatively hyperbolic groups being a relatively hyperbolic group. In [3] (see also [20]), the authors introduce the notion of (finite) qi-embedded graph of groups and spaces G. Then, assuming the Gromov hyperbolicity of the vertex spaces and the quasiconvexity of the edge spaces in the vertex spaces, they give a criterion for the hyperbolicity of the fundamental group of G. Since then different proofs have appeared, which treat the so-called ‘acylindrical case’: see, among others, [18, 22]. Acylindrical means that the fixed set of the action of any element of the fundamental group of the graph of groups on the universal covering has uniformly bounded diameter. The nonacylindrical case is less common: see [21] which relies on [3] but clarifies its consequences when dealing with a certain class of mapping-tori of injective, non surjective free group endomorphisms, or [12] which, by an approach similar to the one presented here, gives a new proof of [3] in the case of mapping-tori of free group endomorphisms. Nowadays the attention has drifted from hyperbolic groups to relatively hyperbolic groups. A notion of relative hyperbolicity was already defined by Gromov in his seminal paper [19]. Since then it has been revisited and elaborated on in many papers. Two distinct definitions now coexist. In parallel to the Gromov relative hyperbolicity, sometimes called strong relative hyperbolicity, there is the sometimes called weak relative hyperbolicity introduced by Farb [11]. Bowditch [5] and Osin [26] give alternative definitions, but which are equivalent either to Farb’s or to Gromov’s definition. In fact, it has been proved [8, 26] (also [5]) that Gromov definition is equivalent to Farb definition plus an additional property termed Bounded Coset Penetration property (BCP in short), due to Farb [11]. Relatively 2000 Mathematics Subject Classification. 20F65, 20F67, 20E08, 20E06, 05C25, 37B05. Key words and phrases. Gromov-hyperbolicity, Farb and Gromov relative hyperbolicity, graphs of groups and spaces, mapping-tori, free and cyclic extensions, combination theorem. Acknowledgements. This paper benefited from discussions of the author with F. Dahmani, V. Guirardel, M. Heusener, I. Kapovich and M. Lustig. Warm thanks are particularly due to I. Kapovich for his explanations about Bestvina-Feighn paper. At that time, the author was Assistant at the University of Geneva, whereas I. Kapovich visited this university thanks to a funding of the Swiss National Science Foundation. The author is glad to acknowledge support of these institutions. Finally, P. de la Harpe also deserves a great share of these acknowledgements for his help. 1

hyperbolic groups in the strong (that is Gromov) sense form a class encompassing hyperbolic groups, fundamental groups of geometrically finite orbifolds with pinched negative curvature, groups acting on CAT(0)-spaces with isolated flats among many others. First combination theorems in some particular (essentially acylindrical) cases have been given in the setting of the relative hyperbolicity: [1], [10] or [27, 28]. One result [15] treats a particular non-acylindrical case, namely the relative hyperbolicity of one-ended hyperbolic by cyclic groups. However it still lacks for relatively hyperbolic groups an analog of the Bestvina-Feighn combination theorem for hyperbolic groups 1. This is one of the questions (attributed to Swarup) raised in Bestvina’s list [2]. We offer here an answer, as an application of our work on geodesics in trees of spaces. We would like to emphasize at once that we do not appeal to the Bestvina-Feighn combination theorem, but instead give a new proof of it as a particular case. Where they use “second-order” geometric characterization of hyperbolicity via isoperimetric inequalities, we use “first-order” geometric characterization, via approximations of geodesics and the thin triangle property. At the expense of heavier and sometimes tedious computations, this na¨ıve approach allows us to engulf in a same setting (at least when dealing with combination theorems) both absolute and relative hyperbolicity. We feel moreover that an extension to the case of 2-complexes of groups as defined by Bridson and Haefliger (see [7] for instance) should be at hand with only a little bit of additional work. 2. Statement of results Let us stress some particular cases of our results (the theorems we give essentially concern non-acylindrical cases): Theorem 2.1. Let G be a finitely generated group, let α be an automorphism of G and let Gα = Goα Z be the associated mapping-torus group. Let H be a finite family of subgroups of G such that α is hyperbolic relative to H. Then, if G is weakly hyperbolic relative to H, Gα is weakly hyperbolic relative to H, and if G is strongly hyperbolic relative to H, Gα is strongly hyperbolic relative to the mapping-torus of H. The relatively hyperbolic automorphims we refer to in Theorem 2.1 above first appeared in [13] where we announced a (weak) version of the results of the present paper. They generalize the Gromov hyperbolic automorphisms [3]. Definition 2.2. Let G = hSi be a finitely generated group and let H = {H1 , · · · , Hk } be a finite family of subgroups of G. (a) An automorphism α of G is a relative automorphism of (G, H) if H is α-invariant up to conjugacy, that is there is a permutation σ of {1, · · · , k} such that for any Hi ∈ H there is gi ∈ G with α(Hi ) = gi−1 Hσ(i) gi . (b) The mapping-torus of H under a relative automorphism α of (G, H) is a maximal family Hα of subgroups Hj ⊂ Gα satisfying the following properties: • Hj = hHij , tnij gi−1 i, where nij is the minimal integer such that there is gij ∈ G j nij with α (Hij ) = gi−1 Hij gij ; j i are two distinct subgroups i, Hk = hHik , tnik gi−1 • whenever Hj = hHij , tnij gi−1 j k in Hα , no power of t conjugates Hij to Hik in Gα . 1Since

a first version of this paper was written a long time ago, a paper [24] appeared on the arXiv, also giving a combination theorem dealing with non-acylindrical cases. At the difference of the work presented here, the paper [24] relies upon Bestvina-Feighn’s theorem 2

(c) The H-word metric |.|H is the word-metric for G equipped with the (usually infinite) set of generators which is the union of S with the elements of G in the subgroups of the collection H. (d) An automorphism α of G is hyperbolic relative to H if α is a relative automorphism of (G, H) and there exist λ > 1 and M, N ≥ 1 such that for any w ∈ G with |w|H ≥ M : λ|w|H ≤ max(|αN (w)|H , |α−N (w)|H ). The definition of relatively hyperbolic automorphism given above is slightly more general than the definition given in [13]. The constant M did not appear there. It is however more natural: thanks to this additional constant M , the definition is obviously invariant under conjugacy2. It is easy to prove that a hyperbolic group is weakly hyperbolic relative to any finite family of quasi convex subgroups, and strongly hyperbolic relative to any almost malnormal finite family of quasi convex subgroups. By this we mean the following: Definition 2.3. A finite family {H1 , · · · , Hk } of subgroups of a group G is almost malnormal if: (a) for any i = 1, · · · , k, Hi is almost malnormal in G. (b) for any i, j ∈ {1, · · · , k} with i 6= j, the cardinality of the set {w ∈ Hj ; ∃g ∈ G s.t. w ∈ g −1 Hi g} is finite. Of course, if the family of subgroups consists of only one subgroup, the definition above is nothing else than the definition of almost malnormality of this subgroup. Thus as a corollary of the previous theorem we have: Corollary 2.4. Let G be a hyperbolic group, let H be a finite family of subgroups of G and let α be an automorphism of G which is hyperbolic relative to H. If H is quasi convex in G then Gα is weakly hyperbolic relative to H and if in addition H is almost malnormal in G then Gα is strongly hyperbolic relative to the mapping-torus of H. This corollary may be specialized to torsion free one-ended hyperbolic groups, and so in particular to fundamental groups of surfaces. Using the JSJ-decomposition of Sela or Bowditch [6], this allows us to re-prove the result of [15]. Since there we gave only an idea for the statement and the proof in the Gromov relative hyperbolicity case, we include here the full statement of this result: Corollary 2.5. Let G be a torsion free one-ended hyperbolic group and let α be an automorphism of G. Let H be a maximal family of maximal subgroups of G which consist entirely of elements on which α acts up to conjugacy periodically or with linear growth. Then Gα is weakly hyperbolic relative to H, and strongly hyperbolic relative to the mapping-torus of H. If G is the fundamental group of a compact surface S (possibly with boundary) with negative Euler characteristic and h a homeomorphism of S inducing α on π1 (S) (up to inner automorphism), then the subgroups in H are: (i) the cyclic subgroups generated by the boundary curves, (ii) the subgroups associated to the maximal subsurfaces which are unions of components on which h acts periodically, pasted together along reduction curves of the NielsenThurston decomposition, 2The

author is grateful to F. Dahmani, V. Guirardel and M. Lustig for this observation. 3

(iii) the cyclic subgroups generated by the reduction curves not contained in the previous subsurfaces. The passage from Corollary 2.4 to Corollary 2.5 needs to prove first that the subgroups given in Corollary 2.5 are quasi convex and malnormal in G and second that the automorphism α is hyperbolic relative to this collection of subgroups. The first property comes from the JSJ-decomposition of torsion free one-ended hyperbolic groups and the fact that maximal subgroups of surfaces consisting only of elements with linear growth are malnormal. The second property is proved by “standard” computations: consider a closed curve in the surface not contained (up to isotopy) in the relative part given by the corollary. It passes through pseudo-Anosov components. The length of the parts in the pseudo-Anosov components (defined as the total variation with respect to the stable and unstable measures associated to the invariant geodesic laminations of the pseudo-Anosov) are uniformly bounded away from zero and are dilated either by the homeomorphism or its inverse. The length of the other parts either remain unchanged equal to 1 or increases. This gives the dilatation of the curve by h or h−1 , and so the relative hyperbolicity of α follows. We won’t provide the details in this paper because on the one hand they have nothing to do with the core of the paper and on the other hand they appear as a particular case of a subsequent paper [16]. There is a more general statement about free extensions of relatively hyperbolic groups, this is Theorem 2.8 below. We need two more definitions, which in some sense generalize the definitions of the mapping-torus of a family and of the relatively hyperbolic automorphisms. Definition 2.6. Let G be a finitely generated group and let H = {H1 , · · · , Hk } be a finite family of subgroups of G. Let Fr = hAi, with A = {α1±1 , · · · , αr±1 }, be a rank r free group of relative automorphisms of (G, H). A Fr -extension HA of H is a maximal family of subgroups of G oA Fr of the form −1 −1 hHi , ai,1 gi,1 , · · · , ai,m gi,m , · · · i such that: −1 Hi gi,m and hai,1 , ai,2 , · · · i generates the sub• each ai,m ∈ Fr satisfies ai,m Hi = gi,m group of all the elements of Fr leaving Hi invariant up to conjugacy; • if Hj , Hj 0 are two distinct subgroups in HA with Hi ⊂ Hj , Hk ⊂ Hj 0 then no element of hAi conjugates Hi to Hk in G oA Fr .

Definition 2.7. Let G be a finitely generated group and let H be a finite family of subgroups of G. A uniform free group of relatively hyperbolic automorphisms 3 of (G, H) is a rank r free group Fr of relative automorphisms of (G, H) such that there exist, for some (and hence any) basis A of Fr , λ > 1 and M, N ≥ 1 such that, for any element w ∈ G with |w|H ≥ M , any pair of automorphisms α, β with |α|A = |β|A = N and dA (α, β) = 2N satisfies: λ|w|H ≤ max(|α(w)|H , |β(w)|H ). Theorem 2.8. Let G be a finitely generated group and let H be a finite family of subgroups of G. Let Fr be a uniform free group of relatively hyperbolic automorphisms of (G, H), with basis A, and let GA = G oA Fr . Then, if G is weakly hyperbolic relative to H, GA is weakly hyperbolic relative to H, and if G is strongly hyperbolic relative to H, GA is strongly hyperbolic relative to the Fr -extension HA of H. 3The

author would like to thank M. Heusener for inciting him to correct a previous formulation of this definition, which was unnecessarily more restrictive. 4

To give a concrete application, take a free group of pseudo-Anosov isotopy-classes of a compact surface with boundary and with negative Euler characteristic. Then this free group is a uniform free group of automorphisms which are hyperbolic relative to the cyclic groups generated by the boundary curves (see [25] for the case of closed surfaces, the arguments there should be easily adapted to the case we are speaking about here). Thus the semi-direct product of the fundamental group of the surface with this free group is weakly hyperbolic relative to the above cyclic subgroups, and strongly hyperbolic relative to the free extension of their family. Up to now, we only treated extensions of relatively hyperbolic groups via semi-direct products. However such a product is only a particular case of HNN-extension. Alibegovic in [1], Dahmani in [10] or Osin in [28] treat acylindrical HNN-extensions and amalgated products. In order to obtain a full combination theorem in the Gromov relative hyperbolicity setting, we need a theorem about non-acylindrical HNN-extensions. As an important, particular case of such HNN-extensions, Theorem 2.1 below deals with injective, not necessarily surjective, endomorphisms of relatively hyperbolic groups. We first introduce a notion of relative malnormality. Definition 2.9. Let G be a group and let H = {H1 , · · · , Hk } be a finite family of subgroups of G. A subgroup H 0 of G is almost malnormal relative to H if there is an upperbound on the H-word length of the elements in the set {w ∈ H 0 ; ∃g ∈ G − H 0 with w ∈ g −1 H 0 g}. If H is empty, the definition above is nothing else than the usual notion of almost malnormality and if in addition there is no torsion, we get the notion of malnormality. Whereas the definitions of a relative automorphism and of a mapping-torus of a family of subgroups given in Definition 2.2 remain valid for injective endomorphisms, the definition of relative hyperbolicity for automorphisms is easily adapted to the more general case of injective endomorphisms: Definition 2.10. Let G be a finitely generated group and let H be a finite family of subgroups of G. An injective endomorphism α of G is hyperbolic relative to H if α is a relative endomorphism of (G, H) and there exist λ > 1 and M, N ≥ 1 such that, for any w ∈ Im(αN ) with |w|H ≥ M , if |αN (w)|H ≥ λ|w|H does not hold then w = αN (w0 ) with |w0 |H ≥ λ|w|H . Theorem 2.11. Let G be a finitely generated group, let α be an injective endomorphism of G and let Gα be the associated mapping-torus group, i.e. the associated ascending HNNextension. Let H be a finite family of subgroups of G such that α is hyperbolic relative to H. Assume that Im(α) is almost malnormal relative to H. Then, if G is strongly hyperbolic relative to H, Gα is weakly hyperbolic relative to H and strongly hyperbolic relative to the mapping-torus of H. The reader will notice at once that, at the difference of the previous results, the above theorem does not treat the extension of weakly relatively hyperbolic groups. The reason is that the condition of almost malnormality does not imply in this case the so-called “exponential separation property” which is the real condition we require further in the paper (see below in the Introduction - see also Definition 5.1 and Lemma 11.11 further in the paper). This is due to the fact that Farb relative hyperbolicity is very loose and not to the fact that this exponential separation property is very restrictive: it is indeed equivalent to Bestvina-Feighn “annuli flare property” [3]. Moreover it is optimal in the surface case, as we observed in [15] the collection of subgroups we put in the relative part in this case is minimal. In fact, like the Bestvina-Feighn condition, this exponential 5

separation property is also a necessary condition, although we were not able to give here a direct proof (in the absolute hyperbolicity case, Gersten gives the converse to the combination theorem, using cohomological arguments [17], we adapt his arguments in [14]). We now define graphs of weakly, and of strongly relatively hyperbolic groups and we will thereafter give our most general combination theorems. Definition 2.12. A graph of weakly (resp. strongly) relatively hyperbolic groups is a graph of groups (G, Hv , He ) such that: (a) Each edge group Ge and each vertex group Gv is weakly (resp. strongly) hyperbolic relative to a specified (possibly empty) finite family of subgroups He and Hv . (b) In the case of strong relative hyperbolicity, the edge collections He are required to be (possibly empty) families of conjugates of the subgroups in the families Hi(e) and Ht(e) , where i(e) and t(e) are the initial and terminal vertices of e. (c) For any edge e, (Ge , |.|He ) is quasi isometrically embedded in (Gi(e) , |.|Hi(e) ) and in (Gt(e) , |.|Ht(e) ). The definition of a graph of strongly relatively hyperbolic groups is slightly more restrictive than the equivalent definition for weakly relatively hyperbolic groups. This is because the description of the subgroups to put in the relative part is heavier in the former case than in the latter. For the sake of clarity of the theorem, we adopted Item (b), hoping that this is a not too bad compromise between clarity and generality. Theorem 2.13. Let (G, Hv , He ) be a finite graph of weakly relatively hyperbolic groups. If the universal covering of G satisfies the exponential separation property with respect to the He ’s and Hv ’s, then the fundamental group of G is weakly hyperbolic relative to union of the families Hv ’s and He ’s. Theorem 2.14. Let (G, Hv , He ) be a finite graph of strongly relatively hyperbolic groups and let T be a maximal tree in G. We denote by R a maximal collection of subgroups Hi in the Hv ’s such that two distinct Hi , Hj in R are not conjugate by an edge-path in T . We denote by Rc the family of subgroups Hi ’s in R which are conjugated to themselves by non-empty maximal free subgroups generated by the edges in T c , the complement of T in G. We denote by Rcc the complement of Rc in R. If the universal covering of G satisfies the strong exponential separation property with respect to the He ’s and Hv ’s, then the fundamental group of G is strongly hyperbolic relative to the family formed by Rcc together with the extension of the family Rc by the free group generated by the edges in T c . In order to explain what is the exponential separation property, we have to go back for a while to our main object, which is the description of the geodesics in a tree of hyperbolic ˜ T , π) be a tree of hyperbolic qi-embedded spaces [3, 23], that groups or spaces. Let (X, ˜ is X is isometric to Xe × (0, 1) over each open edge e of T , where Xe is the edge space of e, the edge spaces are (u, v)-quasi isometrically embedded in the vertex spaces Xv , and edge and vertex spaces are δ-hyperbolic, where u, v, δ are uniform constants. The ˜ → T is such that π −1 (x) = Xe × {t} for x in an open edge e and t ∈ (0, 1), map π : X −1 and π (v) = Xv for v a vertex of T . For more details about this now classical notion we refer the reader to [3] or [23]. In this paper, the pre-images under π of the points of T are termed strata and paths contained in strata are horizontal paths. For the sake of clarity of the explanations, let us restrict for a while to the case where the quasi isometric ˜ is in the image of a (v, 0)embeddings are in fact quasi isometries. Then, any point x ∈ X ˜ quasi isometric embedding σ : T → X which is a section of π. We temporarily call such a 6

section a quasi leaf (later in the paper, such sections do not always exist and when they do, are only subsets of quasi leaves). A telescopic path is a concatenation of non trivial quasi leaf segments and horizontal paths. A diagonal between two quasi leaves is a horizontal geodesic between these quasi leaves, which minimizes the horizontal length. A dynamical notion of the behaviour of these quasi leaves comes in a natural way, their exponential separation. When the property of exponential separation is satisfied, the diagonals are dilated in all directions, in other words there is essentially only one diagonal between two quasi leaves. Another quite natural notion is the notion of a corridor between two points ˜ this is a union of horizontal geodesics, exactly one in each stratum, which x, y of X: connects a quasi leaf of x to a quasi leaf of y. The definition given here is easier than the general definition given further in the paper because of our assumption here that the quasi isometric embeddings between edge and vertex spaces are in fact quasi isometries. In the general case, in addition, a corridor does not necessarily exists between two given points, but only what we call further a pseudo-corridor. The following theorem gives a fairly accurate description of the geodesics in a tree of hyperbolic spaces: ˜ be a tree of hyperbolic spaces with exponential separation of the Theorem 2.15. Let X ˜ given any pseudo-corridor C between the endquasi leaves. Given any geodesic g in X, points of g, g is C(d)-close to a telescopic (κ(d), κ(d))-quasi geodesic in C, the horizontals of which have horizontal length d and, at the exception of at most one, are diagonals. The constants C(d) and κ(d) increase with d as soon as d is sufficiently large enough. The combination theorem for hyperbolic spaces follows in a straightforward way from the above theorem, see Theorem 5.3. As in [3], in the acylindrical cases, the condition we give, the exponential separation of the quasi leaves, is vacuously satisfied. Indeed the acylindrical case corresponds to the case where there is an uniform bound on the length of the quasi leaves. When dealing with graphs of relatively hyperbolic groups, in particular the metric involved is the relative metric of the strata, see Definitions 11.1 and 11.4. The combination theorems for relatively hyperbolic groups follow almost as easily as in the hyperbolic case. Speaking of Farb’s relative hyperbolicity, this is quite obvious since this weak relative hyperbolicity is nothing else than the hyperbolicity of a certain “coned” space constructed from the original Cayley graph by making each subgroup in the relative part of diameter 1. This is achieved by putting a cone over each right coset of the subgroups one wants to put in the relative part. The only difference with Gromov’s relative hyperbolicity is that when a subgroup of the relative part is preserved up to conjugacy when taking the extension, then the whole extension of this subgroup has to be put in the relative part of the extension. Plan of the paper: Throughout Sections 4 and 5 are introduced the basis, namely trees of

hyperbolic spaces, quasi leaf segments and quasi leaves, corridors, exponential separation and telescopic metric. In Sections 6 and 7 is proved the thin-property for quasigeodesic bigons in corridors. The main step is Proposition 6.8. Section 6 is devoted to its proof. Section 8 gives the quasiconvexity of the corridors. We mean that any geodesic lies in a bounded neighborhood of any corridor containing its endpoints. The proof decomposes as follows: in subsection 8.2 is given the “local quasiconvexity” of corridors, in other words a quasi geodesic with small vertical deviation remains in a bounded horizontal neighborhood of a corridor between its endpoints; then in subsection 8.3 we introduce the notion of a stair relative to a corridor and prove that a stair with “large vertical deviation” cannot go back to the corridor that it left before; finally in subsection 8.4 7

we approximate by a stair a quasi geodesic leaving then reentering a sufficiently large horizontal neighborhood of a corridor, which allows us to conclude the quasi convexity thanks to the first two subsections. In Section 9 we gather the previous facts and prove the combination theorem in the absolute case, together with Theorem 2.15. More generally we get all that we need about the behaviour of quasi geodesics in trees of hyperbolic spaces with exponentially separated quasi leaves to deduce the combination theorem in the relative hyperbolic setting. Sections 10 and 11 are about relative hyperbolicity, the combination theorems are proved in Section 11. The case of Farb relative hyperbolicity amounts to proving a combination theorem for trees of non-proper hyperbolic spaces. This is exactly what was done before so that no real additional work is needed. Roughly b b speaking, the case of Gromov relative hyperbolicity is treated as follows. Let X (resp. b X) be coned spaces associated to the coned Cayley graphs given by the theorems in b is a tree of hyperbolic the Gromov relative case (resp. Farb relative case). The space X b b b b spaces, but not X. First one proves that quasi geodesic bigons of X are thin and satisfy the b stay in a bounded Bounded Coset Penetration property provided that their “traces” in X horizontal neighborhood of a corridor between their endpoints; second one proves that b b b b b is short the passage of a quasi geodesic of X through a cone of X which is not a cone of X b in X provided the entrance and exit points are outside a certain horizontal neighborhood b b of a corridor between the endpoints of the quasi geodesic. Therefore a quasi geodesic of X is close to another one (of course with different constants of quasi geodesicity), the trace of which lies in a bounded horizontal neighborhood of a corridor between its endpoints. The conclusion follows from the first step. 3. Preliminary notions If (X, d) is a metric space with distance function d, and x a point in X, we set Bx (r) = {y ∈ X ; d(x, y) ≤ r}. If A and B are any two subsets of (X, d), di (A, B) = inf d(x, y) x∈A,y∈B

and ds (A, B) =

sup d(x, y). We set also Ndr (A) = {x ∈ X ; di (x, A) ≤ r}. And

x∈A,y∈B

dH (A, B) = sup{r ≥ 0 ; A ⊂ Ndr (B) and B ⊂ Ndr (A)} is then the usual Hausdorff distance between A and B. 3.1. Quasi isometries, quasi geodesics and hyperbolic spaces. A (λ, µ)-quasi isometric embedding from (X1 , d1 ) to (X2 , d2 ) is a map f : X1 → X2 such that, for any x, y in X1 : 1 d1 (x, y) − µ ≤ d2 (f (x), f (y)) ≤ λd1 (x, y) + µ λ A (λ, µ)-quasi isometry f : (X1 , d1 ) → (X2 , d2 ) is a (λ, µ)-quasi isometric embedding such that for any y ∈ X2 there exists x ∈ X1 with d2 (f (x), y) ≤ µ. A (λ, µ)-quasi geodesic in a metric space (X, d) is the image of an interval of the real line under a (λ, µ)-quasi isometric embedding. We work with a version of the Gromov hyperbolic spaces which is slightly extended with respect to the most commonly used. We do not require first that they be geodesic, and second that they be proper, that is closed balls are not necessarily compact. Instead of geodesicity, we require quasi-geodesicity. We say that a metric space (X, d) is a (r, s)quasi geodesic space if, for any two points x, y in X there is a (r, s)-quasi geodesic between x and y. We then denote by [x, y] such a (r, s)-quasi geodesic (and of course in a geodesic space, [x, y] denotes any geodesic between x and y). A quasi geodesic metric space is 8

a metric space which is (r, s)-quasi geodesic for some non negative real constants r, s. The (r, s)-quasi geodesic triangles in a (r, s)-quasi geodesic metric space (X, d) are thin if there exists δ ≥ 0 such that any (r, s)-quasi geodesic triangle in (X, d) is δ-thin, that is any side is contained in the δ-neighborhood of the union of the two other sides. In this case, X is a δ-hyperbolic space. A metric space (X, d) is a Gromov hyperbolic space if there exists δ ≥ 0 such that (X, d) is a δ-hyperbolic space. The sligth “generalization” from geodesic to quasi geodesic spaces is only a technical point. But not requiring our spaces to be proper is important in order to deal with relatively hyperbolic groups, the definitions of which involve non-proper metric graphs. Let (X, d) be a geodesic space and let Y be a closed subspace of X. If x is any point in X, we denote by PY (x) any point in Y which satisfies d(x, PY (x)) ≤ di (x, Y ) + 1. Such a point PY (x) is a projection of x on Y . We will say that Y is a geodesic subspace of X if given any two points x, y in Y , some geodesic of X between x and y is contained in Y . Lemma 3.1. With the notations above, assume that (X, d) is a δ-hyperbolic space and that Y is a geodesic subspace of X. There exists C(δ) ≥ 0 such that, for any two points x, y in X, d(PY (x), PY (y)) ≤ d(x, y) + C(δ). See [9], Corollary 2.2 page 109.

¤

Remark 3.2. Throughout all the text, the constants appearing in each lemma, corollary or proposition will be denoted by C, D, · · · and thereafter they will be referred to by the same letter with the number of the lemma, corollary or proposition in subscript. For instance, if Lemma 3.4 introduces the constants C and D, for referring afterwards to these constants, we will write C3.4 and D3.4 . 4. Trees of hyperbolic spaces 4.1. Trees of spaces. A metric tree is a simplicial tree with all edges isometric to (0, 1). If T is a metric tree, we denote by |.|T the length of a path in T and by dT the associated distance. For any path p in T , there is an unique path homotopic to p relative to its endpoints, which minimizes the length function. Such a path is called a geodesic and we denote by Geod(T ) the set of geodesics in T . ˜ T , π) is a metric space X ˜ Definition 4.1. (compare [3, 23]) A tree of metric spaces (X, ˜ equipped with a projection π : X → T onto a metric tree T which satisfy the following properties for some λ ≥ 1 and µ ≥ 0: (a) If me is the midpoint of the edge e, then π −1 (me ) = Xe is a geodesic metric space and π −1 (e) is isometric to Xe × (0, 1). (b) If v is a vertex of T , if Ts is the tree T subdivided at the midpoints of the edges and Sv is the closed star of v in Ts , then: • π −1 (v) is a geodesic metric space Xv ; • π −1 (Sv ) is obtained from the disjoint union of Xv with the spaces Xe ×[0, 1/2], e the edges of Ts in Sv , by identifying each Xe ×{0} with a subset of Xv under a (λ, µ)-quasi isometric embedding. −1 A set π (x), x ∈ T , is a stratum. A tree of hyperbolic spaces is a tree of metric spaces such that there is δ ≥ 0 for which the strata are δ-hyperbolic spaces. By definition, each stratum in a tree of metric spaces comes with a distance, termed horizontal distance. A path contained in a stratum is a horizontal path and we will also speak of the horizontal length of a horizontal path. 9

4.2. The telescopic metric. ˜ T , π) be a tree of metric spaces. Definition 4.2. Let (X, ˜ is (the image of) a section σw of π over a geodesic w of T A v-quasi leaf segment in X which is a (v, 0)-quasi isometric embedding. ˜ The T -length |w|T is the vertical length of the v-quasi leaf segment σw : w → X. ˜ and w is a geodesic of T starting at π(x), the notation wx will If x is a point in X ˜ such that some v-quasi leaf segment s with π(s) = w denote the set of points y ∈ X connects x to y (in particular any such y belongs to π −1 (t(w))). ˜ such that there exists The v-quasi leaf Lv (x) of a point x is the set of points y ∈ X w ∈ Geod(T ) with y ∈ wx. ˜ T , π) be a tree of metric spaces. Definition 4.3. Let (X, (a) A v-telescopic path is a path which is the concatenation of horizontal paths in the strata over the vertices of T and of non-trivial v-quasi leaf segments. (b) The vertical length |p|vvert of a v-telescopic path p is the sum of the vertical lengths of the maximal v-quasi leaf segments. The horizontal length |p|vhor is the sum of the horizontal lengths of the maximal horizontal subpaths in the complement of the maximal v-quasi leaf segments. (c) The telescopic length |p|vtel of a v-telescopic path p is the sum of its horizontal and vertical lengths. (d) The v-telescopic distance dvtel (x, y) between two points x and y is the infimum of the telescopic lengths of the v-telescopic paths between x and y. For the sake of simplification, we will often forget the exponents in the vertical, horizontal and telescopic lengths, unless some ambiguity might exist. Remark 4.4. Let p be a v-telescopic path. At the possible exception of the first and last ones, the vertical length of each maximal v-quasi leaf segment in p is greater or equal to 1. ˜ T ) be a tree of metric spaces. We will only consider telescopic paths Convention: Let (X, with endpoints in the strata over the vertices. This implies in particular that the vertical ˜ is at length of each maximal v-quasi leaf segment is at least 1. Since any point in X 1 vertical distance smaller than 2 from a stratum over a vertex of T , there is no harm in adopting this convention. ˜ T , π) be a tree of hyperbolic spaces. Lemma 4.5. Let (X, (a) There exist λ+ (v) ≥ 1, µ(v) ≥ 0 such that, if w0 and w1 are any two v-quasi leaf segments, with initial (resp. terminal) points x0 , x1 (resp. y0 , y1 ) and such that π(w0 ) = π(w1 ) = [a, b] then: 1

dhor (x0 , x1 ) d (a,b) λ+T (v)

d (a,b)

− µ(v) ≤ dhor (y0 , y1 ) ≤ λ+T

(v)dhor (x0 , x1 ) + µ(v)

The constants λ+ (v), µ(v) will be referred to as the constants of quasi-isometry. (b) The identity maps (Xα , dhor ) → (Xα , dvtel ) and (Xα , dvtel ) → (Xα , dhor ) are proper maps. ˜ dv ) is quasi isometric to (X, ˜ dv0 ). (c) For any v, v 0 ≥ 0, (X, tel tel (d) For any α, β ∈ T and v ≥ 0 there exists C(v, dT (α, β)), increasing in both variables, such that for any x, y, z ∈ Xα with z ∈ [x, y], for any horizontal geodesic h C(v,d (α,β)) (h). between Lv (x) ∩ Xβ and Lv (y) ∩ Xβ , Lv (z) ∩ Xβ ⊂ Nhor T 10

(e) For any 0 ≤ w, there is D(w) such that, if s is a v-quasi leaf segment, then s is a (D(w), D(w))-quasi geodesic for the w-telescopic distance. Proof: Item (a) is a straightforward consequence of the definition of a quasi leaf segment.

Item (b) means, in other words, that the telescopic distance tends toward infinity with the horizontal distance, and conversely. This is a consequence of the existence of the constants of quasi isometry given by the first item. The same is true for Item (c). Item (d) amounts to saying that the image of a geodesic under a (a, b)-quasi isometric embedding is C(a, b)-close to any geodesic between the images of the endpoints. This is a well-known assertion, see for instance [9]. Like Item (a), Item (e) is checked by a straightforward computation. ¤ 5. Exponential separation, corridors and first combination theorem 5.1. Exponential separation of quasi leaves. ˜ T , π) be a tree of hyperbolic spaces. Definition 5.1. Let (X, ˜ are M -exponentially separated, or more simply exponentially The v-quasi leaves of X separated, if there exist positive integers t0 , M and a constant λ > 1 such that, for any α ∈ T , for any (x, y) ∈ Xα × Xα with dhor (x, y) ≥ M , any pair of points β, γ ∈ ∂Bα (t0 ) with dT (β, γ) = 2t0 satisfies: max(dshor (Lv (x) ∩ Xβ , Lv (y) ∩ Xβ ), dshor (Lv (x) ∩ Xγ , Lv (y) ∩ Xγ )) ≥ λdhor (x, y) The constants λ, M, t0 will be referred to as the constants of hyperbolicity. Remark 5.2. In Definition 5.1, the sets Lv (x) ∩ Xβ , Lv (y) ∩ Xβ might be empty for a given β ∈ ∂B(t0 ). In this case, we consider that their horizontal distance is infinite, so that the given assertion is satisfied for any pair β, γ (β fixed). A first result of this paper is now stated as follows: ˜ be a tree of hyperbolic spaces. If the v-quasi leaves are exponentially Theorem 5.3. Let X ˜ is a Gromov-hyperbolic metric separated for some v greater than a critical constant then X space. Remark 5.4. The critical constant referred to in Theorem 5.3 above is C5.6 . This constant is chosen so that, if no v-quasi leaf segment can be prolonged past some stratum, then to go past this stratum, one has to prolong them by horizontal geodesics which come close one to each other. This constant would not have appeared if we had restricted ourselves on the one hand to continuous gluing-maps between the strata (which would not have been a great deal) and on the other hand to quasi-isometries instead of tolerating quasi isometric embeddings. With this last restriction, we could have treated semi-direct products, but not the whole world of HNN-extensions and amalgated products. This theorem generalizes Bestvina-Feighn theorem for trees of hyperbolic spaces [3] in the sense that we do not require a Gromov hyperbolic space be a proper space. This difference is the key-point for extending Bestvina-Feighn result to the relatively hyperbolic case. 5.2. Corridors. ˜ T , π) be a tree of hyperbolic spaces. Definition 5.5. Let (X, 11

˜ be two sections of π over T which (a) Let T be a subtree of T and let σ1 , σ2 : T → X are (v, 0)-quasi isometric embeddings, v ≥ 1. A union of horizontal geodesics, at most one in each stratum, connecting each point of σ1 (T ) to a point of σ2 (T ) is a simple v-corridor. (b) A v-corridor C is a concatenation of simple v-corridors {Ci }i=1,··· such that π(C) is a subtree of T and which satisfies the following conditions of “maximality”: (a) for any i 6= j, π(Ci ) ∩ π(Cj ) is either empty or reduced to a single point; (b) if v is a vertex of T in π(C) and e is an edge of T which is incident to v but does not belong to π(C), then there is no v-quasi leaf segment over e starting from C; (c) if x is a point in the horizontal boundary of some Ci such that some v-quasi leaf segment s with π(s) ⊂ π(Cj ), j 6= i, starts at x, then x is in Cj . ˜ a tree of δ-hyperbolic spaces, C a v-corridor. Notations: X From the second item in the definition of a v-corridor, by choosing v sufficiently large enough with respect to δ and to the constants of quasi isometric embeddings of the edge spaces into the vertex spaces, we get the following lemma: Lemma 5.6. There exists C such that if v ≥ C, if C 0 is any v-corridor with π(C)∩π(C 0 ) = {α} then the horizontal diameter of PC (C 0 ) in Xα is less or equal to 2δ. Lemma 5.7. There exists C(v) ≥ v such that: (a) If s is a v-quasi leaf segment, then PC (s) is a C(v)-quasi leaf segment. (b) For any w ≥ C5.7 (v), (C, dw tel ) is a quasi geodesic metric space. Proof: Item (a) is a consequence of Item (d) of Lemma 4.5 and of the hyperbolicity of

the strata. Item (b) is then easily deduced.

¤

˜ are exponentially separated Lemma 5.8. Assume v ≥ C5.6 . If the v-quasi leaves of X with constants of hyperbolicity λv > 1, Mv , t0 ≥ 0 then for any w ≥ v, the w-quasi leaves are exponentially separated, with constants of hyperbolicity λw (λv ) > 1, Mw (Mv ) ≥ 0 and t0 . This is in particular true for the w-quasi leaves of C, with w ≥ C5.7 (v). The proof is easy and left to the reader.

¤

Lemma 5.9. Let g be a v-telescopic path, which is a (a, b)-quasi geodesic for the telescopic r distance dvtel . Then there exists C(a, b, r) ≥ 1 such that, if g ⊂ Ntel (C) then PC (g) is a C5.7 (v) C5.7 (v)-telescopic (C(a, b, r), C(a, b, r))-quasi geodesic of (C, dtel ). Proof: Lemma 5.7 implies in a straightforward way that PC (g) is a C5.7 (v)-telescopic

path. Let us consider any two points x, y in G = PC (g). There are r-close to two points x0 , y 0 in g. We denote by gx0 y0 the subpath of g between these last two points and by Gxy the subpath of G between x and y. Since we now consider the C5.7 (v)-telescopic C5.7 (v) = |gx0 y0 |vvert . From Lemma 3.1 and since any two maximal horizontal distance, |Gxy |vert subapths of G are separated by a quasi leaf segment of vertical length at least 1, we C (v) then get |Gxy |tel5.7 ≤ 2C3.1 (δ)|gx0 y0 |vtel , where δ is the constant of hyperbolicity of the strata. Since g is a v-telescopic (a, b)-quasi geodesic, |gx0 y0 |vtel ≤ advtel (x0 , y 0 ) + b. But dvtel (x0 , y 0 ) ≤ 2r + dvtel (x, y). Therefore: C

|Gxy |tel5.7

(v)

≤ 2C3.1 (δ)(a(2r + dvtel (x, y)) + b).

Since all the telescopic distances are quasi isometric, see Lemma 4.5, the conclusion follows. ¤ 12

6. Approximation of quasi geodesics in corridors We begin with a very general lemma about the constants of hyperbolicity. ˜ T ) be a tree of hyperbolic spaces with exponentially separated v-quasi Lemma 6.1. Let (X, leaves. (a) The constants of hyperbolicity and quasi isometry can be chosen arbitrarily large enough. (b) If λ, M, t0 are the constants of hyperbolicity and M is chosen sufficiently large enough, then there exists C such that, for any α ∈ T , for any (x, y) ∈ Xα × Xα with dhor (x, y) ≥ M , if β ∈ ∂Bα (t0 ) satisfies: 1 dhor (x, y) < dihor (Lv (x) ∩ Xβ , Lv (y) ∩ Xβ ) < λdhor (x, y), λ then, for any n ≥ 1, for any w ∈ Geod(T ) starting at α with [α, β] ⊂ w and |w|T ≥ C + nt0 : dihor (wx, wy) ≥ λn dihor (x, y). Of course the same inequality holds if β satisfies dihor (Lv (x) ∩ Xβ , Lv (y) ∩ Xβ ) ≥ λdhor (x, y). Convention: Throughout the paper, the constants of hyperbolicity and of quasi isometry

are chosen sufficiently large enough to satisfy the conclusions of Lemma 6.1, and also sufficiently large enough so that computations make sense. Moreover the horizontal subpaths of the (a, b)-quasi geodesics considered will be assumed to be horizontal geodesics. The hyperbolicity of the strata gives a constant C(a, b) such that any (a, b)-quasi geodesic g may be substituted by another one g 0 satisfying this 0 property and with dH tel (g, g ) ≤ C(a, b). Remark 6.2. We won’t always indicate the dependence of the various constants on v (where the telescopic distance we work with is dvtel ). However the constants appearing in Lemmas 6.3, 6.4, 6.7, in Corollary 6.5 and in Proposition 6.8 depend on v. This is not important because at the end the constant v will be just chosen sufficiently large enough with respect to the original constants given with the tree of hyperbolic spaces. We now state a lemma about quasi geodesics. It holds not only in a corridor but in the whole tree of hyperbolic spaces. ˜ T , π) be a tree of hyperbolic spaces with exponentially separated Lemma 6.3. Let (X, ˜ There exist C(a, b) and v-quasi leaves. Let g be a v-telescopic (a, b)-quasi geodesic in X. D such that, if [x, y] ⊂ g ∩ Xα satisfies dhor (x, y) ≥ C(a, b) then for any w ∈ Geod(T ) starting at α with |w|T ≥ D + nt0 , n ≥ 1, we have dihor (wx, wy) ≥ λn dhor (x, y). Proof: We denote by λ > 1, M, t0 ≥ 1 the constants of hyperbolicity and by λ+ , µ the

constants of quasi isometry. Let us choose n? (a) such that λan? < 1. Solving the inequality ? t0 +b e > a( λn1? e + 2n? t0 ) + b gives us e(a, b) ≥ 2an . 1−a λn1 ? 0 0 Claim: If dhor (x, y) ≥ e(a, b), if x , y are the endpoints of two v-quasi leaf segments s, s0 of vertical length n? t0 , starting at x and y and with π(s) = π(s0 ), then for any w0 ∈ Geod(T ) with w0 π(s) ∈ Geod(T ) and |w0 |T = t0 , dihor (w0 x0 , w0 y 0 ) ≥ λdhor (x0 , y 0 ) holds. Proof of Claim: Assume the existence of w with |w|T = n? t0 such that for some x0 , y 0 with x ∈ wx0 , y ∈ wy 0 and dhor (x0 , y 0 ) ≥ M , dhor (x, y) ≥ λn? dhor (x0 , y 0 ) holds. Then λn1? e+2n? t0 is the telescopic length of a telescopic path between x and y. But the inequality given at the beginning of the proof tells us that the existence of such a telescopic path is a 13

contradiction with the fact that g is a v-telescopic (a, b)-quasi geodesic. Therefore, if dhor (x, y) ≥ e(a, b) and dhor (x, y) ≥ λn+? (M + µ) (this last inequality is to assert that dhor (x0 , y 0 ) ≥ M - see above), then dhor (x0 , y 0 ) does not increase after t0 in the direction of the v-quasi leaf segments s, s0 . The claim follows from the exponential separation of the v-quasi leaf segments. −n? From the inequality given by the Claim, since dhor (x0 , y 0 ) ≥ λ+ (dhor (x, y) + µ), we easily compute an integer N? such that, if w0 is as in the Claim but with length N? t0 then dihor ([w0 π(s)]x, [w0 π(s)]y) ≥ λdhor (x, y). Setting D = N? t0 and C(a, b) = e(a, b), the constant computed above, we get the lemma. ¤ ˜ T , π) a tree of δ-hyperbolic spaces, C a Notations: δ a fixed non negative constant, (X, corridor with exponentially separated v-quasi leaves, λ > 1, M, t0 ≥ 1 the associated constants of hyperbolicity, λ+ , µ the associated constants of quasi isometry, g a v-telescopic (a, b)-quasi geodesic of C. The above constants are chosen sufficiently large enough to satisfy the conclusions of Lemma 5.7. Lemma 6.4. There exists C(a, b) such that, if the endpoints x, y of g both lie in a same stratum Xα , if g lies in the closure of a single connected component of C − Xα and if dhor (x, y) ≥ C(a, b) then, for any w ∈ Geod(T ) starting at α with |w|T ≥ C(a, b) + nt0 , n ≥ 1, and w ∩ π(g) = {α}, we have: dihor (wx, wy) ≥ λn dhor (x, y). Proof: Let us observe that, if [p, q] is any horizontal geodesic in g then the v-quasi leaves

of p and q bound a horizontal geodesic [p0 , q 0 ] in [x, y]. Claim: If dhor (p0 , q 0 ) ≥ Cte, with Cte ≡ λt+0 (C6.3 (a, b) + t0 + µ), then for any w as given by the current Lemma with |w|T ≥ D6.3 + t0 , dihor (wp0 , wq 0 ) ≥ λdhor (p0 , q 0 ). Proof of Claim: If p0 and q 0 are not exponentially separated in the direction of p, q after t0 , then, because of the exponential separation of the quasi leaves, they are exponentially separated after t0 in the direction of w, which yields the announced inequality. Let us assume that p0 , q 0 are separated after t0 in the direction of [π(p0 ), π(p)]. Thus dihor (rp0 , rq 0 ) ≥ λn dhor (p0 , q 0 ) for r ∈ Geod(T ) with |r|T = nt0 , r ∩ w = {α}. Therefore dhor (p, q) ≥ C6.3 (a, b) + t0 . Lemma 6.3 then implies that p, q are exponentially separated in the direction of [π(p), π(p0 )] after D6.3 + t0 , and the claim is proved. There is a finite decomposition of [x, y] ⊂ Xα in subgeodesics [p0j , qj0 ] with disjoint interiors such that each [p0j , qj0 ] connects the v-quasi leaves of the endpoints of a maximal horizontal geodesic in g. We denote by ID the set of [p0j , qj0 ]’s with dhor (p0j , qj0 ) ≥ Cte and by IC the set of the others. Let us choose an integer n ≥ 1. We consider a stratum Xβ with dT (β, α) = D6.3 + nt0 . Let h be the horizontal geodesic in C ∩ Xβ which connects the v-quasi leaves of x and y. Assume that the endpoints of h are exponentially separated after t0 in the direction of [β, α]. Then: (1)

λn |ID |hor ≤ |h|hor ≤ λ−n (|ID |hor + |IC |hor )

so that

λn − λ−n |ID |hor λ−n and consequently, since dhor (x, y) = |ID |hor + |IC |hor , |IC |hor ≥

|IC |hor ≥

X(n) dhor (x, y) 1 + X(n) 14

X(n) = 1, there is n? ≥ 0 such that for any n ≥ n? , n→+∞ 1 + X(n) 1 |IC |hor ≥ dhor (x, y). 2 But, by definition, the horizontal length of each subgeodesic in IC is smaller than Cte. 1 dhor (x, y) + 1. Thus the number of elements in IC is at least the integer part of 2Cte Furthermore, since g is a v-telescopic path, the telescopic length of any subpath of g containing j maximal horizontal geodesics is at least (j − 1). We so obtain: 1 |g|vtel ≥ dhor (x, y). 2Cte On the other hand: dvtel (x, y) ≤ λ−n dhor (x, y) + 2nt0 . since there is a v-telescopic path between x and y the telescopic length of which is given by the right-hand side of the above inequality. Since g is a (a, b)-quasi geodesic, the last two inequalities give n?? ≥ 0 such that for n ≥ n?? : 2ant0 + b . dhor (x, y) ≤ 1 − aλ−n 2Cte with X(n) =

λn −λ−n . λ−n

Since lim

Taking the maximum of n? , n?? and the above upper-bound for dhor (x, y), we get the announced constant in the case where the endpoints of the horizontal geodesic h above are exponentially separated in the direction of [β, α]. If not, there are in all the other directions so that we easily get a constant N ≥ 0 such that dihor (wx, wy) ≥ λdhor (x, y) for any w ∈ Geod(T ) with |w|T = N t0 and [π(x), π(h)] ⊂ w. Lemma 6.4 is then easily deduced. ¤ As a consequence we have: Corollary 6.5. With the assumptions and notations of Lemma 6.4, there exists C(a, b, d) ≥ d such that, if x, y are the endpoints of two v-quasi leaf segments s, s0 with dihor (s, s0 ) ≤ d and which do not lie in the same side of Xα than g then dhor (x, y) ≤ C(a, b, d). ¤ Remark 6.6. At this point, we would like to notice that Lemma 6.4 is similar to Lemma 6.7 of [12]. However in addition of some misprints, a slight mistake took place there in the proof of the Lemma. Indeed the inequality (1) in the proof of Lemma 6.4 is true here, in the corridor, but there the constant λ should have been modified to take into account the so-called “cancellations”. Lemma 6.7. Let x and y be the endpoints of a r-quasi leaf segment s in C. There exists C(r) such that, if z ∈ Lv (y) ∩ Xπ(x) satisfies dhor (x, z) ≥ C(r), then for any w ∈ Geod(T ) with |w|T = t0 and w ∩ π(s) = {π(x)}, dihor (wx, wz) ≥ λdhor (x, z). Proof: If |s|vert ≤ t0 , the existence of the constants of quasi isometry, Item (a) of Lemma

4.5, and the definition of r-quasi leaf segment give an upper-bound for dhor (x, z). Let us thus assume |s|vert > t0 . Choose d such that λd − r0 ≥ 2r0 , where r0 is the above upperbound when |s|vert = t0 . Then set C = max(d, M ). Assume dhor (x, z) ≥ C and that x and z are exponentially separated in the direction given by s. If [π(x), π(y)] = w0 w0 with |w0 |T = t0 , then dihor (w0 x, w0 z) ≥ λdhor (x, z). The inequality used to defined d and an easy induction then tell us that the horizontal distance between s and Lv (y) increases along s when going from x to y which of course cannot happen. The conclusion follows from the exponential separation of the quasi leaf segments in C. ¤ 15

We are now in position to prove the following proposition (we recall that M is a constant of hyperbolicity): Proposition 6.8. Let C be a corridor with exponentially separated v-quasi leaves in a tree of hyperbolic spaces. There exists C(dhor (x, y), a, b) such that, if G is a v-telescopic (a, b)quasi geodesic of (C, dvtel ), which connects the v-quasi leaves Lv (x) and Lv (y) of two points x, y ∈ C ∩Xα , then G is contained in the C(dhor (x, y), a, b)-neighborhood of the v-quasi leaf segments joining its endpoints to x and y. If L0 > L ≥ M , then C(L0 , a, b) > C(L, a, b). Proof: Let us set d = dhor (x, y). From Corollary 6.5, by suitably choosing maximal

subapths of G which satisfy the assumptions of Lemma 6.4, and substituting each one of them by a horizontal geodesic connecting its endpoints, we construct a telescopic path G 0 which is a C6.5 (a, b, d)-quasi leaf segment or the concatenation of two C6.5 (a, b, d)quasi leaf segments. From Lemma 6.7, G 0 lies in a bounded neighborhood of the v-quasi 0 leaf segments connecting its endpoints to x and y. From the construction, dH tel (G, G ) ≤ aC6.5 (a, b, d) + b + 1. The proposition follows. ¤ 7. Quasi geodesic bigons in corridors are thin We prove here the following result, which is a crucial step toward the proof of Theorem 5.3: Proposition 7.1. Let C be a w-corridor (w ≥ C5.6 ) with exponentially separated v-quasi leaves (v ≥ C5.7 (C5.6 )) in a tree of hyperbolic spaces. There exists C(a, b) such that (a, b)-quasi geodesic bigons of (C, dvtel ) are C(a, b)-thin. We first introduce a notion which will be used for the proof: Definition 7.2. A diagonal is any horizontal geodesic which minimizes the horizontal distance among all the horizontal geodesics which connect the v-quasi leaves of its endpoints. If [x, y] is a diagonal then we say that dhor (x, y) is the diagonal distance between x and y (or between their v-quasi leaves). The diagonal distance is denoted by diag(x, y) or diag(Lv (x), Lv (y)). Proof of Proposition 7.1: We denote by G and G 0 the sides of the bigon. We denote by

λ, M, t0 the constants of hyperbolicity. We choose L ≥ M . Let x be any point in G. We consider two points y, z ∈ G on both sides of x such that: (2)

2C

x lies in the closed complement of Ntel 6.8

(a,b,L)

2C

(Lv (y)) ∪ Ntel 6.8

(a,b,L)

(Lv (z)),

i and such that y (resp. z) minimizes dtel (x, Lv (u)) among all u’s in G lying in the same side of x as y (resp. as z) and satisfying the above property (2). Let us assume for a while that such points y, z exist. Obviously there is ² depending only on the tree of hyperbolic spaces (the constants of quasigeodesicity) such that ditel (x, Lv (y)) ≤ ² + 2C6.8 (a, b, L) and ditel (x, Lv (z)) ≤ ² + 2C6.8 (a, b, L). Moreover, from Proposition 6.8, diag(Lv (y), Lv (z)) > L. Let h be a diagonal between Lv (y) and Lv (z). Since v-quasi leaves in C are exponentially separated, it is straightforward from the definition of a diagonal and from the inequality |h|hor > L ≥ M that the endpoints of the diagonal h are exponentially separated in all directions. From Lemma 4.5, Item (b), the telescopic distance increases with the horizontal distance. Since ditel (x, Lv (y)) ≤ ² + 2C6.8 (a, b, L) and ditel (x, Lv (z)) ≤ ² + 2C6.8 (a, b, L), this simple observation followed D(a,b,L) (h). by a straightforward computation gives a constant D(a, b, L) such that x ∈ Ntel

16

Since G 0 connects the same v-quasi leaves as G, some subpath p0 of G 0 connects Lv (y) to Lv (z). Once again, from Proposition 6.8 and since the endpoints of the diagonal h with |h|hor ≥ M are exponentially separated in all directions, some point κ ∈ p0 ⊂ G 0 C (a,b,|h|hor ) satisfies κ ∈ Ntel6.8 (h). We saw above that Lemma 4.5 gives an upper bound i v i for dhor (x, L (y)) and dhor (x, Lv (z)). Whence an upper-bound for the horizontal length between Lv (y) and Lv (z) in the stratum containing x, whence an upper-bound L0 for the horizontal length of the diagonal h. The constant L0 depends only on a, b and L (which was chosen at the beginning of the proof). Since |h|hor ≥ M , we conclude from C (a,b,L0 ) Proposition 6.8 that κ ∈ Ntel6.8 (h). Therefore dvtel (x, κ) ≤ D(a, b, L) + L0 (a, b, L). This is what we wanted. The case where there exists no pair of points y, z satisfying (2) means that x is close to the v-quasi leaf of some endpoint of G. This case is treated by considering a horizontal geodesic with horizontal length L and with one endpoint in this v-quasi leaf. Proposition 6.8 then gives the conclusion following similar, but simpler, arguments than above. ¤ 8. Quasiconvexity of corridors In this section we prove the following proposition: ˜ be a tree of hyperbolic spaces with exponentially separated vProposition 8.1. Let X quasi leaves, v ≥ C5.6 , and equipped with the v-telescopic distance. There exists C(a, b) ˜ if C is a corridor containing such that, if g is a v-telescopic (a, b)-quasi geodesic in X, the endpoints of g then C(a,b) g ⊂ Ntel (C). The following lemma reduces Proposition 8.1 to the case where the image of g under ˜ onto T ) is contained in the image of C under π. π (the projection of X ˜ T ) be a tree of hyperbolic spaces. If v ≥ C5.6 then there exists C(a, b) Lemma 8.2. Let (X, such that, if g is any telescopic (a, b)-quasi geodesic the endpoints of which lie in a vcorridor C, then there is a telescopic (a, b + 2δ)-quasi geodesic G with dH tel (g, G) ≤ C(a, b) and π(G) ⊂ π(C). Proof: Let γ ∈ T be an endpoint of π(C). Assume that g 0 is a maximal subpath of g

with endpoints in Xγ and such that π(g 0 ) ∩ π(C) = γ. Then, since v ≥ C5.6 , Lemma 5.6 tells us that the endpoints of g 0 are 2δ-close with respect to the horizontal distance. Since g is a (a, b)-quasi geodesic, g 0 is (2aδ + b)-close to Xγ with respect to the telescopic distance. Substituting g 0 by a horizontal geodesic between its endpoints and repeating this substitution for all the subpaths of g as g 0 yields a quasi geodesic as announced. ¤ Convention: In what follows, in the course of the proof of Proposition 8.1, it will be tacitly

assumed that the quasi geodesics g and corridors C we work with satisfy the conclusion of Lemma 8.2, i.e. π(g) ⊂ π(C). 8.1. Two basic lemmas. We need first a very general lemma about Gromov hyperbolic spaces. Lemma 8.3. Let (X, d) be a Gromov hyperbolic space. There exists an increasing affine function D(r) ≥ 0, and C ≥ 0 such that, if [x, y] is a diameter of a ball Bx0 (r) with r ≥ C and w is any path in X with w ∩ Bx0 (r) = {x, y}, then |w|d ≥ eD(r) . This lemma is a rewriting of Lemma 1.6 page 26 of [9]. 17

¤

˜ be a tree of δ-hyperbolic spaces with exponentially separated v-quasi Lemma 8.4. Let X leaves. There exists C such that, if x, y, z, t are the vertices of a geodesic quadrilateral in some stratum Xα , with dhor (x, z) ≤ 2δ, dhor (y, t) ≤ 2δ, and dhor (x, y) ≥ C, dhor (z, t) ≥ C, then for any w ∈ Geod(T ) with |w|T ≥ C6.1 + nt0 , starting at π(x): dihor (wx, wy) ≥ λn dhor (x, y) ⇔ dihor (wz, wt) ≥ λn dhor (z, t) Proof: Let us consider any w ∈ Geod(T ) with |w|T = t0 starting at α. From Lemma 4.5,

dshor (wx, wz) ≤ λt+0 (2δ + µ) and dshor (wy, wt) ≤ λt+0 (2δ + µ). Assume dihor (wx, wy) ≥ λdhor (x, y) but dihor (wz, wt) < λdhor (z, t). We take dhor (x, y) ≥ M and dhor (z, t) ≥ M . Assume dshor (wz, wt) ≤ λ1 dhor (z, t). But dhor (z, t) ≤ 4δ + dhor (x, y). Putting together these inequalities we get 1 λdhor (x, y) ≤ 2λt+0 (2δ + µ) + (4δ + dhor (x, y)). λ Whence an upper bound for dhor (x, y) and thus for dhor (z, t). If dshor (wz, wt) > λ1 dhor (z, t) then the lemma follows from the definition of the constant C6.1 , see the corresponding lemma. ¤ The above two lemmas are not needed if one only considers trees of 0-hyperbolic spaces, the proof in this last case being much simpler. 8.2. Approximation of quasi geodesics with bounded vertical deviation. ˜ T ), a quasi geodesic Proposition 8.5 below states that, in a tree of hyperbolic spaces (X, with bounded image in T lies close to a corridor between its endpoints. This is intuitively obvious and nothing is new neither surprising in the arguments of the proof: they heavily rely upon the δ-hyperbolicity of the strata and the fact that strata are quasi isometrically embedded into each other. For the sake of brevity, we do not develop them here. ˜ T , π) be a tree of hyperbolic spaces. There exists C(κ, a, b) Proposition 8.5. Let (X, such that, if g is any v-telescopic (a, b)-quasi geodesic with dsT (π(g), π(g)) ≤ κ, if C is a C(κ,a,b) corridor between its endpoints then g ⊂ Ntel (C). 8.3. Stairs. The sign '1 stands for an equality up to ±1. ˜ T , π). Definition 8.6. Let C be a corridor in a tree of hyperbolic spaces (X, A r-stair relative to C, r ≥ M , is a telescopic path S the maximal quasi leaf segments of which have vertical length greater than C6.1 and such that, for any maximal horizontal geodesic [ai , bi ] in S: (a) dhor (ai , bi ) ≥ r and dihor ([ai , bi ], C) '1 dhor (ai , PC (ai )), (b) any pair of points a, b ∈ [ai , bi ] with dhor (a, b) ≥ r are exponentially separated in the direction of the T -geodesic [π(ai ), π(ai+1 )]. Lemma 8.7. With the notations of Definition 8.6: there exist C ≥ C8.4 such that for any r ≥ C, if S is a r-stair relative to C, if U is a corridor between a quasi leaf of the terminal point of S and a quasi leaf in the boundary of C, then r+2δ (U). S ⊂ Nhor

18

Proof: Let ai , bi ∈ S as given in Definition 8.6 and let z be a point at the intersection of

the stratum Xπ(ai ) with a quasi leaf of some point farther in the stair. Then: Claim 1: There exists K > 0 not depending on ai nor z such that, if r is sufficiently large enough then dihor ([ai , z], C) ≥ dhor (ai , PC (ai )) − K. Proof of Claim 1: Choose K such that eD8.3 (K) > 4δ + 1 and assume dihor ([ai , z], C) < dhor (ai , PC (ai )) − K. Then Lemma 8.3 implies that [bi , z] descends at least until a 2δneighborhood of ai . Assume r ≥ C8.4 + 2δ. Then Lemma 8.4 gives an initial segment of [bi , z] of horizontal length greater than r − 2δ which is dilated in the direction of [π(ai ), π(ai+1 )]. If r is chosen sufficiently large enough with respect to the constants of hyperbolicity for a corridor (see Lemma 5.8), we get z 0 at the intersection of the quasi leaf of z with the stratum Xπ(ai+1 ) such that dihor ([ai+1 , z 0 ], C) < dhor (ai+1 , PC (ai+1 )) − K. The repetition of these arguments show that the horizontal distance between S and the quasi leaf of z does not decrease along S. This is an absurdity since z was chosen in the quasi leaf of a point farther in S. The proof of Claim 1 is complete. Claim 2: There exists K(r) not depending on bi nor z such that, if r is sufficiently large enough then dihor ([bi , z], C) ≥ dhor (bi , PC (bi )) − K(r). Proof of Claim 2: Let z? ∈ [bi , z] with dhor (z? , PC (z? )) '1 max(dihor ([bi , z], C), dhor (ai , PC (ai ))). From the δ-hyperbolicity of the strata, [bi , z? ] lies in the horizontal 2δ-neighborhood of [ai , bi ]. Assume dhor (bi , z? ) ≥ r and is sufficiently large enough to apply Lemma 8.4. Then there is K(r) such that, if z? satisfies dhor (z? , PC (z? )) < dhor (bi , PC (bi )) − K(r), the points bi and z? are exponentially separated in the direction of [π(ai ), π(ai+1 )]. We thus obtain at ai+1 a situation similar to that of Claim 1. The proof of Claim 2 follows. Lemma 8.7 is easily deduced from the above two claims, we leave the reader work out the easy details. ¤ Lemma 8.8. There exists C > 0 such that, for any r ≥ C8.7 , if S is a r-stair relative to C, which is not contained in the vertical C-neighborhood of the stratum containing its initial point, then the terminal point of S does not belong to the telescopic r-neighborhood of C. Proof: Decompose S in maximal substairs S0 · · · Sk such that π(Sj ) is a geodesic of T .

Let [ai , bi ] be the first maximal horizontal geodesic in Sj , let x be the initial point of Sj and let z be any point in Sj with nt0 ≤ dT (π(z), π(x)) ≤ (n + 1)t0 . The inequality (3)

dhor (z, PC (z)) ≥ Cteλn dhor (ai , bi )

is an easy consequence of the definition of a stair and of Lemma 8.4 as soon as r ≥ C8.4 . Indeed, the initial segment of horizontal length r in [bi , PC (bi )] lies in the horizontal 2δneighborhood of [bi , ai ]. The assertion then follows from Item (b) of Definition 8.6 and Lemma 8.4. The inequality (3) readily gives the announced result. ¤ 8.4. Approximation of a quasi geodesic by a stair.

˜ T ) a tree of δ-hyperbolic spaces with exponentially separated v-quasi Notations: (X, leaves, v ≥ C5.6 , C a corridor, g a v-telescopic (a, b)-quasi geodesic. Lemma 8.9. Assume that the endpoints of g are in a horizontal r-neighborhood of C and that g lies in the closed complement of this horizontal neighborhood. Suppose moreover that the maximal quasi leaf segments in g have vertical length greater than 3(C6.1 + D6.3 ). Then there exist C(r, a, b), D(a, b), E(r, a, b) such that for any r ≥ D(a, b), either g lies in the telescopic C(r, a, b)-neighborhood of a E(r, a, b)-stair relative to C, where E(r, a, b) is affine in r, or g is contained in the telescopic C(r, a, b)-neighborhood of C. 19

Proof: We decompose the proof in two steps. The first one is only a warm-up, to present

the ideas in a particular, but important, case. The general case, detailed in the second step, is technically more involved but no new phenomenon appears. Step 1: Proof of Lemma 8.9 when the horizontal length of any maximal horizontal subpath in g is greater than some constant (depending on a et b). The endpoints of any horizontal subpath h of g with horizontal length greater than C6.3 (a, b) are exponentially separated under every geodesic w of T with length D6.3 . If |h|hor ≥ C8.4 , this is also true for any horizontal geodesic h0 in the 2δ-neighborhood of h. Finally, if |h|hor is sufficiently large enough, by Lemma 5.8 the endpoints of h are also exponentially separated in any v-corridor containing h. If e(a, b) is the maximum of the above constants, we now assume |h|hor ≥ 3e(a, b). Let us consider two consecutive maximal horizontal geodesics h1 , h2 in g, separated by a quasi leaf segment s. Let D be a corridor containing h1 and s. Then:

(4)

2δ (D)|hor ≤ e(a, b). |h2 ∩ Nhor

Otherwise we have a contradiction with the fact that the endpoints of any subgeodesic of h2 whose length is greater than C6.3 (a, b) are exponentially separated in the direction of h1 . From the inequality (4), the concatenation of h1 , s and h2 is e(a, b)-close, with respect to the horizontal distance, of a 2e(a, b)-stair relative to C if dihor (h1 , C) '1 dhor (a1 , PC (a1 )) where a1 is the initial point of h1 . Let us now set r ≥ 3e(a, b) and assume that the maximal horizontal geodesics in g have horizontal length greater than r. Let x be the initial point of g (in particular dhor (x, PC (x)) '1 r). Let s be the quasi leaf segment starting at x and ending at y in g. Let h be the maximal horizontal geodesic following s along g. Let n ≥ 1 be the greatest integer with n(C6.1 + D6.3 ) ≤ |s|vert . By assumption x and PC (x) are exponentially separated in the direction of s. Since the strata are quasi isometrically embedded one into each other, this gives κ > 1 such that, any two points a, b ∈ [x, PC (x)] with dhor (a, b) ≥ max( κ1 r, M ) satisfy dhor (π(s)a, π(s)b) ≥ λn dhor (a, b). Thus the same arguments as those exposed above when working with h1 , h2 2δ show that |h ∩ Nhor ([y, PC (y)])|hor ≤ max(e(a, b), λn1 κ r, M ). If n is greater than some critical constant n∗ , this last maximum is equal to e(a, b). Thus, in this case, we can take h1 = [x, PC (x)] and h2 = h: the above arguments prove that the concatenation of h1 , s and h2 is e(a, b)-close to a e(a, b)-stair. If n is smaller than n∗ , then we substitute r by n (C +D ) λ+∗ 6.1 6.3 r, modify g by taking the starting point at the endpoint y of s and take h1 as the first maximal horizontal geodesic. In both cases, by repeating the arguments above at any two consecutive maximal horizontal geodesic following the first two ones along g, we show that g is e(a, b)-close, with respect to the horizontal distance, of a e(a, b)-stair relative to C. ¤ Step 2: Adaptation of the argument to the general case: The boundary leaves of C are denoted by L1 and L2 , and g goes from L1 to L2 . We choose a positive constant r, which when necessary will be set sufficiently large enough with respect to the constants C8.7 , M, δ and C8.4 . Let x0 be the initial point of g. It lies in the boundary of the horizontal r-neighborhood of C. We denote by Ci and xi , i = 1, · · · , a sequence of corridors and points of g defined inductively as follows: 20

(a) Ci is a corridor with boundary leaves a v-quasi leaf of xi−1 and the v-quasi leaf L2 in ∂C, (b) xi is the first point following xi−1 along g such that dhor (xi , PCi (xi )) ≥ r. The subpath of g between xi−1 and xi is denoted by gi−1,i . Obviously gi−1,i is contained in the horizontal r-neighborhood of Ci . We project it to Ci . From Lemma 5.9, C (v) we get a C5.7 (v)-telescopic (C5.9 (a, b, r), C5.9 (a, b, r))-quasi geodesic of (Ci , dtel5.7 ). We set X(a, b, r) = C6.8 (r, C5.9 (a, b, r), C5.9 (a, b, r)). From Proposition 6.8, PCi (gi−1,i ) is contained in the X(a, b, r)-neighborhood of the concatenation of a subpath of [xi−1 , PCi−1 (xi−1 )] with a quasi leaf segment in Ci (and is followed by [PCi (xi ), xi ]). Consider in this approximation of (a subpath of) g a maximal collection of points yi which defines a r-stair relative to C. The points yi do not necessarily agree with the xi ’s, because it might happen that, after xi−1 for instance, the approximation constructed above reenters in the r-neighborhood of Ci−1 before leaving the r-neighborhood of Ci . We proceed as in Step 1 and choose the yi ’s so that: (a) either yi is contained in a maximal horizontal geodesic, and from the observations in Step 1, this horizontal geodesic may be included in a stair, (b) or the vertical distance from yi to the next horizontal geodesic is at least C6.1 +D6.3 . Either we obtain a non-trivial r-stair relative to C which approximates a subpath g00 of g or the approximation we constructed above exhausts g and is contained in some telescopic neighborhood of C the size of which is obtained from the previously exhibited constants. In this last case, the same assertion holds for the whole path g. This is one of the announced alternatives. We can thus assume that we got y0 , · · · , yk forming a r-stair relative to C. It is denoted by S. Since the strata are quasi isometrically embedded one into each other, there is κ > 1, only depending on the constants of quasi isometry, such that S is in fact a max( κ1 r, M, e(a, b))-stair relative to C. As soon as r > κ(M + e(a, b)), which we suppose from now, this maximum is just κ1 r. Thus S is a κr -stair whose maximal horizontal geodesics have horizontal length at least r. By construction S approximates g00 ⊂ g. We now consider the maximal subpath g10 of g starting at (or near - recall that we constructed an approximation of a subpath of g) yk which lies in the r-neighborhood of Ck . This last corridor plays the rˆole of the corridor U of Lemma 8.7. We project the subpath g10 to Ck , so getting a (C5.9 (a, b, r), C5.9 (a, b, r))quasi geodesic of this corridor. From Lemma 8.7, and because of the hyperbolicity of the strata, each horizontal geodesic of the κr -stair S admits a subgeodesic with horizontal length greater than κ−1 r in the horizontal 2δ-neighborhood of Ck . If r is chosen sufficiently κ large enough, Lemma 8.4 gives horizontal geodesics in Ck with horizontal length greater than M which are dilated in the same directions than the horizontal geodesics of S. Now Proposition 6.8 applies and allows us to approximate the projection of g10 on Ck by a sequence of these horizontal geodesics. But each one of these horizontal geodesics is close to a point in g00 ⊂ g. Thus, since g is a (a, b)-quasi geodesic, the vertical length of g10 , and so its telescopic length, is bounded above by a constant depending on a and b. So we can forget g10 and continue the construction of our κr -stair relative to C at the point where the approximation of g10 leaves the r-neighborhood of Ck . We eventually exhaust g and obtain a κr -stair relative to C. ¤ 8.5. Proof of Proposition 8.1. Let g and C be as given by this proposition. Lemma 8.2 gives a telescopic (a, b + 2δ)-quasi geodesic g0 with dH tel (g, g0 ) ≤ C8.2 (a, b) such that π(g0 ) ⊂ π(C) (this property was implicitly assumed before). Assume that some subpath 21

g00 of g0 leaves and then reenters the horizontal D8.9 (a, b + 2δ)-neighborhood of C. Assume that g00 is not contained in the telescopic C8.9 (D8.9 (a, b + 2δ), a, b)-neighborhood of C. Suppose for the moment that the vertical quasi leaf segments in g00 have vertical length greater than 3(C6.1 + D6.3 ). Then Lemma 8.9 gives G, a R(a, b)-stair relative to C, 0 where R(a, b) ≡ E8.9 (D8.9 (a, b + 2δ), a, b + 2δ), with dH tel (g0 , G) ≤ C8.9 (D8.9 (a, b + 2δ), a, b). From Lemma 8.8, G does not leave the vertical C8.8 (R(a, b))-neighborhood of the stratum containing the initial point of G. Therefore, setting V (a, b) = C8.8 (R(a, b)) + C8.9 (D8.9 (a, b + 2δ), a, b), g00 does not leave the vertical V (a, b)-neighborhood of this stratum. From Proposition 8.5, g00 lies in the telescopic C8.5 (V (a, b), a, b)-neighborhood of C. It remains to consider the case where the vertical quasi leaf segments in g00 are not sufficiently large enough. Let s be a quasi leaf segment in g with |s|vert < 3(C6.1 + D6.3 ) = X. If s is contained in a quasi leaf segment s0 of vertical length greater than X, then we modify g00 by sliding, along s0 , a horizontal geodesic in g00 incident to s until getting a quasi leaf segment with vertical length X. This yields a new telescopic (a0 , b0 )-quasi geodesic in a bounded neighborhood of g, where the constants a0 , b0 only depend on a, b and on the constants of quasi isometry. After finitely many such moves, either we obtain a quasi geodesic as desired, and we are done, or we obtain a quasi leaf segment s from bi to ai+1 satisfying the following properties (we still denote by g00 the (a0 , b0 )-quasi geodesic eventually obtained, we denote by s0 the quasi leaf segment of g00 ending at ai and by s1 the one starting at bi+1 ): (a) there is no quasi leaf segment starting at ai (resp. at ai+1 ) over the edge π(s) (resp. over π(s1 )); (b) there is no quasi leaf segment ending at bi over π(s0 ). Consider horizontal geodesics αi = [ai , PC (ai )], βi = [bi , PC (bi )], αi+1 = [ai+1 , PC (ai+1 )] and βi+1 = [bi+1 , PC (bi+1 )]. By the δ-hyperbolicity of the strata, there is a0i ∈ [ai , bi ] ∩ 2δ 2δ Nhor (αi ∪ βi ) and b0i ∈ [ai+1 , bi+1 ] ∩ Nhor (αi+1 ∪ βi+1 ). Because the strata are quasi isometrically embedded one into each other, we get two points a00i , b00i which satisfy: (A) they are Y -close (with respect to the horizontal distance) respectively to a0i and b0i , where the constant Y only depends on δ and on the constants of quasi isometry; (B) there is a v-quasi leaf segment from a00i to b00i which is contained in a larger v-quasi leaf segment going over π(s0 ) and π(s1 ). We modify g00 by going from ai to a00i then to b00i and eventually end at bi+1 . The resulting path is a (a00 , b00 )-quasi geodesic, where the constants a00 , b00 only depends on δ and on the constants of quasi isometry. Moreover this new path is in a bounded neighborhood of g00 . Thanks to Item (B), we can modify it by enlarging the quasi leaf segment from a00i to b00i . Since the vertical distance between two strata is uniformly bounded away from zero, after finitely many such substitutions, we eventually get a quasi geodesic, in a bounded neighborhood of g, which satisfies the assumptions required by Lemma 8.9. This completes the proof of Proposition 8.1. ¤ 9. Conclusion The following section is devoted to the proof of: ˜ T ) be a tree of hyperbolic spaces with exponentially separated Theorem 9.1. Let (X, v-quasi leaves, v ≥ C5.7 (C5.6 ). There exists C(a, b) such that v-telescopic (a, b)-quasi geodesic bigons are C(a, b)-thin. 22

Indeed, Theorem 5.3 is easily deduced from this theorem thanks to the following lemma, first indicated to the author by I. Kapovich: Lemma 9.2. [12] Let (X, d) be a (r, s)-quasi geodesic space. If for any r0 ≥ r, s0 ≥ s, there exists δ(r0 , s0 ), such that (r0 , s0 )-quasi geodesic bigons are δ(r0 , s0 )-thin, then (X, d) is a 2δ(r, 3s)-hyperbolic space. 9.1. Hyperbolicity. ˜ T ) be a tree of hyperbolic spaces. If v ≥ C5.6 and g is any Lemma 9.3. Let (X, telescopic (a, b)-quasi geodesic, then there exist a constant C(a, b), independent of g, a collection of v-corridors Ci , i = 0, · · · , k, and a collection of horizontal geodesics hj , j = 0, · · · , k + 1, such that g lies in the telescopic C(a, b)-neighborhood of the concatenation h0 C0 · · · hi Ci · · · Ck hk+1 , termed a pseudo-corridor. The pseudo-corridor only depends on the endpoints of g. Proof: Let Xα , Xβ be the strata containing the initial and terminal points of g. There

is a unique sequence of γi ∈ [α, β], i = 1, · · · , k, γ0 = α, γk = β such that γi maximizes dT (γi−1 , φ) among all φ’s in [α, β] such that some v-quasi leaf segment connects Xγi−1 to Xφ . We denote by Yi the maximal region of Xγi for which v-quasi leaf segments are defined from Yi to Xγi+1 . Since v ≥ C5.6 , from Lemma 5.6, for any γi , [γi γi+1 ]Yi is connected to Yi+1 by a horizontal rectangle Ri of width at most 2δ. We denote by ai (resp. bi ) a point in Ri ∩ [γi γi+1 ]Yi (resp. in Ri ∩ Yi+1 ). We then denote by Ci a v-corridor between bi and ai+1 (that is for a horizontal geodesic [bi , Xγi+1 ∩ Lv (ai+1 )]). And hi = [ai , bi ]. The (a, b)-quasi geodesic g connects a point in the horizontal 2δ-neighborhood of bi to a point in the horizontal 2δ-neighborhood of ai+1 . Let us denote by gi such a subpath of g. Thus, by connecting the endpoints of gi to bi and ai+1 we obtain a (a, b + 2δ)-quasi geodesic between bi and ai+1 , still denoted by gi . From Lemma 8.2, there is a (a, b + 4δ)-quasi geodesic Gi with dH tel (gi , Gi ) ≤ C8.2 (a, b), π(Gi ) = π(Ci ). From Proposition 8.1, Gi lies in the telescopic C8.1 (a, b + 4δ)-neighborhood of Ci . Obviously, since the width of Ri is less or equal to 2δ, the subpath of g between two gi ’s is in the 2δ-neighborhood of hi . This completes the proof of Lemma 9.3. ¤ ˜ Proof of Theorem 9.1: Let g0 , g1 be a C5.6 -telescopic (a, b)-quasi geodesic bigon of X. From Lemma 9.3, both g0 and g1 lie in the C9.3 (a, b)-neighborhood of a concatenation C (a,b) of C5.6 -corridors Ci and horizontal geodesics hi . From Lemma 5.9, PCi (gi ∩ Ntel9.3 (Ci )) C5.7 (C5.6 ) is a C5.7 (C5.6 )-telescopic (κ(a, b), κ(a, b))-quasi geodesic of (Ci , dtel ) with κ(a, b) = C5.9 (a, b, C9.3 (a, b)). So that we obtain a (κ(a, b), κ(a, b))-quasi geodesic bigon in each Ci . From Proposition 7.1, and thanks to Lemma 5.8, (κ(a, b), κ(a, b))-quasi geodesic bigons C (C ) in (Ci , dtel5.7 5.6 ) are C7.1 (κ(a, b), κ(a, b))-thin. It follows that C5.6 -telescopic (a, b)-quasi geodesic bigons are 2C9.3 (a, b) + C7.1 (κ(a, b), κ(a, b))-thin. The proof of Theorem 9.1 is now complete. ¤ At this point we have proven Theorem 5.3 and Theorem 2.15 is easily deduced. 10. Relative hyperbolicity: definitions The motivation of this section is to deal with weak (Farb’s) and strong (Gromov’s) relative hyperbolicity. In [8] and [26] is proved, for finitely generated groups, the equivalence between strong relative hyperbolicity and weak relative hyperbolicity + Bounded Coset Penetration property (or BCP in short) (see also [5]). We first recall these definitions. 23

If S is a discrete set, the cone with base S is the space S × [0, 12 ] with S × {0} collapsed to a point, the vertex of the cone. This cone is considered as a metric space, with distance function dS ((x, t), (y, t0 )) = t + t0 . Let (X, d) be a quasigeodesic space. Putting a cone over a discrete subset S of X consists of pasting to X a cone with base S by identifying ˆ S , dS ) is S × {1/2} with S ⊂ X. The resulting metric space, called the coned space, (X 1 such that all the points in S are now at distance 2 from the vertex of the cone and so at distance 1 one from each other. The metric of the coned space is the coned, or relative, metric. If S is a disjoint union of sets, then the coned space XS is the space obtained by putting a cone over each set in S. Definition 10.1. [11] A quasi geodesic space (X, d) is weakly hyperbolic relative to a ˆ S , dS ) is Gromov hyperbolic. family of subsets S if the coned space (X Let G be a group with finite generating set S and associated Cayley graph ΓG , and let H = {H1 , · · · } be a (possibly infinite) family of subgroups Hi of G. The group G is weakly hyperbolic relative to H if ΓG is weakly hyperbolic relative to the family of the right classes xHi . Remark 10.2. The definition of Farb relative hyperbolicity given above is the original one [11]. It is equivalent to require that G equipped with the relative metric, i. e. the metric associated to the system of generators S ∪ H, be hyperbolic. However the introduction of the cones and of the coned Cayley graphs above is needed to introduce below the Bounded Coset Penetration property. ˆ S , dS ) be a coned space and let gˆ be a (u, v)-quasi geodesic in (X ˆ S , dS ). A trace Let (X g of gˆ in (X, d) is obtained by substituting each subpath of gˆ not in (X, d) by a subpath of (X, d) in S with same endpoints, which is a geodesic for the metric induced by X on S. We say that g (or gˆ) backtracks if g reenters a subset in S that it left before. ˆ S , dS ) satisfies the Bounded-Coset Penetration Definition 10.3. [11] A coned space (X property (BCP) if there exists C(u, v) such that, for any two (u, v)-quasi geodesics gˆ0 , gˆ1 ˆ S , dS ) with traces g0 , g1 in (X, d), which have the same initial point, which have of (X terminal points at most 1-apart and which do not backtrack, the following two properties are satisfied: (a) if both g0 and g1 intersects a set Si ∈ S then their first intersection points with Si are C(u, v)-close in (X, d), (b) if g0 intersects a set Si that g1 does not, then the length in (X, d) of g0 ∩ Si is smaller than C(u, v). Definition 10.4. [11] A quasi geodesic space (X, d) is strongly hyperbolic relative to a ˆ S , dS ) is Gromov hyperbolic and satisfies the family of subsets S if the coned space (X BCP. Let G be a group with finite generating set S and associated Cayley graph ΓG , and let H = {H1 , · · · , } be a (possibly infinite) family of subgroups Hi of G. The group G is strongly hyperbolic relative to H if ΓG is strongly hyperbolic relative to the union of the right classes xHi . 11. Combination theorems: the relative cases 11.1. Setting. In this subsection we introduce the objects, then in the next two ones, we prove Theorems 2.13 and 2.14. Finally in Subsection 11.4, we derive the other results of the Introduction from the former two theorems. 24

Let J be a finite graph of finitely generated groups. Each edge (resp. vertex) group Ge (resp. Gv ) is the fundamental group of a standard 2-complex Ke (resp. Kv ): its 1skeleton is a rose, the petals of which are in bijection with the generators of the group; its 2-cells are glued along the petals by simplicial maps of their boundaries, which represent the relations. We denote by G a graph of spaces, with fundamental group J , defined as follows: the vertex (resp. edge) spaces are the above standard 2-complexes Kv (resp. Ke ); each edge space Ke is glued to the vertex spaces Ki(e) and Kt(e) by simplicial maps ψe,i(e) , ψe,t(e) which induce, on the level of the fundamental groups, the injections of Ge into Gi(e) and into Gt(e) coming with J . ˜ →T Let us consider the universal covering of G. This is a tree of metric spaces π : X as defined in 4.1. The vertex (resp. edge) spaces are the universal covering of the Kv ’s (resp. Ke ’s), these are just Cayley complexes for the edge and vertex groups of J . They are equipped with the usual simplicial metric. In the case where J is a graph of weakly relatively hyperbolic groups, the edge and vertex groups are weakly hyperbolic relative to certain subgroups. Associated to these subgroups is a relative metric. We equip the strata of the above constructed tree of spaces ˜ (the 1-skeleton of a stratum is the Cayley graph of the corresponding edge or vertex X b the space obtained. This is a tree of group) with these coned metrics. We denote by X hyperbolic spaces. Definition 11.1. Let (G, Hv , He ) be a graph of weakly relatively hyperbolic groups. The b of G satisfies the exponential separation property relative to the He ’s universal covering X and Hv ’s if, after equipping the strata with the relative metrics associated to the Hv ’s b are exponentially separated for some and He ’s, the v-quasi leaves of the vertices of X v ≥ C5.6 . Now, we assume given a graph of strongly relatively hyperbolic groups J . As before, the edge and vertex groups are denoted by Ge and Gv . Each one comes with a family He or Hv relative to which it is hyperbolic. We construct as before the tree of hyperbolic b If g ∈ Gv and H ⊂ Hv , we denote by v(gH) the exceptional vertex of X b spaces X. associated to the right-class gH in the stratum considered. It might happen that some subgroup H ⊂ Hv be left invariant, up to conjugacy, by a maximal free subgroup of the fundamental group of the graph. That is, for any geodesic in T (the tree onto which b projects) between two vertices in the lift of v, which represents an element of this X maximal free group, there is an injective endomorphism α with domain a subgroup of Gv containing H and an element h ∈ Gv with α(H) = h−1 Hh (of course both α and h depend on the geodesic considered). Definition 11.2. With the notations above: an exceptional leaf is a maximal set S of b such that there is a vertex v in G and a subgroup H ∈ Hv exceptional vertices in X fulfilling the following property: if v(gH 0 ) ∈ S and v(g 0 H 00 ) ∈ S then H 0 = H 00 = H, π(v(gH)) and π(v(g 0 H)) are in the lift of v, and g 0 = α(g)h−1 where α and h are the endomorphism and element of Gv defined by the T -geodesic [π(v(gH))π(v(g 0 H))]. Lemma 11.3. With the notations above, let L be an exceptional leaf. There exists C (not depending on L) such that any two points in L are the endpoints of a C-quasi leaf b (equivalently: L is the image of a subtree of T under a (C, 0)-quasi isometric segment of X embedding which is a section of π). Proof: There is only a finite number of subgroups preserved up to conjugacy, so that

there is only a finite number of conjugacy elements. The maximum of the word-lengths 25

b to X of these conjugacy elements plus 32 ( 12 for going from an exceptional vertex of X b gives the announced constant. ¤ plus 1 for going through a right H-class coned in X) Definition 11.4. Let (G, Hv , He ) be a graph of strongly relatively hyperbolic groups. b of G satisfies the strong exponential separation property relative The universal covering X to the He ’s and Hv ’s if, after equipping the strata with the relative metrics associated to the Hv ’s and He ’s: b are M -exponentially separated for some (a) the v-quasi leaves of the vertices of X v ≥ C5.6 , (b) there is T ≥ 0 such that the vertical width of any region where two exceptional leaves remain at horizontal distance smaller than M one from each other is smaller than T . The second condition in the above definition is needed for the BCP. 11.2. Farb relative hyperbolicity. By definition, Farb relative hyperbolicity amounts to hyperbolicity of the coned Cayley graphs so that no further work is needed to obtain Theorem 2.13. 11.3. Gromov relative hyperbolicity. This is Farb relative hyperbolicity plus BCP (see Section 10). The following lemma is easily deduced from the definitions. b b b by Lemma 11.5. With the notations above: let X be the metric space obtained from X b b putting a cone over each exceptional leaf. If X is hyperbolic and satisfies the BCP with respect to the exceptional leaves, then the fundamental group of J is strongly hyperbolic relative to the subgroups given by Theorem 2.14. Remark 11.6. Assume that H1 , H2 are subgroups of G such that α(H1 ) is a conjugate of b b H2 and α(H2 ) is a conjugate of H1 . Then, in X, cones are put above the right Hi -classes, and their exceptional vertices all belong to a same exceptional leaf. However, only one of the two subgroups H1 , H2 appears in the subgroups of the relative part described by Theorem 2.14 because otherwise the condition of malnormality is violated. The spaces are nevertheless quasi isometric because, in the context of Theorem 2.14, the metric induced by the ambiant space on each stratum is quasi isometric to the coned-metric put on this b b stratum when defining the space X. The BCP with respect to the exceptional leaves is also preserved despite this seemingly dissemblance. The following lemma is a straightforward consequence of the strong exponential separation property: b satisfies the strong Lemma 11.7. With the notations above, assume furthermore that X exponential separation property. Then two exceptional leaves through two distinct points b are connected by a diagonal (see Definition 7.2) of horizontal in a same stratum of X length greater or equal to 1 and which is dilated in all the directions. b b The hyperbolicity of the coned space X easily follows from the quasi convexity of the exceptional leaves implied by Lemma 11.3 and from the arguments developped for proving Proposition 1 of [29]. However we re-prove it when listing below the arguments for checking the BCP. 26

b satisfies the exponential sepLemma 11.8. With the notations above: assume that X 0 b between two exceptional aration property. Let g, g be two (a, b)-quasi geodesics of X leaves L1 , L2 . There exists C(a, b) such that g, g 0 admit decompositions g = g1 g2 g3 C(a,b) C(a,b) and g 0 = g10 g20 g30 with the following properties: g1 ⊂ Ntel (L1 ), g10 ⊂ Ntel (L1 ), C(a,b) C(a,b) 0 0 g3 ⊂ Ntel (L2 ), g30 ⊂ Ntel (L2 ) and dH tel (g2 , g2 ) ≤ C(a, b). If g and g have the H 0 same endpoints then dtel (g, g ) ≤ C(a, b). Proof: This is easily deduced from the approximation of the quasi geodesics in trees of

hyperbolic spaces with exponential separation of the leaves (Propositions 6.8 and 8.1). Indeed, if C is a corridor between the given two exceptional leaves, both g and g 0 remain in the horizontal C8.1 (a, b)-neighborhood of C. We project them to C so getting two (C5.9 (a, b, C8.1 (a, b)), C5.9 (a, b, C8.1 (a, b)))-quasi geodesics. Both are approximated by a same concatenation of: • diagonals with fixed horizontal length, this is Proposition 6.8, • leaves in C between these diagonals, • leaves between the first and last diagonals and the endpoints of g and g 0 . These last leaves are where g and g 0 are not necessarily close one to each other (if they don’t have the same endpoints) but are close to the given exceptional leaves. The extension to the general case where there is not a corridor, but only a pseudo-corridor, between the two exceptional leaves, is easily dealt with, in the same way than in the preceding section. ¤ b satisfies the strong expoProposition 11.9. With the notations above: assume that X b b nential separation property. Let g1 , g2 be two (a, b)-quasi geodesics of X, the terminal b b points of which are at most 1-apart in X, and with same initial point in X. Let C be a corridor between the leaves of the endpoints of g1 . There exists C(a, b, r) such that, if the b satisfy gbi ⊂ N r (C) for i = 1, 2 then dH (g1 , g2 ) ≤ C(a, b, r). traces gbi ’s of the gi ’s in X b b X b X Furthermore, if g1 and g2 do not backtrack then they satisfy the two conditions required for the BCP with a constant D(a, b, r). We emphasize that this proposition is false if one only requires a bound on the distance b b in X from the gi ’s to C. Proof: We consider the horizontal projections on C of the maximal subpaths of g1 , g2

b and of the various exceptional leaves g1 , g2 go through. From Lemma which belong to X 5.9, the projections of the above maximal subpaths of g1 and g2 are (C5.9 (a, b, r), C5.9 (a, b, r))quasi geodesics. From Lemmas 11.3, 11.7 on the one hand and Lemma 5.8 on the other hand, there is K, depending on r and C5.7 (C11.3 ), such that the projections of the exceptional leaves are (K, 0)-quasi isometric embeddings of subtrees of T , for which there exists a constant L playing the rˆole of the constant T11.4 . It is equivalent to prove the announced properties for the bigon g1 , g2 with respect to the exceptional leaves than to prove them for the above projections on C. If g1 , g2 go through the same exceptional leaves, then their projections on C satisfy the same property with respect to the projections of the exceptional leaves. From Lemma 11.8, the “bigon” obtained by projection to the corridor is thin. And moreover the points where the projections of g1 and g2 penetrate a given exceptional leaf are close, because either they are close to the diagonal preceding this exceptional leaf, or they leave a same exceptional leaf: in this last case we are done by the existence of the constant L above (the analog on the corridor of the constant T11.4 ). Let us now assume that g1 enters in an 27

exceptional leaf S but g2 does not. Of course this also holds for the respective projections on C. Two cases then: First case: If the exit point of g1 is followed by a diagonal with horizontal length greater than some constant (depending on the constants of hyperbolicity and exponential separation) then (the projection of) g2 has to go to a bounded neighborhood of this diagonal, this is Proposition 6.8. And it remains before in a bounded horizontal neighborhood of the exceptional leaf, the bound depending on a, b and r (since the constants of quasigeodesicity of the projections depend on r). Thus the vertical length of the passage of g1 through this exceptional leaf is bounded above by a constant depending on a, b and r. Second case: If the exit point of g1 is followed by another exceptional leaf: the same arguments than above apply, using Proposition 6.8 thanks to the existence of the constant L and Lemma 11.7. We leave the reader work out details and computations. The proof of Proposition 11.9 now follows in an easy way: to conclude for the BCP, we need of course the fact that the horizontal metrics on the strata satisfy the BCP. ¤ Proposition 11.10. With the assumptions of Proposition 11.9: there exist C(a, b) ≥ 1 and D(a, b) > 0 such that, if x0 , x1 , · · · , xn are consecutive points in some exceptional leaf L, which lie outside the horizontal D(a, b)-neighborhood of a corridor C, and if the vertical distance between the strata of x0 and xn is greater than C(a, b), then no non-backtracking b b (a, b)-quasi geodesic of X with both endpoints in the horizontal D(a, b)-neighborhood of C contains as subpath the cone over {x0 , xn }. Proof: The arguments are similar to those exposed for proving the quasi convexity of

the corridors. We give here only a sketch of the proof. The horizontal deviation of an exceptional leaf with respect to C depends linearly on the vertical variation of the leaf (Lemma 11.3). Thus, if a sufficiently large segment of the leaf remains outside a sufficiently large horizontal neighborhood of C, the exponential separation of the leaves implies that the horizontal distance between the leaf and C exponentially increases with the vertical length of the leaf. Assume now that the exceptional leaf considered is followed by another exceptional one. The strong exponential separation gives the same consequence: this second exceptional leaf does not go back to C and the horizontal distance with respect to C exponentially increases with its vertical length, as soon as this length is sufficiently large enough. Here the arguments are similar to those used for proving Lemmas 8.7 and 8.8. Finally, if the exceptional leaf is followed by a quasi geodesic b then the approximation by a stair as was done before, yields the same conclusion. ¤ in X, b

b Proof of Theorem 2.14: Let g, g 0 be two non-backtracking (a, b)-quasi geodesics of X b Let C be a with same initial point, and with terminal points at most 1-apart in X. pseudo-corridor (see Lemma 9.3) between the leaves of the initial and terminal points of g. This is a concatenation of corridors C0 , · · · , Cn connected by horizontal geodesics h1 , · · · , hn . From this decomposition, g and g 0 inherit a decomposition g0 · · · gn , g00 · · · gn0 b such that the initial and terminal points of gi , gi0 lie in the (2δ + 1)-neighborhood, in X, 0 of Ci . Up to increasing the constant b to a constant b = b + 2δ + 1, and extending a little bit gi and gi0 , we can assume that each pair gi , g 0 , i forms a (a, b0 )-quasi geodesic bigon. Proving the BCP for any of the gi , gi0 , is enough to prove the BCP for g, g 0 . b From Proposition 11.10, substituting, outside the D11.10 (a, b0 )-neighborhood of C in X, 0 each passage of gi and gi through the cone over a subset of an exceptional leaf by this b b with κ(a, b) = subset yields non-backtracking (κ(a, b), κ0 (a, b))-quasi geodesics hi , h0 of X, 28

i

C11.10 (a, b)C11.3 a and κ0 (a, b) = C11.10 (a, b)C11.3 b + C11.10 (a, b)C11.3 , such that dH (gi , hi ) ≤ b b X

1, dH (gi0 , h0i ) ≤ 1. b b X By Proposition 8.1, the subpaths of hi and h0i between two exceptional leaves are contained in the horizontal C8.1 (κ(a, b), κ0 (a, b))-neighborhood of a corridor between these leaves. Thus hi and h0i are contained in the D11.10 (a, b0 ) + C8.1 (κ(a, b), κ0 (a, b))b From Proposition 11.9, hi , h0i satisfy the BCP. The conclusion neighborhood of Ci in X. 0 for g, g follows. The proof of the hyperbolicity follows the same scheme. If g, g 0 form a (a, b)-quasi b b geodesic bigon of X, it is first substituted by a non-backtracking (a, b)-quasi geodesic bigon g0 , g00 with dH (g, g0 ) ≤ b, dH (g 0 , g00 ) ≤ b. The line of the arguments thereafter is the b b b b X X same than above: at the end, Proposition 11.9 gives the thinness of the quasi geodesic bigons instead of the BCP. As in Section 9, the hyperbolicity follows from Lemma 9.2. ¤ 11.4. From Theorems 2.13 and 2.14 to group-theoretical theorems. All the theorems about semi-direct products come from these two theorems thanks to the following observation: the definitions of hyperbolic relative automorphisms and of uniform free groups of hyperbolic relative automorphisms imply the (strong) exponential separation property. For Theorem 2.11, we need the following lemma: Lemma 11.11. Let G be a finitely generated group which is strongly hyperbolic relative to a finite family of subgroups H. Let K be a finitely generated subgroup of G, which is almost malnormal relative to H, which is strongly hyperbolic relative to a (possibly empty) finite family H0 the subgroups of which are conjugated to subgroups in H, and such that (K, |.|H0 ) is quasi isometrically embedded in (G, |.|H ). There exists C > 0 such that, if x, y (resp. z, t) are any two vertices in a same right-class gK (resp. hK) with g 6= h then dH (P[z,t] (x), P[z,t] (y)) ≤ C. Proof: Since (G, |.|H ) is hyperbolic, there is a constant δ ≥ 0 such that the geodesic

triangles of (G, |.|H ) are δ-thin. Thus, geodesic rectangles are 2δ-thin. This implies the existence of a quadruple of vertices x0 , y0 , z0 , t0 with x0 , y0 ∈ [x, y], z0 , t0 ∈ [z, t] and dH (x0 , z0 ) ≤ 2δ + 1, dH (y0 , t0 ) ≤ 2δ + 1. Since (K, |.|H0 ) is (λ, µ)-quasi isometrically embedded in (G, |.|H ), and (G, |.|H ) is δ-hyperbolic, there exist c0 (λ, µ, δ) and x1 , y1 , z1 , t1 such that g −1 x1 , g −1 y1 ∈ K, h−1 z1 , h−1 t1 ∈ K and dH (x0 , x1 ) ≤ c0 (λ, µ, δ), dH (y0 , y1 ) ≤ c0 (λ, µ, δ), dH (z0 , z1 ) ≤ c0 (λ, µ, δ), dH (t0 , t1 ) ≤ c0 (λ, µ, δ). We choose x1 , y1 , z1 , t1 to minimize the distance in (G, S), that is the distance associated to the given finite set of generators S of G. We denote by [x1 , y1 ]K (resp. [z1 , t1 ]K ) the images, under the embedding of K in G, of geodesics between the pre-images of x1 , y1 (resp. z1 , t1 ) in K. Both [x1 , y1 ]K and [z1 , t1 ]K are (λ, µ)-quasi geodesics. Moreover [x1 , z1 ][z1 , t1 ]K [t1 , y1 ] is a (λ, 4δ + 2 + 4c0 (λ, µ, δ) + µ)-quasi geodesic between x1 and y1 . Since G is strongly hyperbolic relative to H, G satisfies the BCP with respect to H. This gives a constant c1 (λ, µ, δ) such that the H-classes [x1 , z1 ] and [t1 , y1 ] go through correspond to geodesics in (G, S) with length smaller than c1 (λ, µ, δ): indeed, the H-classes crossed by [x1 , z1 ] and [t1 , y1 ] are not crossed by [x1 , y1 ]K . Therefore the distance in (G, S) between x1 and z1 on the one hand, and between y1 and t1 on the other hand is less or equal to (2δ + 1 + 2c0 (λ, µ, δ))c1 (λ, µ, δ). There are a finite number of elements in G with such an upper-bound on the length, measured with a word-metric associated to a finite set of generators. Whence, by the almost normality of K relative to H, an upper-bound on the length between x1 and y1 , and so also between x0 and y0 . Lemma 11.11 is proved. ¤ 29

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