GEODESICS IN TREES OF HYPERBOLIC AND RELATIVELY HYPERBOLIC GROUPS FRANC ¸ OIS GAUTERO Abstract. We present a careful approximation of the quasi 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.

1. Introduction The main part of this paper is devoted to give a precise description of the (quasi) geodesics in trees of hyperbolic and relatively hyperbolic groups. As an application of this description we prove 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 [24]), the authors introduce the notion of (finite) graph of qi-embedded 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, [21, 26]. 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 non-acylindrical case is less common: see [25] 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 [14] 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 [22]. 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 [12]. Bowditch [5] and Osin [29] give alternative definitions, but which are equivalent either to Farb’s or to Gromov’s definition. In fact, it has been proved [8, 29] (see also [5]) that Gromov’s definition is equivalent to Farb’s definition plus an additional property termed Bounded Coset Penetration property (BCP in short), due to Farb [12]. Relatively 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 relative hyperbolicity: [1], [10] or [28, 30]. One result [17] treats a particular non-acylindrical Date: January 9, 2011. 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. 1

case, namely the relative hyperbolicity of one-ended hyperbolic by cyclic groups. In [27] the authors give a combination theorem dealing with more general non-acylindrical cases than [17]. This last paper heavily relies upon [3], which is used as a “black-box”. In the current paper, as an application of our work on geodesics in trees of spaces we offer a quite general combination theorem in relative hyperbolicity. We emphasize at once that we do not appeal to [3], but instead give a new proof of it as a particular case. Where the authors of [3] 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. Acknowledgements. This paper benefited from discussions of the author with F. Dahmani, V. Guirardel, M. Heusener, I. Kapovich and M. Lustig. In particular the author is grateful to F. Dahmani, V. Guirardel and M. Lustig for their observations about the definition of relative automorphisms, and to M. Heusener for inciting him to correct a previous formulation of the definition of relatively hyperbolic automorphism, which was unnecessarily more restrictive. Warm thanks are due to I. Kapovich for his numerous explanations about the combination theorem. 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, as well as the support of the University Blaise Pascal (ClermontFerrand) where this work was finalized. Professor P. de la Harpe (University of Geneva) also deserves a great share of these acknowledgements for his help. Last but not least the referee provided an invaluable help to correct some mistakes and get a better presentation. 2. Statements of results Since relatively hyperbolic groups are the objects of study in this paper, we begin with recalling the definitions of weak and strong relative hyperbolicity. Both notions were defined in [12]. However the approach of strong relative hyperbolicity is taken from [5]. Definition 2.1. [12, 5] Let G be a group with finite generating set S and Cayley graph ΓS (G). Let Λ be a finite set and let H = {Hi }i∈Λ be a a finite family of subgroups Hi of G. The H-coned graph ΓH S (G) is the graph obtained from ΓS (G) by: • adding an exceptional vertex v(gHi ) for each left Hi -class, • putting an edge of length 21 between v(gHi ) and each vertex of ΓS (G) associated to an element in the class g(Hi ), for each Hi ∈ H. The group G is weakly hyperbolic relative to H if and only if ΓH S (G) is Gromov hyperbolic. The group G is strongly hyperbolic relative to H if and only if: (a) The graph ΓH S (G) is Gromov hyperbolic; (b) For any positive integer n, any edge in ΓH S (G) is contained in a finite number of embedded loops of length n. The subgroups Hi in the family H are the parabolic subgroups of (G, H). It was observed in [12] that the family of subgroups H is necessarily almost malnormal: 2

Definition 2.2. Let G be a group, let Λ be a finite set and let H = {Hi }i∈Λ be a a finite family of subgroups of G. The family H is almost malnormal if and only if: (a) any subgroup Hi is almost malnormal in G, i.e. for any g ∈ / Hi , g −1 Hi g ∩ Hi is finite. (b) for any two Hi , Hj ∈ H with i 6= j, for any g ∈ G, g −1 Hi g ∩ Hj is finite. We now give some group-theoretic results that will be deduced from our - perhaps more technical - theorems about trees of spaces. We begin with a result about free extensions of relatively hyperbolic groups. It is more easily explained than the more general combination theorems we prove later in the paper. The relatively hyperbolic automorphims we define below first appeared in [15] where we announced a (weak) version of the results of the present paper. They generalize the Gromov hyperbolic automorphisms [3]. Definition 2.3. Let G = hSi be a finitely generated group, let Λ be a finite set and let H = {Hi }i∈Λ be a finite family of subgroups of G. The H-word metric |.|H is the word-metric for G equipped with the generating set obtained by adding to S all the elements of G which belong to the subgroups in the family H. In the above definition, if some subgroup Hi in H is infinite, the generating set considered for the H-word metric is infinite: this will usually be the case. Definition 2.4. With the notations of Definition 2.3: A relative automorphism of (G, H) is an automorphism α of G such that there is a permutation σ(α) ∈ Sym(Λ) satisfying the following property: For any Hi ∈ H there is gi ∈ G with α(Hi ) = gi−1 Hσ(α)(i) gi . A relative automorphism α of (G, H) fixes H, or fixes each Hi , up to conjugacy if and only if σ(α) is the identity. We denote by Aut(G, H) the set of all relative automorphisms of (G, H): this is a subgroup of Aut(G, ∅) := Aut(G), the group of automorphisms of G. The map σ : Aut(G, H) → Sym(Λ) is a morphism. α 7→ σ(α) Definition 2.5. A relatively hyperbolic automorphism of (G, H) is a relative automorphism α ∈ Aut(G, H) satisfying the following property: 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 ). We also say in this case that α is hyperbolic relative to H. Remark 2.6. This definition is slightly more general than the definition given in [15]. Definition 2.5 is invariant: • Under conjugacy of α in Aut(G, H). Indeed, any β ∈ Aut(G, H) acts as a bilipschitz map on (G, H), i.e. there is a ≥ 1 such that for any w ∈ G 1 |w|H ≤ |β(w)|H ≤ a|w|H . a Hence by choosing M 0 ≥ aM we get for any w ∈ G with |w|H ≥ M 0 either |(β −1 ◦ α ◦ β)jN (w)|H ≥ λj |w|H for any j ≥ 1 or |(β −1 ◦ α ◦ β)−jN (w)|H ≥ λj |w|H for any j ≥ 1. By choosing j such that λj ≥ a2 λ we get N 0 = jN such that for any w ∈ G with |w|H ≥ M 0 we have 0

0

λ|w|H ≤ max(|αN (w)|H , |α−N (w)|H ). 3

More generally, if β ∈ Aut(G) and α ∈ Aut(G, H) is relatively hyperbolic then β −1 ◦ α ◦ β ∈ Aut(G, β −1 (H)) is relatively hyperbolic: just observe that β acts as a bi-lipschitz map from (G, β −1 (H)) to (G, H) and applies the same computation as above. • Under the substitution of any subgroup Hi in H by a conjugate g −1 Hi g with g ∈ G. Indeed, if H0 denotes the new family of subgroups, the metric spaces (G, H) and (G, H0 ) are bi-lipschitz equivalent with Lipschitz constant a = 2max(|g|H , |g|H0 )+1. Then choose M 0 and N 0 as above. Definition 2.7. Let G be a finitely generated group and let H be a finite family of subgroups of G. Let ı : A ,→ Aut(G) be a monomorphism from a finitely generated group A into the group of automorphisms of G. The pair (ı, A) defines a group of uniformly relatively hyperbolic automorphisms of (G, H) if and only if ı(A) < Aut(G, H) and for any finite generating set A of A there exist λ > 1 and M, N ≥ 1 such that for any element w ∈ G with |w|H ≥ M , for any a1 , a2 ∈ A with |a1 |A = |a2 |A = N and dA (a1 , a2 ) = 2N : λ|w|H ≤ max(|ı(a1 )(w)|H , |ı(a2 )(w)|H ). In Definition 2.7 the existence of the constants λ, M, N holds for any finite generating set A if and only if it holds for some such generating set. However λ, M, N depend on the choice of A. With the notations of Definitions 2.3 and 2.4, let r be a positive integer, let Fr be the free group of rank r and let ı : Fr ,→ Aut(G, H) be a monomorphism. For each i ∈ Λ let Ki := (σ◦ı)−1 (StabSym(Λ) (i)). Since a subgroup of a free group is free (Schreier’s theorem) each Ki admits a free basis Bi = {ui,j }j=1,2,··· . Since the subgroups StabSym(Λ) (i) are finite, each Ki is of finite index in Fr and so is a finitely generated free group (see [13][Proposition 3.9]), the rank of which is denoted by ri . Definition 2.8. With the notations of Definitions 2.3 and 2.4, assume that H is a finite family of almost malnormal, infinite subgroups of G. Let r be a positive integer, let Fr be the free group of rank r and let ı : Fr ,→ Aut(G, H) be a monomorphism. For each i ∈ Λ let Ki := (σ ◦ ı)−1 (StabSym(Λ) (i)) and let Bi = {ui,1 , · · · , ui,ri } be a free basis of Ki . −1 For each ui,j ∈ Bi let gi,j ∈ G such that ı(ui,j )(Hi ) = gi,j Hi gi,j . −1 Then the subgroup hui,j , ui,j gi,j ; j ∈ {1, · · · , ri }i of G oı Fr is the (Fr , ı)-extension of Hi and is denoted by Hiı (even if its definition depends on the choices of the ui,j and of the gi,j we do not include them in the notation - see Remark 2.9 below). Let K ⊂ {Ki }i∈Λ be composed of exactly one representative of each conjugacy-class defined by the Ki ’s. Then the family {Hiı }Ki ∈K is a (Fr , ı)-extension of H over K and is denoted by Hı,K . When r = 1 in Definition 2.8, i.e. when Fr = hti, a Fr -extension of H is preferably called mapping-torus of H under α ∈ Aut(G, H), where α = ı(t). Remark 2.9. Since the subgroups in H are assumed almost malnormal and infinite, −1 −1 any two elements gi,j , ki,j which satisfy ui,j (Hi ) = gi,j Hi gi,j = ki,j Hi ki,j also satisfy gi,j ∈ Hi ki,j . This is why the choices of the ui,j and of the gi,j do not matter and we can speak of the (Fr , ı)-extension of Hi . Moreover let Hı,K and Hı,K0 be any two (Fr , ı)-extension of H. By definition K is obtained from K0 by substituting some of its subgroups with conjugates in Fr . Therefore, since here the subgroups in H are assumed almost malnormal and infinite, Hı,K is obtained from Hı,K0 by substituting some of its 4

subgroups by conjugates in G oı Fr . This is why we will later make a slight abuse of language and write the (Fr , ı)-extension of H. Theorem 2.10. Let G be a finitely generated group and let H be a finite family of infinite subgroups of G. Let (Fr , ı) define a free group of uniformly relatively hyperbolic automorphisms of (G, H). If G is weakly hyperbolic relative to H, then G oı Fr is weakly hyperbolic relative to H. If G is strongly hyperbolic relative to H, then G oı Fr is strongly hyperbolic relative to the (Fr , ı)-extension of H. When r = 1 in the above theorem, that is when the considered free group is just Z, we get the classical “mapping-torus” case, that is the case of semi-direct products G o Z with G a relatively hyperbolic group. We now give a corollary for this case. From [20], a hyperbolic group is weakly hyperbolic relative to any finite family of quasi convex subgroups. From [5] or [31], a hyperbolic group G is strongly hyperbolic relative to any almost malnormal finite family of quasi convex subgroups. We so get: Corollary 2.11. Let G be a hyperbolic group, let H be a finite family of infinite subgroups of G and let α ∈ Aut(G, H) be hyperbolic relative to H. If H is quasi convex in G then the mapping-torus group Gα = G oα Z is weakly hyperbolic relative to H. If H is quasi convex and almost malnormal in G then the mapping-torus group Gα = Goα Z 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. We so re-prove the result of [17]. Since there we gave only an idea for the statement and the proof in the Gromov’s (i.e. strong) relative hyperbolicity case, we include here the full statement of this result: Corollary 2.12. Let G be a torsion free one-ended hyperbolic group and let α be an automorphism of G. Let H be a family of maximal subgroups of G which consist entirely of elements on which α acts up to conjugacy periodically or with linear growth and such that each infinite-order element on which α acts up to conjugacy periodically or with linear growth is conjugate to an element in a subgroup in H. Then Gα = G oα Z 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 if h is 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, (iii) the cyclic subgroups generated by the reduction curves not contained in the previous subsurfaces. Since the ultimate goal is a theorem about graphs of relatively hyperbolic groups, we introduce some notations for graphs and graphs of groups. If Γ is a graph, V (Γ) (resp. E(Γ)) denotes its set of vertices (resp. of oriented edges). For e ∈ E(Γ) we denote by e−1 the same edge with opposite orientation. If p is an edge-path in Γ, in particular if p is an edge, i(p) (resp. t(p)) denotes the initial (resp. terminal) vertex of p. In a tree, given any two vertices x, y, we denote by [x, y] the unique reduced edge-path from x to y. In a metric graph Γ (i.e. a graph equipped with a positive length le on each edge e and an isometry from e to the real interval (0, le ) - for instance a Cayley graph or a 5

H-coned graph), dΓ (x, y) denotes the geodesic distance between two vertices x, y, [x, y] any geodesic, i.e. length-minimizing, edge-path between x and y and |p|Γ the length of an edge-path p in Γ. Let us now consider a graph of groups G = (Γ, {Ge }, {Gv }, {ıe }) where • Γ is a graph, • for each oriented edge e, Ge denotes the group associated to the edge e (an edgegroup) and Ge = Ge−1 , • for each vertex v, Gv denotes the group associated to the vertex v (a vertex-group), • for each oriented edge e, ıe : Ge → Gt(e) is a monomorphism. The group we are interested in when considering a finite graph of groups G = (Γ, {Ge }, {Gv }, {ıe }) is its fundamental group π(G). Once chosen a maximal tree TM ⊂ Γ, π(G) is the group with the following presentation: • the generators are the vertex-groups and elements te associated to the oriented edges e in Γ; • the relations are the relations of the vertex-groups together with relations t−1 e ıe−1 (g)te = ıe (g) for each oriented edge e in Γ and te = 1 for each edge e in TM . It is of course independent of the maximal tree TM . Moreover, if N is the normal closure in π(G) of the subgroup generated by the vertex-groups, then π(G)/N is a finitely generated free group F. See [32] or [11] for a complete account about graph of groups. Hence π(G) is a semi-direct product N oı F. The universal covering of G = (Γ, {Ge }, {Gv }, {ıe }) is a tree of groups T = (T, {Te }, {Tv }, {eıe }), and N is equivalently defined as the subgroup of π(G) generated by the vertex-groups of T (which are the conjugates of the vertex-groups of G). Definition 2.13. A graph of weakly relatively hyperbolic groups (resp. of strongly relatively hyperbolic groups), denoted by (G, Hv , He ), is a finite graph of finitely generated groups (Γ, {Ge }, {Gv }, {ıe }) satisfying the following properties: (a) Each edge-group Ge and each vertex-group Gv is weakly hyperbolic (resp. strongly hyperbolic) relative to a specified, possibly empty, finite family of infinite subgroups He = He−1 and Hv . (b) There are a ≥ 1, b ≥ 0 such that for any oriented edge e of Γ, ıe is a (a, b)-quasi isometric embedding from (Ge , |.|He ) to (Gt(e) , |.|Ht(e) ). (c) For each parabolic subgroup H ∈ He of each edge-group Ge , ıe (H) is conjugate to a subgroup of a parabolic subgroup H 0 ∈ Ht(e) of Gt(e) . In order to state the combination theorem for graphs of weakly relatively hyperbolic groups we are now going to introduce a new property termed property of exponential separation and for this purpose we need preliminary definitions. Definition 2.14. Let (G, Hv , He ) be a graph of (weakly or strongly) relatively hyperbolic groups (recall that G = (Γ, {Ge }, {Gv }, {ıe })). A C-quasi orbit segment is a (finite or infinite) sequence g = (gn )n=0,1,··· of elements in π(G) which satisfies the following properties: (a) any two elements g2j , g2j+1 ∈ g belong to a same left-class of a vertex-group Gv(j) −1 and the relative length of g2j+1 g2j in this vertex-group is smaller than C, (b) for any two consecutive elements g2j+1 , g2j+2 ∈ g the vertices v(j) and v(j + 1) are the initial and terminal vertices of an oriented edge e(j) and g2j+2 = g2j+1 te(j) , (c) For any two consecutive edges e(j − 1)e(j) (j ≥ 1) in the edge-path e(0)e(1) · · · , −1 if g2j+1 g2j ∈ v(j) = t(e(j − 1)) = i(e(j)) is trivial then e(j) 6= e(j − 1)−1 . 6

A C-quasi orbit is a C-quasi orbit segment which is maximal in the sense that it is properly contained in no other C-quasi orbit segment. The length of a C-quasi orbit segment g is infinite if g is infinite and is otherwise equal to the length of the edge-path e(0)e(1) · · · . If g is a quasi orbit segment with length 2N , the midpoints of g are the elements gN , gN +1 in g: they are in a same left-class of the group of the terminal vertex of e(N − 1), that is in a same left-class of the vertex-group at the middle of the edge-path e(0)e(1) · · · e(2N − 1). Definition 2.15. The graph of (weakly or strongly) relatively hyperbolic groups (G, Hv , He ) satisfies the property of exponential separation if and only if, for any non-negative constant C, there exist λ > 1 and integers M, N ≥ 1 such that if g and h are any two C-quasi orbit segments with length 2N , with initial (resp. terminal) elements g0 , h0 in a same left-class of a vertex-group Gα (resp. g4N +1 , h4N +1 in a same left-class of a vertex-group Gβ ) and midpoints g ∈ g, h ∈ h in a same left-class of a vertex-group Gv satisfying |h−1 g|Hv ≥ M then −1 λ|h−1 g|Hv ≤ max (|h−1 0 g0 |Hα , |h4N +1 g4N +1 |Hβ ). Remark 2.16. Definition 2.15 requires that the given condition be satisfied for any nonnegative constant C. However, in order to be satisfied for any non-negative constant C, it is in fact sufficient that it be satisfied for some sufficiently large enough constant C. The minimal constant C necessary for Definition 2.15 only depends on the constant of “relative hyperbolicity” of the edge- and vertex-groups and on the constants a, b of Definition 2.13. See Lemmas 3.16 and 3.18. The combination theorem for graphs of weakly relatively hyperbolic groups is then stated as follows (we recall that a parabolic subgroup in a relatively hyperbolic group (G, H) is a subgroup in the family H): Theorem 2.17. If a graph of weakly relatively hyperbolic groups satisfies the property of exponential separation then its fundamental group is weakly hyperbolic relative to the family composed of all the parabolic subgroups of the vertex-groups. We pass to the strong relative hyperbolicity. The combination theorem we get in this case only deals with a restricted class of graphs of strongly relatively hyperbolic groups. This is because the description of the subgroups to put in the relative part is heavier in this setting than in the setting of weak relative hyperbolicity. We hope that the restriction we put is a not too bad compromise between clarity and generality. We first need the notion of parabolic orbit. Definition 2.18. Let (G, Hv , He ) be a graph of strongly relatively hyperbolic groups (recall that G = (Γ, {Ge }, {Gv }, {ıe })). Two parabolic subgroups H ∈ Hv and H 0 ∈ Hw belong to a same parabolic orbit if and only if there exists a reduced edge-path p = ei11 · · · eikk in Γ with i(p) = v and t(p) = w and there exist g1 ∈ Gv , g2 ∈ Gt(ei 1 ) , · · · , gk ∈ Gt(ek−1 ) , gk+1 ∈ Gw and h ∈ H, h0 ∈ H 0 ik−1

1

such that

−1 t−1  g ei k k k

−1 · · · t−1 hg1 tei 1  g ei 1 1 1 1

· · · gk t

 ei k k

=

−1 0 h gk+1 . gk+1

Definition 2.19. A graph of strongly relatively hyperbolic groups (G, Hv , He ) is fine if and only if for each oriented edge e of Γ (recall that G = (Γ, {Ge }, {Gv }, {ıe })): (a) If there is a vertex v of Γ, a reduced loop p = ei11 · · · eikk in Γ with i(p) = t(p) = v, elements g1 , gk+1 ∈ Gv , g2 ∈ Gt(ei 1 ) , · · · , gk ∈ Gt(ek−1 ) , a parabolic subgroup H of ik−1

1

−1 −1 −1 0 (Gv , Hv ) and h, h0 ∈ H such that t−1 · · · t−1 hg1 tei 1 · · · gk tei k = gk+1 h gk+1  g  g e1 1 ek k ik

7

i1

1

k

−1 −1 −1 then t−1 · · · t−1 Hg1 tei 1 · · · gk tei k = gk+1 Hgk+1 : the parabolic subgroup H  g  g e1 1 ek k 1

i1

ik

k

belongs to a periodic parabolic orbit. (b) If two parabolic subgroups H ∈ Hv and H 0 ∈ Hw belong to a same parabolic orbit (see Definition 2.18) then either they belong to the same periodic parabolic orbit or neither one nor the other belongs to a periodic parabolic orbit. The notion of periodic parabolic orbit appearing in Item (a) of Definition 2.19 amounts to asking some kind of “maximality” condition: if a sequence of morphisms and conjugacies in the vertex-groups associated to a reduced loop in Γ conjugates an element in a parabolic subgroup to another element of this same parabolic subgroup then it conjugates the whole subgroup to itself. Item (b) of Definition 2.19 amounts to asking a dichotomy for the parabolic subgroups of the vertex groups in the fundamental group of G: either a given parabolic subgroup belongs to a periodic parabolic orbit, or roughly speaking it is malnormal with respect to the periodic parabolic orbits, i.e. none of its conjugates meets a periodic parabolic orbit in a non-trivial element. Here is the strengthening of the property of exponential separation which is needed to deal with strong relative hyperbolicity: Definition 2.20. A fine graph of relatively hyperbolic groups (G, Hv , He ) satisfies the property of strong exponential separation if and only if, for any non-negative constant C, there exist λ > 1 and integers M, N ≥ 1 such that: (a) for any two C-quasi orbit segments g and h with length 2N , with initial (resp. terminal) elements g0 , h0 in a same left-class of a vertex-group Gα (resp. g4N +1 , h4N +1 in a same left-class of a vertex-group Gβ ) and midpoints g ∈ g, h ∈ h in a same left-class of a vertex-group Gv which satisfy |h−1 g|Hv ≥ M , we have −1 λ|h−1 g|Hv ≤ max (|h−1 0 g0 |Hα , |h4N +1 g4N +1 |Hβ ).

(b) for any vertex v of Γ (recall that G = (Γ, {Ge }, {Gv }, {ıe })), for any parabolic subgroup Hi ∈ Hv in a periodic parabolic orbit, for any reduced loop p = ei11 · · · eikk in Γ −1 −1 with i(p) = t(p) = v and k = |p|Γ ≥ N such that t−1 · · · t−1 Hg1 tei 1 · · · gk tei k =  g  g e1 1 ek k −1 gk+1 Hgk+1

ik

(g1 , gk+1 ∈ Gv , g2 ∈ G

 t(ei 1 ) 1

, · · · , gk ∈ G

 t(ei k−1 ) k−1

1

i1

k

), for any x, y in a same

−1 Gv -left class, x ∈ / yHi and |x−1 y|Hv ≤ M which satisfy that x0 = xg1 tei 1 · · · gk tei k gk+1 1

k

−1 and y 0 = yg1 tei 1 · · · gk tei k gk+1 also belong to a same Gv -left class we have M ≤

|x0 −1 y 0 |Hv .

1

k

Item (a) of Definition 2.20 gives the exponential separation property of Definition 2.15. Roughly speaking, Item (b) amounts to asking that any two distinct orbits of left Hi classes with Hi in a periodic parabolic orbit separate exponentially since, by Item (a) and with the notations of Definition 2.20, the length of x0 is necessarily exponentially dilated in all the directions with the exception of the one given by p. We now want to precise the notion of a free extension of a parabolic subgroup in this graph of groups setting. We summarize what we got up to now: The fundamental group of G is a semi-direct product N oı F with F a finitely generated free group, and each parabolic subgroup is a subgroup of N. For each parabolic subgroup H ∈ Hv there is a maximal (possibly trivial) free subgroup KH of F which fixes H up to conjugacy in its vertex-group Gv . At the difference of the case treated in Definition 2.8, KH might be trivial, and in particular is not necessarily of finite index in F. However, as it was the case in Definition 2.8, KH still is finitely generated (see the proof of Lemma 7.5). The 8

parabolic subgroups H for which KH is non-trivial are exactly the parabolic subgroups belonging to periodic parabolic orbits. Thus it makes sense, for each such subgroup, to consider a (F, ı)-extension of it, as in Definition 2.8. Moreover, since these subgroups are parabolic subgroups of strongly relatively hyperbolic subgroups, each one is almost malnormal in its vertex group. Thus, since parabolic subgroups are assumed to be infinite it follows from Remark 2.9 that the (F, ı)-extension of H is well-defined as in Definition 2.8 whenever H is a parabolic subgroup in a periodic parabolic orbit: we term it the free extension of H. Theorem 2.21. If a fine graph of strongly relatively hyperbolic groups satisfies the property of strong exponential separation then its fundamental group is strongly hyperbolic relative to any family composed of: • exactly one representative in each non-periodic parabolic orbit, • the free extension of exactly one representative in each periodic parabolic orbit. Any two such families are obtained one from the other by substituting some of their subgroups by conjugates. In the semi-direct product case G o Fr over a group of relatively hyperbolic automorphisms, there is no non-periodic parabolic orbit and taking the free extension of exactly one representative in each periodic parabolic orbit amounts to taking the free extension of the family of the parabolic subgroups of G as it was defined before. The corollaries of the above two theorems we gave before only concern the case of the semi-direct product of a free group Fr acting on a finite type relatively hyperbolic group (G, H). However a semi-direct product is only a particular case of HNN-extension. Alibegovic in [1], Dahmani in [10] or Osin in [30] treat acylindrical HNN-extensions and amalgated products. Let us now give a corollary about non-acylindrical HNN-extensions. Corollary 2.24 below deals with injective, not necessarily surjective, endomorphisms of relatively hyperbolic groups. Definition 2.22. 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 and only if there is an upper-bound 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 in the definition above, we get the usual notion of almost malnormality of a subgroup. 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 2.3 remain valid for injective endomorphisms, the definition of relative hyperbolicity for automorphisms is easily adapted to the more general case of injective endomorphisms: Definition 2.23. Let G be a finitely generated group and let H be a finite family of infinite subgroups of G. An injective endomorphism α of G is hyperbolic relative to H if and only 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 . Corollary 2.24. 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 HNN-extension. Let H be a finite family of infinite 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. 9

The reader will notice at once that the above theorem does not treat the extension of weakly relatively hyperbolic groups. The reason is that the condition of relative almost malnormality does not imply in this case the strong exponential separation property. This last property is however also a necessary condition, although we do not give here a direct proof: in the absolute hyperbolicity case, Gersten was the first to give the converse to the combination theorem, using cohomological arguments [19] and we adapt his arguments in [16]. Bowditch exposed a more direct proof in [6]. Up to now, all the stated corollaries dealt with HNN-extensions. We conclude this list of statements by a result about fundamental groups of 3-manifolds, which generalizes the case of 3-manifolds fibering over S1 treated by Corollary 2.12, and where less rudimentary graphs of groups appear. Let us recall that a closed 3-manifold is a connected, compact 3-manifold without boundary. A Seifert fibred space is a connected, compact, orientable, irreducible 3-manifold which is a union of disjoint circles Cα , called the fibers of M 3 , such that each Cα admits a neighborhood T (Cα ), homeomorphic by a fiber-preserving homeomorphism hα to a fibred solid torus T2p,q , i.e. the suspension D2 × [0, 1]/(x, 1) ∼ (r(x), 0) of a rotation r of the disc D2 centered at the origin O and of angle 2πp . The fibers of q 2 Tp,q are the orbits of the rotation r, the homeomorphism hα is required to carry Cα to the r-orbit of O and each circle Cβ ⊂ T (Cα ) to a fiber of T2 (p, q). A graph-manifold is a compact, orientable, irreducible 3-manifold M 3 which admits a finite union of disjointly embedded incompressible tori T1 , · · · , Tr such that the closure of each connected compoS nent of M 3 \ ri=1 Ti is a Seifert-fibred manifold. Given any closed, irreducible, orientable 3-manifold M 3 there exists a family of maximal graph-submanifolds S GM1 , · · · , GMr in M 3 such that the closure of each connected component of M 3 \ ri=1 GMi is a compact 3-manifold with hyperbolic, finite volume interior, the boundary of which is a union of tori. Corollary 2.25. Let M 3 be a closed, irreducible, orientable 3-manifold. Then the fundamental group of M 3 is strongly hyperbolic relative to the family composed of (a) the subgroups Gi corresponding to conjugates of the fundamental groups of the maximal graph-submanifolds GM1 , · · · , GMr in M 3 , S (b) the Z ⊕ Z-subgroups corresponding to the incompressible tori in M 3 \ ri=1 GMi . 2.1. Plan of the paper: The results above are consequences of Theorems 4.8 and 5.5 about the behavior of quasi geodesics in trees of hyperbolic spaces. Section 3 contains the basis, from quasi isometries to the “hallways-flare” property (a tree of spaces version of the exponential separation property given above). Section 4 deals with the approximation of quasi geodesics in the particular case where all the attaching-maps of the considered tree of hyperbolic spaces are quasi isometries. Section 5 contains the adaptations to the general case. The important notions appearing in these two sections are the corridors in Section 4, and the generalized corridors in Section 5. These two sections appeal to two important Propositions whose proofs are delayed: Proposition 4.9 is proved in Section 8; Proposition 4.10 is proved in Section 9 whereas its adaptation to generalized corridors (Proposition 5.6) is dealt with in subsection 9.6. Section 6 presents the results about the hyperbolicity whereas Section 7 deals with the consequences about strong relative hyperbolicity of graphs of strongly relatively hyperbolic groups. This last section contains another proposition whose proof is postponed to subsection 9.7. Subsection 7.1 contains the proofs of the statements (about both weak and strong relative hyperbolicity) given in Section 2. 10

3. Preliminaries 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). We set also Ndr (A) = {x ∈ X ; di (x, A) ≤ r} and dH (A, B) = sup{r ≥ x∈A,y∈B

0 ; A ⊂ Ndr (B) and B ⊂ Ndr (A)} is then the usual Hausdorff distance between A and B. Finally, diamX (A) stands for the diameter of A: diamX (A) = sup{d(x, y) ; (x, y) ∈ A × A}. 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. Since a quasi isometric embedding is not necessarily a continuous map, a quasi geodesic as defined above is not a path in the usual sense, but a chain where by “chain” we mean an ordered family of points and oriented continuous paths. As for edge-paths in graphs, if c is a chain or path in a metric space we denote by i(c) (resp. t(c)) its initial (resp. terminal) point. 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 geodesic spaces, we consider quasi geodesic spaces: 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 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 slight “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. 3.2. Trees of spaces. Definition 3.1. (compare [3]) A graph of qi-embedded metric spaces (Γ, {Xe }, {Xv }, {e }) is a metric graph Γ with length 1 edges, together with a collection of geodesic metric spaces {Xe }e∈E(Γ) (resp. {Xv }v∈V (Γ) ) associated to the oriented edges e (resp. to the vertices v) of Γ, with Xe = Xe−1 , and a collection of (a, b)-quasi isometric embeddings e : Xe → Xt(e) from the edgespaces Xe = Xe−1 to the vertex-spaces Xv , where a ≥ 1 and b ≥ 0 are two fixed real constants. A tree of qi-embedded metric spaces is a graph of qi-embedded metric spaces (Γ, {Xe }, {Xv }, {e }) such that Γ is a tree. 11

A graph of hyperbolic spaces is a graph of qi-embedded metric spaces such that there is δ ≥ 0 for which each edge- and vertex-space is a δ-hyperbolic space. Let (Γ, {Xe }, {Xv }, {e }) be a graph of qi-embedded metric spaces. If E + (Γ) denotes the subset of E(Γ) composed of exactly one representative in each couple (e, e−1 ) then ˜ obtained from the metric space X G G (Xe × [0, 1]) t Xv e∈E + (Γ)

v∈V (Γ)

by identifying (x, 1) ∈ Xe × [0, 1] with e (x) ∈ Xt(e) and (x, 0) ∈ Xe × [0, 1] with e−1 (x) ∈ Xi(e) for each e ∈ E + (Γ), and equipped with the quotient pseudo-metric, is called the geometric realization of Γ. ˜ → Γ the map which identifies each subset Xe × {t} ⊂ X ˜ to the We denote by π : X point in e ∈ E(Γ) with coordinate t ∈ [0, 1] (recall that each edge e comes with an ˜ to the vertex v of Γ. The sets Xe × {t} isometry with [0, 1]) and each subset Xv ⊂ X ˜ and Xv are the strata of X. See Figure 1. -1 π (v)

-1

e 2)

π(

π-1(e1)

π-1(e3)

π e1

e2 v

e3

Figure 1. Remark 3.2. It is obvious from the definition that a graph of qi-embedded metric spaces is a quasi geodesic metric space as defined in Section 3.1. We could also only require that strata be quasi geodesic spaces, instead of geodesic ones. As soon as the fundamental group of each edge- and vertex-space is well-defined, and the quasi isometric embeddings e : Xe → Xt(e) induce monomorphisms ıe : Ge ,→ Gt(e) between the corresponding fundamental groups Ge and Gv , the fundamental group of a graph of qi-embedded metric spaces (Γ, {Xe }, {Xv }, {e }) is well-defined, and is isomorphic to the fundamental group of the graph of groups (Γ, {Ge }, {Gv }, {ıe }). The universal ee }, {X ev }, {e covering of (Γ, {Xe }, {Xv }, {e }) is a tree of qi-embedded metric spaces (T , {X e }), on which the fundamental group acts cocompactly, isometrically and properly discontinuously: the tree T is the same as the tree appearing in the universal covering of ee }, {X ev }, {e (Γ, {Ge }, {Gv }, {ıe }) and the vertex- and edge-spaces of (T , {X e }) are the universal coverings of the vertex- and edge-spaces of (Γ, {Xe }, {Xv }, {e }). Conversely, starting from a graph of groups (Γ, {Ge }, {Gv }, {ıe }), we construct a graph of qi-embedded metric spaces (Γ, {Xe }, {Xv }, {e }) whose fundamental group is isomorphic to the fundamental group of (Γ, {Ge }, {Gv }, {ıe }). The universal coverings are related as before, we leave the reader work out the (easy) details. 12

ee }, {X ev }, {e When considering a tree of qi-embedded spaces (T , {X e }), we will essen˜ π, T ). The tially be interested in its geometric realization which will be denoted by (X, edge- and vertex-spaces, and the quasi isometric embeddings will be implicit. Example 3.3. Consider the mapping-torus group Fαn := Fn oα Z of an automorphism α of the rank n free group Fn . This is the fundamental group of a graph of groups which is a loop: the vertex group Gv and the edge group Ge are Fn , the morphism ıe−1 : Fn → Fn is the identity whereas ıe : Fn → Fn is the automorphism α. This group acts cocompactly, isometrically and properly discontinuously on a tree of 0-hyperbolic ˜ T , π) defined as follows: Let Γ be the Cayley graph of Fn with respect to some spaces (X, basis {x1 , · · · , xn }, hence a tree, and let Γi := Γ × [i, i + 1]. We denote by fα : Γ → Γ a continuous map which realizes α on Γ, that is which sends any edge labeled with ˜ = F Γi / ∼, where the generator xi to the edge-path corresponding to α(xi ). Then X i∈Z

(x, i + 1) ∼ (y, i + 1) if and only if (x, i + 1) ∈ Γi , (y, i + 1) ∈ Γi+1 and y = fα (x). The tree T is homeomorphic to the real line R as a topological space: this is the graph with ˜ →T one vertex at each integer i and one edge for each interval [i, i + 1]. The map π : X sends each Γi to a point so that each stratum is an isomorphic copy of Γ. The embedding [i+1,i] : X[i,i+1] → Xi is the identity whereas [i,i+1] : X[i,i+1] → Xi+1 is fα . Example 3.4. Consider the amalgamated product F2 ∗Z F2 with F2 = hx1 , x2 i and Z = hx1 i. This is the fundamental group of a graph of groups G with one edge e and one edge-group Ge := Z = hti, with two vertices v, w and vertex-groups Gv := F2 , Gw := F2 and with morphisms ıe−1 , ıe : Z ,→ F2 defined by ıe−1 (t) = ıe (t) = x1 . This group acts cocompactly, isometrically and properly discontinuously on a tree of 0-hyperbolic spaces ˜ T , π) defined as follows. The tree T is the tree of the universal covering of G. Over (X, ˜ is isometric to R × (0, 1) and the map π : X ˜ → T sends each R × {t} to a an open edge, X point, i.e. Xe = R and the strata over the points interior to the edges are real lines. The strata over the vertices are the copies of a tree Γ, which is the Cayley-graph of F2 with respect to {x1 , x2 }. Each attaching-map e is the identity onto its image. The domain and the images of e are the axis of some conjugates of x1 < F2 . By definition, each stratum in a tree of qi-embedded 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. We extend the definition ˜ ×X ˜ by declaring that the horizontal distance between of the horizontal distance to X two points which are not in a same stratum is infinite. 3.3. The telescopic metric. A section of a map π : A → B is a map σ : B → A such that π ◦ σ = IdB . This notion of section is only a set-theoretic notion: unless otherwise specified, we do not require that a section of a continuous map be continuous, of a morphism be a morphism, . . . . ˜ T , π) be a tree of qi-embedded metric spaces. Definition 3.5. Let (X, ˜ is a section σω (resp. σT ) of For v ≥ 0, a v-vertical segment (resp. v-vertical tree) in X π over a geodesic ω of T (resp. over a subtree T of T ) which is a (v + 1, v)-quasi isometric embedding. ˜ The T -length |ω|T is the vertical length of the v-vertical segment σω : ω → X. ˜ If x is a point in X and if ω is a geodesic of T starting at π(x), the notation ωx denotes ˜ such that some v-vertical segment s with π(s) = ω connects x to the set of points y ∈ X y. 13

Remark 3.6. With the notations above, observe that in particular any point in ωx belongs to π −1 (t(ω)). By a slight abuse of terminology we will not distinguish a vertical segment or tree, which by definition is a map, from its image in the tree of spaces. Since a section is not necessarily continuous, this image is of course not a segment nor a tree in the usual sense. But if ω = ei11 · · · eikk is a geodesic edge-path then a v-vertical segment over ω can be approximated by a chain of intervals xi × (0, 1) over the ei ’s and points yi in the π −1 (t(ei i )) (see Definition 3.1), where the Hausdorff distance between the v-vertical segment and this chain only depends on v. Of course a similar approximation exists in the case of a v-vertical tree. ˜ T , π) be a tree of qi-embedded metric spaces. Definition 3.7. Let (X, ˜ which satisfies the following properties: For v ≥ 0, a v-telescopic chain is a chain p in X • π(p) is an edge-path between two vertices of T , • p is a concatenation h0 s0 · · · hj sj · · · sn hn+1 , with t(sj ) = i(hj+1 ) and t(hj ) = i(sj ) for j = 0 · · · n, of horizontal paths hj in the strata over the vertices of T with non-trivial v-vertical segments sj . Definition 3.8. Let v ≥ 0 and let p be a v-telescopic chain in a tree of qi-embedded ˜ T , π). metric spaces (X, (a) The vertical length |p|vvert of p is the sum of the vertical lengths of the maximal vvertical segments. The horizontal length |p|vhor is the sum of the horizontal lengths of the maximal horizontal paths in the complement of the maximal v-vertical segments. (b) The telescopic length |p|vtel of a v-telescopic chain p is the sum of its horizontal and vertical lengths. ˜ T , π) be a tree of qi-embedded metric spaces. For v ≥ 0, the Definition 3.9. Let (X, v v-telescopic distance dtel (x, y) between two points x and y is the infimum of the telescopic lengths of the v-telescopic chains between x and y. Remark 3.10. Let v ≥ 0 and let p be a v-telescopic chain. The vertical length of each maximal v-vertical segment in p is greater or equal to 1. ˜ is at vertical distance smaller than 1 from a stratum over a vertex of Any point in X 2 T . Thus, when dealing with the behavior of quasi geodesics or with the hyperbolicity ˜ there is no harm in requiring that telescopic chains begin and end at strata over of X vertices of T , as was done in Definition 3.7. For the sake of simplification, we will often forget the exponents in the vertical, horizontal and telescopic lengths, unless some ambiguity might exist. ˜ T , π) be a tree of hyperbolic spaces. Lemma 3.11. Let (X, (a) For any v ≥ 0 there exist λ+ ≥ 1, µ ≥ 0 such that, if ω0 and ω1 are any two v-vertical segments, with initial (resp. terminal) points x0 , x1 (resp. y0 , y1 ) and such that π(ω0 ) = π(ω1 ) = [a, b] then: 1

d (x , x1 ) d (a,b) hor 0 λ+T

d (a,b)

− µ ≤ dhor (y0 , y1 ) ≤ λ+T

dhor (x0 , x1 ) + µ

The constants λ+ , µ will be referred to as the constants of quasi isometry. (b) lim dhor (x0 , xn ) = +∞ ⇔ lim dvtel (x0 , xn ) = +∞ whenever (xn )n∈Z+ is a sen→+∞

n→+∞

quence of points in some stratum. 14

(c) For any v, v 0 ≥ 0, there exist A ≥ 1, B ≥ 0 such that the identity-map from ˜ dv ) to (X, ˜ dv0 ) is a (A, B)-quasi isometry. (X, tel tel (d) For any d, v ≥ 0 there exists C ≥ 0 increasing with both d and v such that for any α, β ∈ T with dT (α, β) = d, for any x, y, z ∈ Xα with z ∈ [x, y], whenever x0 , y 0 , z 0 ∈ Xβ are the endpoints of v-vertical segments starting respectively at x, y C and z, then z 0 ∈ Nhor ([x0 , y 0 ]). (e) For any v, w ≥ 0, there is D ≥ 0 such that if s is a v-vertical segment, then s is a (D, D)-quasi geodesic for the w-telescopic distance. Proof. Item (a) is a straightforward consequence of the definition of a vertical segment. Items (b) and (c) are consequences of the existence of the constants of quasi isometry given by the first item. 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.  Remark 3.12. 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 then for referring afterwards to these constants we will write C3.4 and D3.4 . The following lemma relates the telescopic metric to the original one. In the sequel, it allows one to indistinctively use the original metric or a telescopic metric dvtel . ˜ T , π) be a tree of hyperbolic spaces. Lemma 3.13. Let (X, ˜ (a) For any v ≥ 0, there exist C ≥ 1, D ≥ 0 such that the identity-map from X ˜ equipped with the v-telescopic metric is a equipped with its original metric to X (C, D)-quasi isometry. (b) For any r ≥ 1, s, v ≥ 0, there exist C 0 ≥ 1 and D0 ≥ 0 such that any v-telescopic ˜ ˜ dv ) is a (C 0 , D0 )-quasi geodesic of X chain which is a (r, s)-quasi geodesic of (X, tel equipped with its original metric. (c) For any r ≥ 1, s, v ≥ 0, there exist E ≥ 1, F ≥ 0 such that, given any (r, s)˜ equipped with the original metric, there is a v-telescopic quasi geodesic g of X (E, F )-quasi geodesic G whose Hausdorff distance from g is bounded above by E. Observe that, once Item (a) has been proved Item (b) is straightforward and it is useless in Item (c) to precise what is the Hausdorff distance we refer to: the one coming the original metric or the telescopic one. Indeed, by Item (a) the conclusion of Item (c) holds for the former if and only if it holds for the latter. ˜ equipped with its original metric. Let us define two Proof. Consider a geodesic g of X kinds of modifications: • Let g 0 ⊂ g be a maximal subpath with both endpoints in a same stratum over a vertex v of T and such that π(g 0 ) ⊂ e for some edge e incident to v. Then we substitute g 0 by a horizontal geodesic in Xv between its endpoints. • Let g 0 ⊂ g be a subpath of g with π(g 0 ) = e, π(i(g 0 )) = i(e), π(t(g 0 )) = t(e). Then we substitute g 0 by the concatenation of a 0-vertical segment s from i(g 0 ) to Xt(e) with a horizontal geodesic from t(s) to t(g 0 ). 15

Let G be the path obtained after these substitutions. This is a 0-telescopic chain. In both kinds of substitutions we add a horizontal geodesic h in some stratum Xv . Thanks to the fact that strata are Gromov hyperbolic and edge-spaces are (a, b)-quasi isometrically embedded in the vertex-spaces, there is a bound (not depending on g nor G) on the horizontal Hausdorff distance between h and the projection of g 0 to Xv along the 0vertical segments. This observation gives E ≥ 1 and F ≥ 0 (not depending on g nor G) such that for any subpath G0 of G there is a subpath g 0 of g which is at Hausdorff distance smaller than E from G0 and such that |G0 |tel ≤ E|g 0 | + F . From this we get d0tel ≤ EdX˜ + F . Since dX˜ ≤ d0tel is obvious (a 0-telescopic chain is a particular kind of path in the usual sense) we get that the identity realizes a quasi isometry between the 0-telescopic metric and the original metric. Since all telescopic metrics are quasi isometric (see Lemma 3.11, Item (c)) we get Item (a). Item (b) is then obvious and Item (c) is proven in the same way as above by considering a quasi geodesic g instead of a geodesic.  3.4. Exponential separation of vertical segments, hallways-flare property. The term of “hallways-flare property” was introduced in [3]: it designated the main property introduced by the authors for the hyperbolicity of a graph of quasi isometrically embedded hyperbolic groups. Although the presentation here is very different, we use this same denomination for our central property given in Definition 3.14 below and invite the reader to compare with the “hallways-flare property” of [3]. Definition 3.14. (compare [3]) ˜ T , π) satisfies the hallways-flare property if and only if A tree of hyperbolic spaces (X, for any v ≥ 0 there exist λ > 1 and positive integers t0 , M such that, for any α ∈ T , for any two β, γ ∈ ∂Bα (t0 ) with dT (β, γ) = 2t0 , any two v-vertical segments s0 , s1 over [β, γ] with dhor (s0 ∩ Xα , s1 ∩ Xα ) ≥ M satisfy: max(dhor (s0 ∩ Xβ , s1 ∩ Xβ ), dhor (s0 ∩ Xγ , s1 ∩ Xγ )) ≥ λdhor (s0 ∩ Xα , s1 ∩ Xα ) We will sometimes say that the v-vertical segments are exponentially separated. The constants λ, M, t0 will be referred to as the constants of hyperbolicity. The hallways-flare property requires the exponential separation of the v-vertical segments for any v ≥ 0. It suffices in fact that it be satisfied for some v sufficiently large enough as we are now going to check (see Lemma 3.18). ˜ be a tree of hyperbolic spaces and let S be a horizontal subset Definition 3.15. Let X ˜ which is quasi convex in its stratum, for the horizontal metric. If x is any point in X hor then a horizontal quasi projection of x to S, denoted by PS (x), is any point y in S such that dhor (x, y) < dihor (x, S) + 1. If x and S do not belong to a same stratum, such a horizontal quasi projection does not exist, the horizontal distance dhor (x, y) being infinite for any y ∈ S. ˜ T , π) be a tree of δ-hyperbolic spaces. There exists Lemma 3.16. Let δ ≥ 0 and let (X, C ≥ 0 such that if v ≥ C, if e is an edge of T and if h is a horizontal geodesic in Xt(e) , then: • If no v-vertical segment starting at h can be defined over e, then diamXt(e) (Phhor (e (Xe ))) ≤ 2δ 16

• If v-vertical segments can be defined over e starting at the initial and terminal points of h, then v-vertical segments can be defined over e starting at any point in h. We recall that for each oriented edge e, e denotes the (a, b)-quasi isometric embedding of the edge-space Xe into the vertex-space Xt(e) associated to the given tree of spaces. ˜ be a tree of 0-hyperbolic spaces (i.e. of R-trees - see Examples 3.3 Example 3.17. Let X and 3.4). Then one can set C3.16 := 0. Indeed, the 0-hyperbolicity of the strata implies that, if x and y are two points in e (Xe ) then the whole geodesic of Xt(e) between x and y is contained in (Xe ). Thus, if h is a horizontal geodesic in Xt(e) and e an edge of T such that no 0-vertical segment starting from h is defined over e, then any horizontal geodesic of Xt(e) between two points x, y of e (Xe ) is disjoint from h. This readily implies diamXt(e) (Phhor (e (Xe ))) = 0. Problems occur as soon as one deals with trees of δ-hyperbolic spaces with δ > 0. Then the constants δ, a and b come into play. See Figure 2 and the proof below. -1 (e

π

)

h

Figure 2. Proof. By definition of a tree of δ-hyperbolic spaces, each stratum is δ-hyperbolic for the horizontal metric. This gives a constant c depending on δ, a and b such that for any two points x, y ∈ e (Xe ), any horizontal geodesic [x, y] lies in the horizontal c-neighborhood of e (Xe ). Choose v > 2δ + c. Assuming that no v-vertical segment starting at h can be 2δ defined over e, since horizontal geodesic rectangles are 2δ-thin, we get [x, y] ∩ Nhor (h) = ∅ for any two points x, y ∈ e (Xe ) and any horizontal geodesic [x, y]. The conclusion of the first item follows by the 2δ-thinness of the geodesic rectangles. The second item of the lemma is proved in the same way, details are left to the reader.  ˜ T , π) be a tree of hyperbolic spaces. If v ≥ C3.16 is such that the vLemma 3.18. Let (X, ˜ are exponentially separated with constants of hyperbolicity λv > 1, vertical segments of X Mv , t0 ≥ 0 then for any w ≥ 0, the w-vertical segments are exponentially separated, with constants of hyperbolicity λw > 1, Mw ≥ 0 and t0 . Proof. The statement is a tautology if w ≤ v. We thus assume w ≥ v. Consider α, β, γ in T with α ∈ [β, γ] and dT (α, β) = dT (α, γ) = t0 . Consider two w-vertical segments S0 , S1 over [β, γ] with dhor (x0 , x1 ) ≥ M , where xi = Si ∩ Xα and M > Mv . We distinguish two cases: • there exist v-vertical segments s0 , s1 passing through x0 , x1 and defined over [β, γ]. From Item (a) of Lemma 3.11, each endpoint of the si ’s is at bounded horizontal distance from an endpoint of a Si , where the upper-bound only depends on 17

w, t0 and the constants of quasi isometry. Thus choosing M sufficiently large enough with respect to w gives the desired inequality between dhor (x0 , x1 ) and max(dhor (S0 ∩ Xβ , S1 ∩ Xβ ), dhor (S0 ∩ Xγ , S1 ∩ Xγ )). • the other case: since v has been chosen greater than C3.16 , there is some stratum Xµ , µ ∈ [β, γ] such that dhor (S0 ∩ Xµ , S1 ∩ Xµ ) is bounded above by a constant depending on w, δ, t0 and the constants of quasi isometry. By Item (a) of Lemma 3.11, we get an upper-bound on dhor (x0 , x1 ). Setting M greater that this upperbound, we get the lemma.  We end this section by a very general and easy lemma about the constants of hyperbolicity. ˜ T ) be a tree of hyperbolic spaces satisfying the hallways-flare Lemma 3.19. Let (X, property. (a) The constants of hyperbolicity and quasi isometry can be chosen arbitrarily large enough. (b) For any v ≥ C3.16 such that the v-vertical segments are exponentially separated, for any constants of hyperbolicity λ, M, t0 such that M is sufficiently large enough, there exists C ≥ 0 such that for any α ∈ T , for any β, γ ∈ ∂Bt0 (α) with α ∈ [β, γ], for any two v-vertical segments s0 , s1 over [β, γ] such that dhor (x0 , x1 ) ≥ M where xi = si ∩ Xα , if the endpoints y0 , y1 of s0 , s1 in Xβ (resp. in Xγ ) satisfy: 1 dhor (x0 , x1 ) < dhor (y0 , y1 ), λ then, for any n ≥ 1, for any T -geodesic ω starting at α with [α, β] ⊂ ω (resp. [α, γ] ⊂ ω) and |ω|T ≥ C + nt0 : dihor (ωx, ωy) ≥ λn dihor (x, y). 4. Approximation of quasi geodesics: a “simple” case From a group-theoretical point of view, the case treated in this section allows one to deal with semi-direct products of (relatively) hyperbolic groups with free groups but not with HNN-extensions and amalgamated products along proper subgroups. For this we need the similar, but more general, result of Section 5. Beware that the corridors (and later the generalized corridors) defined below are not the hallways of [3]. The reason is that we are interested in exhibiting quasi convex subsets of our trees of hyperbolic spaces and the hallways of [3], in general, are not quasi convex. ˜ T , π) be a tree of hyperbolic spaces, and let v ≥ 0. Definition 4.1. Let (X, A v-vertical tree σ : T → T is maximal if and only if there exists no v-vertical tree 0 0 σ : T 0 → T such that T ⊂ T 0 , T 6= T 0 and σ|T = σ. A v-corridor C is a union of horizontal geodesics which satisfies the following properties: (a) For each α ∈ T , C ∩ Xα either is empty or is equal to a horizontal geodesic. (b) There exists a subtree T of T such that C = t hα where each hα = C ∩ Xα is a α∈T

horizontal geodesic. ˜ such that each hα connects (c) There exist two maximal v-vertical trees σ1 , σ2 : T → X a point of σ1 (T ) to a point of σ2 (T ). The subsets σ1 (T ) and σ2 (T ) are the vertical boundaries of the v-corridor C. 18

Example 4.2. Consider the tree of spaces associated to the mapping-torus Fαn of a free group automorphism as described in Example 3.3. Then any horizontal geodesic h is contained in a 0-corridor C with π(C) = T . If h connects two vertices of the Cayley graph of Fn , then the vertical boundaries of C are the α-orbits of the endpoints of h. Example 4.3. Consider the tree of spaces of Example 3.4. Given a horizontal geodesic h, there will be different kinds of 0-corridors containing h, depending on the position of h in the Cayley graph of F2 . For simplicity, the geodesics we consider always have their endpoints at vertices of this Cayley graph. If h is contained in the axis of a conjugate of x1 (for instance h = x21 ), then h is contained in a 0-corridor C such that π(C) is a bi-infinite geodesic of T . Each horizontal geodesic in C reads the same word, a power of x±1 1 . For any other horizontal geodesic h, for instance h = x2 x21 x2 , the unique 0-corridor C containing h is equal to h. ˜ T , π) be a tree of hyperbolic spaces the attaching-maps of which Remark 4.4. Let (X, are all quasi isometries (and not only quasi isometric embeddings). Then, as soon as ˜ there is a v-corridor C whose vertical boundaries v ≥ C3.16 , given any two points x, y in X pass through x and y. Moreover π(C) = T . Definition 4.5. Let C be a union of horizontal geodesics in a tree of hyperbolic spaces ˜ T ). Assume that for each stratum Xα the intersection C ∩ Xα either is empty or is (X, equal to a horizontal geodesic. If x is a point in a stratum Xα , then PChor (x) stands for the horizontal quasi projection hor (x) of x to C (see Definition 3.15). PC∩X α In the definition above, for instance C might be a corridor. Before stating Lemma 4.6 below, we would like to insist on two points: • The horizontal quasi projection PChor is a projection in the strata which only refers to the horizontal metric defined on each stratum. • Item (b) does not tell anything about the behavior of the telescopic (quasi)geodesics in a tree of hyperbolic spaces. It only allows one to consider a corridor as a quasi geodesic telescopic metric space. ˜ T , π) be a tree of hyperbolic spaces. For any v ≥ C3.16 there exists Lemma 4.6. Let (X, ˜ then: C ≥ v such that, if C is a v-corridor in X (a) For any v-vertical segment s, PChor (s) is a C-vertical segment. (b) For any w ≥ C, C equipped with the length-metric induced by the w-telescopic ˜ is a quasi geodesic metric space, denoted by (C, dw ). metric on X tel Figure 3 illustrates Lemma 4.6. ˜ is the section of π such that s = σ(ω) then P hor (s) is the image of ω Proof. If σ : ω → X C hor under the map PC ◦ σ. This map is a section of π since the horizontal quasi projection PChor is a projection in each stratum. We want to prove the existence of a ≥ 1, b ≥ 0 ˜ independent of ω such that PChor ◦ σ is a (a, b)-quasi isometric embedding of ω into X. Assume that ω is a single edge. Since v ≥ C3.16 and since C is a v-corridor, if it is defined over ω then v-vertical segments can be defined over ω starting at each point of C ∩ Xi(ω) . ˜ be such a v-vertical segment starting at PC (σ(i(ω))). By Items (a) and Let σ0 : ω → X (c) of Lemma 3.11, dhor (PC (σ(t(ω))), σ0 (PC (σ(i(ω)))) is bounded above by a constant. 19

Projection on the v-corridor v-vertical segments Horizontal geodesics

Figure 3. Thanks to Item (b) of Lemma 3.11, this proves Item (a) of the current lemma. Item (b) is now easy.  ˜ T , π) be a tree of qi-embedded metric spaces, let v ≥ 0 and Definition 4.7. Let (X, ˜ be two maximal v-vertical trees in X. ˜ A diagonal between σ1 (T ) and let σ1 , σ2 : T → X σ2 (T ) is any horizontal geodesic D with π(D) ∈ V (T ), with endpoints in σ1 (T ) and σ2 (T ) and such that any other horizontal geodesic h satisfying these two properties also satisfies |D|hor ≤ |h|hor . ˜ is the The diagonal distance between two maximal v-vertical trees σ1 , σ2 : T → X horizontal length of any diagonal between σ1 (T ) and σ2 (T ). In other words, a diagonal is a horizontal geodesic which minimizes the horizontal distance between two maximal vertical trees passing through its endpoints. See Figure 4. It might happen that a diagonal be reduced to a single point, in which case the diagonal distance between the two vertical trees considered vanishes (hence the diagonal distance is in fact a pseudo-distance).

A diagonal

Figure 4. Before the statement of Theorem 4.8, we would like to point out that this is a theorem about trees of hyperbolic spaces whose attaching-maps are quasi isometries, and not only quasi isometric embeddings. The main feature of this theorem is to approximate ˜ by some kind of “canonical” quasi geodesics, which in particular are quasi geodesics of X telescopic chains. We stated the results of this theorem by using the original metric of ˜ But by Lemma 3.13 these results also hold for any telescopic metric and it is easily X. seen that they also hold for (C, dvtel ) when this makes sense. ˜ be a tree of hyperbolic spaces which satisfies the hallways-flare Theorem 4.8. Let X property. Assume that each attaching-map from an edge-space into a vertex-space is a quasi isometry. Then: • for any v ≥ C3.16 , there is E ≥ v, 20

• for any L > 0 greater than some critical constant and for any v ≥ C3.16 there exists D ≥ 1, • for any L > 0 greater than some critical constant, for any v ≥ C3.16 , for any a ≥ 1 and b ≥ 0 there exists C ≥ 0, ˜ for any v-corridor C whose vertical such that for any pair of distinct points x, y in X, boundaries pass through x and y, there is a E-telescopic chain P in C satisfying the following properties: ˜ d ˜ ). (a) This is a (D, D)-quasi geodesic of (X, X (b) At the exception of at most one, each maximal horizontal path in P is a diagonal with horizontal length greater or equal to L whereas the last maximal horizontal path has horizontal length less or equal to L. ˜ with endpoints x and y, the Hausdorff distance (c) For any (a, b)-quasi geodesic g in X ˜ d ˜ ) between g and P is bounded above by C. in (X, X (d) Let P 0 be the closed complement in P of its first and last maximal vertical seg˜ whose endpoints lie in the vertical ments. Let g 0 be any (a, b)-quasi geodesic in X boundaries of C. Let s0 , s1 be the v-vertical segments in the vertical boundaries of C from the endpoints of P 0 to the endpoints of g 0 . Then g 0 is at Hausdorff distance ˜ d ˜ ), from the concatenation of P 0 with s0 and s1 . smaller than C, in (X, X For proving this theorem, we need two important propositions which we state now but the proofs of which are postponed for a while. For the understanding of Proposition 4.9, let us recall that we proved in Lemma 4.6 that a corridor C in a tree of hyperbolic ˜ becomes a quasi geodesic metric space when equipped with the length-metric spaces X ˜ as soon as v is sufficiently large enough. This induced by the v-telescopic metric on X, quasi geodesic metric space is denoted by (C, dvtel ). This notation does not mean that we proved that C is a tree of hyperbolic spaces. ˜ be a tree of hyperbolic spaces which satisfies the hallways-flare Proposition 4.9. Let X property. For any v ≥ C3.16 , there is D ≥ v such that for any w ≥ D, L > 0, a ≥ 1 and b ≥ 0 ˜ if L is the horizontal length of some there exists C ≥ 0 such that if C is a v-corridor in X, horizontal geodesic [x, y] in C, if g is a (a, b)-quasi geodesic of (C, dw tel ) from a w-vertical tree through x to a w-vertical tree though y, then g is contained in the C-neighborhood of the union of the w-vertical segments connecting its endpoints to x and y. The constant C is increasing with L as soon as L is greater than some critical constant. See Section 8 for a proof, and Figure 5 for an illustration.

quasi geodesics

Figure 5. ˜ be a tree of hyperbolic spaces which satisfies the hallways-flare Proposition 4.10. Let X property and the attaching-maps of which are quasi isometries. For any v ≥ C3.16 , a ≥ 1 21

˜ and if C is a and b ≥ 0 there exists C ≥ 0 such that, if g is a (a, b)-quasi geodesic in X v-corridor the vertical boundaries of which pass through the endpoints of g then g ⊂ NXC˜ (C). See Section 9 for a proof. We will also need the following two much easier statements. ˜ be a tree of hyperbolic spaces. There exists C ≥ 0 such that for any Lemma 4.11. Let X ˜ for any two points x, y in a same stratum intersected by v ≥ 0, for any v-corridor C in X, hor hor C, dhor (PC (x), PC (y)) ≤ dhor (x, y) + C. The same inequality holds for the horizontal quasi projections of x and y to the image of the embedding of an edge-space into a vertexspace. Proof. Since there is δ ≥ 0 such that strata are δ-hyperbolic spaces for the horizontal metric and the subspaces to which one projects are quasi convex subsets of their stratum for this horizontal metric, this is a consequence of [9], Corollary 2.2.  ˜ be a tree of hyperbolic spaces. For any v ≥ C3.16 , there is D ≥ v Lemma 4.12. Let X and for any v ≥ C3.16 , a ≥ 1 and b, r ≥ 0 there is C ≥ 1 such that for any (a, b)-quasi ˜ and for any v-corridor C, if g ⊂ N r (C) then any horizontal quasi geodesic g of X hor projection PChor (g) is a D-telescopic (C, C)-quasi geodesic of (C, dD tel ). Proof. By Lemma 3.13 we can assume that g is a v-telescopic chain. By Lemma 4.6 PChor (g) is a C4.6 (v)-telescopic chain. Let us consider any two points x, y in G = PChor (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 subset of G between x and y. Since we now consider C4.6 (v) the C4.6 (v)-telescopic distance, |Gxy |vert = |gx0 y0 |vvert . From Lemma 4.11 and since any two maximal horizontal paths in G are separated by a vertical segment of vertical length C (v) at least 1, we then get |Gxy |tel4.6 ≤ 2C4.11 |gx0 y0 |vtel . 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 |tel4.6

(v)

≤ 2C4.11 (a(2r + dvtel (x, y)) + b).

Since all telescopic distances are quasi isometric (Item (c) of Lemma 3.11), we so get the right inequality for the quasi geodesicity of PChor (g). We leave the reader work out the similar proof of the left inequality.  ˜ be a tree of hyperbolic spaces which satisfies the hallwaysProof of Theorem 4.8. Let X flare property and such that each attaching-map from an edge-space into a vertex-space ˜ and let C is a quasi isometry. Let v ≥ C3.16 . Let x, y be any two distinct points in X be any v-corridor whose vertical boundaries pass through x and y. Since the attaching maps of the tree of hyperbolic spaces are all quasi isometries, π(C) = T . From Lemmas 4.6 3.18 and 4.6, (C, dC tel ) is a quasi geodesic metric space and the C4.6 -vertical segments are exponentially separated. From Item (b) of Lemma 3.19, this implies in particular that the endpoints of any diagonal with horizontal length greater than some constant M are exponentially separated in all the directions of T outside a region with vertical width bounded by 2C3.19 . Let L ≥ M . Consider a diagonal h0 with horizontal length L from a vertical boundary B0 of C to some maximal C4.6 -vertical tree T0 in C. Then another diagonal h1 from T0 to another maximal C4.6 -vertical tree T1 , and so on until arriving at a maximal C4.6 -vertical tree Tr which is at diagonal distance smaller than L from the other v-vertical boundary tree B1 of C. Then the concatenation of: 22

• the diagonals h0 , h1 , · · · , hr , • the C4.6 -vertical segments in T0 , T1 , · · · , Tr−1 between the endpoints of the hi ’s, • a horizontal geodesic hr+1 with |hr+1 |hor ≤ M between Tr and B1 which is closest to hr with respect to the vertical distance, • the C4.6 -vertical segment in Tr between hr and hr+1 , gives to us the telescopic chain denoted by P 0 in the last item of Theorem 4.8, whereas of course the concatenation with the v-vertical segments in B0 and B1 between the horizontal geodesics h0 and hr+1 and the points x and y gives the announced telescopic chain P as we are now going to check. ˜ between x and y. By Proposition 4.10, Let g be any (a, b)-quasi geodesic of X C4.10 g ⊂ Ntel (C). From Lemma 4.12, G := PChor (g) is a D4.12 -telescopic (C4.12 , C4.12 )-quasi 4.12 geodesic of (C, dD tel ). The quasi geodesic G intersects the vertical trees T0 , T1 , · · · of C: let G0 be the smallest subset of g between x and T0 . From Proposition 4.9, G0 is contained in the C4.9 neighborhood of the union of the vertical segments s0 , s1 from the endpoints of G0 to those of h0 . From our observation above about the exponential separation of the endpoints of h0 , there is some κ > 0 such that, outside the region in C centered at h0 with vertical width κ, the horizontal geodesics between the vertical trees of the endpoints of h0 have horizontal length greater than 3C4.9 . We so get a constant K(v, L, a, b) > 0, not depending of the quasi geodesic nor on the corridor considered, such that dH tel (G0 , s0 ∪h0 ∪s1 ) ≤ K(v, L, a, b). The same arguments apply for the subset Gi between Ti−1 and Ti until i = r. Since |hr+1 |hor ≤ L and hr+1 has been chosen to minimize the vertical distance between hr and all horizontal geodesics h satisfying |h|hor ≤ L, we easily get a constant K 0 (v, L, a, b) such that the concatenation of hr+1 with • the v-vertical segment in B1 between hr+1 and y (the terminal point of g), • the C4.6 -segment in Tr between hr and hr+1 , is at Hausdorff distance smaller than K 0 (v, L, a, b) from the subset of G following the concatenation of the Gi ’s. It follows that P is a D4.12 -telescopic chain between x and y with dH tel (g, P) ≤ max(K(v, L 0 , a, b), K (v, L, a, b)). Since the construction of the horizontal geodesics hi does not depend on the endpoints x and y, we also have the conclusion for P 0 . It remains to check that P is a (D, D)-quasi geodesic, with D only depending on v and L. It suffices to choose a = 1 and b = 0 and then apply what was proved just above: the chain P is at Hausdorff distance smaller than max(K(v, L, 1, 0), K 0 (v, L, 1, 0)) from a geodesic. From this observation, we easily get by classical arguments and computations that P is a quasi geodesic as announced.  5. Approximation of quasi geodesics: the general case In order to give a simple statement, we added in Theorem 4.8 the restriction that the attaching-maps of the tree of spaces be quasi isometries, instead of requiring that they be quasi isometric embeddings. In this way, the elementary notion of corridor (Definition 4.1) was sufficient to describe the quasi geodesics of the space. In Example 4.2, we saw that the notion of corridor was too rough to capture the geometry of the amalgamated product F2 ∗Z F2 . We now define the generalized v-corridors that we will substitute to the corridors of Theorem 4.8 in order to obtain the more general statement we are looking for. 23

˜ T , π) be a tree of qi-embedded metric spaces. Let v ≥ 0. Definition 5.1. Let (X, ˜ is a union of horizontal geodesics which satisfies the A generalized v-corridor C in X following properties: (a) For each α ∈ T , C ∩ Xα either is empty or is a horizontal geodesic. (b) The image T := π(C) is a subtree of T . (c) If w is a vertex of T in the above subtree T and e is an edge of T which is incident to w but does not belong to T , then there is no v-vertical segment over e starting from C. (d) The subtree T ⊂ T admits a decomposition in subtrees (Ti )i∈I such that: (a) For any (i, j) ∈ I × I with i 6= j, Ti ∩ Tj either is empty or is reduced to a single vertex. ˜ such that each (b) For each i ∈ I, there are two v-vertical trees σ1i , σ2i : Ti → X maximal horizontal geodesic in the subcorridor Ci := C ∩ π −1 (Ti ) has its endpoints in σ1i (Ti ) and σ2i (Ti ). (c) If Ti ∩ Tj is non-empty then Ci ∩ Cj is a horizontal geodesic. S S With the notations above, the subset σ1i (Ti ) ∪ σ2i (Ti ) is the vertical boundary of i∈I

i∈I

the generalized v-corridor. Let h be a horizontal geodesic. A minimal generalized v-corridor through h is any ˜ such that C(h) ∩ Xπ(h) = h and such that any other v-corridor C v-corridor C(h) in X containing h satisfies π(C(h)) ⊂ π(C). ˜ T , π) associated to the Example 5.2. We consider the tree of 0-hyperbolic spaces (X, amalgamated product F2 ∗Z F2 already evoked in Examples 3.4 and 4.3. Let h be a horizontal geodesic reading x2 x21 x2 in the Cayley graph of F2 (a tree) over some vertex of T . Figure 6 represents a minimal generalized 0-corridor C(h) through h. x2 x1 x1

x1 x1

x2

π

Figure 6. With the notations of Definition 5.1, the projection T = π(C(h)) = T1 ∪ T2 ∪ T3 is the union of three bi-infinite geodesics Ti which intersect in a unique common vertex. These are the three thick lines in Figure 6. We have σ11 (T1 ) = σ21 (T1 ) and σ13 (T3 ) = σ23 (T3 ). In 24

other words each horizontal geodesics of C1 = π −1 (T1 ) ∩ C and C3 = π −1 (T3 ) ∩ C is reduced to a single point and the subcorridors C1 and C3 are two bi-infinite 0-vertical segments. They pass through the endpoints of h. Finally the subcorridor C2 = π −1 (T2 ) ∩ C is a minimal generalized 0-corridor containing the horizontal geodesic x21 , see Example 4.3. Observe that the notions of 0-corridors and generalized 0-corridors are very different since we saw in Example 4.3 that the unique 0-corridor containing h = x2 x21 x2 is h. Example 5.3. With the notations of Example 5.2, let x be an endpoint of the horizontal ˜ such that π(y) ∈ geodesic h. Let y be any other vertex in X / T . Then of course y ∈ / C(h). 1 Consider the unique shortest T -geodesic from π(y) to T . This is an edge-path ei1 · · · eikk . For simplicity, assume there is a single edge e. Then there are 0-vertical segments over e from some bi-infinite geodesic A reading x±∞ in Xi(e) = Xπ(y) to another bi-infinite 1 geodesic A0 reading x±∞ in X (t(e) ∈ T ). Let h 0 (resp. h1 ) be the shortest horizontal t(e) 1 (i.e. F2 -) geodesic in Xi(e) (resp. in Xt(e) ) from y (resp. from C(h) ∩ Xt(e) ) to A (resp. to A0 ). Then the union C(h) ∪ C(h0 ) ∪ C(h1 ) is a generalized 0-corridor the vertical boundaries of which pass through x and y. Beware however that, since the projections to T of the C(hi )’s overlap, they do not form the subcorridors of Definition 5.1. One has to further decompose to get the subcorridors. Figure 7 illustrates what a “typical” generalized corridor looks like.

Three subcorridors

Figure 7. A slight generalization of the construction above allows one to get the following statement: ˜ be a tree of hyperbolic spaces. There exist D ≥ C3.16 such that, for Lemma 5.4. Let X ˜ there is a generalized v-corridor the any v ≥ D and for any two distinct points x, y in X vertical boundaries of which pass through x and y. ˜ be a tree of hyperbolic spaces which satisfies the hallways-flare Theorem 5.5. Let X property. Then: • for any v ≥ C3.16 , there is E ≥ v, • for any L > 0 greater than some critical constant and for any v ≥ C3.16 there exists D ≥ 1, • for any L > 0 greater than some critical constant, for any v ≥ C3.16 , for any a ≥ 1 and b ≥ 0 there exists C ≥ 0, ˜ for any generalized v-corridor C such that for any pair of distinct points x, y in X, whose vertical boundaries pass through x and y, there is a E-telescopic chain P in C satisfying the following properties: 25

(a) This is a (D, D)-quasi geodesic. (b) At the exception of at most one in each subcorridor (see Definition 5.1), each maximal horizontal path in P is a diagonal with horizontal length greater or equal to L. ˜ with endpoints x and y, the Hausdorff distance (c) For any (a, b)-quasi geodesic g in X between g and P is bounded above by C. (d) Let P 0 be the closed complement in P of the first and last maximal vertical seg˜ whose endpoints lie in the vertical ments. Let g 0 be any (a, b)-quasi geodesic in X boundaries of C. Let s0 , s1 be the v-vertical segments in the vertical boundaries of C from the endpoints of P 0 to the endpoints of g 0 . Then g 0 is at Hausdorff distance smaller than C from the concatenation of P 0 with s0 and s1 . Proof of Theorem 5.5. We first need an adaptation to this more general setting of some of the lemmas and propositions given for proving Theorem 4.8: Proposition 5.6. Lemma 4.6, Proposition 4.9 and Proposition 4.10 remain true for generalized v-corridors with v ≥ D5.4 . There is nothing to prove with respect to Lemma 4.6 and Proposition 4.9. We refer the reader to Section 9.6 for the proof of the adaptation of Proposition 4.10 to generalized corridors. ˜ be a tree of hyperbolic spaces. For any v ≥ D5.4 , for any a ≥ 1 and Lemma 5.7. Let X b ≥ 0 there exists C ≥ 0 such that if g is any (a, b)-quasi geodesic, if C is any generalized v-corridor the vertical boundaries of which pass through the endpoints of g, then there is a (a, b + 2δ)-quasi geodesic G with dH ˜ (g, G) ≤ C and π(G) ⊂ π(C). X Proof. Let γ ∈ T be an endpoint of π(C). Assume that g 0 is a maximal subset of g with endpoints in Xγ and such that π(g 0 ) ∩ π(C) = γ. Then, since v ≥ C3.16 , Lemma 3.16 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 subsets of g like g 0 yields a quasi geodesic as announced.  With the above adaptations in mind, the proof of Theorem 5.5 is now a duplicate of the proof of Theorem 4.8.  6. Hyperbolicity Theorem 6.1 generalizes Bestvina-Feighn’s combination to non-proper hyperbolic spaces. Bowditch proposed such a generalization in [6]. ˜ be a tree of hyperbolic spaces which satisfies the hallways-flare Theorem 6.1. Let X ˜ property. Then X is a Gromov-hyperbolic metric space. Proof. We begin by proving the ˜ be a tree of hyperbolic spaces which satisfies the hallways-flare Theorem 6.2. Let X property. For any a ≥ 1 and b ≥ 0 there exists C ≥ 0 such that (a, b)-quasi geodesic bigons are C-thin. Proof of Theorem 6.2. By Lemma 3.13, it suffices to prove Theorem 6.2 for D5.4 -telescopic (a, b)-quasi geodesic bigons. Thus let g0 , g1 be the two sides of a D5.4 -telescopic (a, b)-quasi geodesic bigon. By Theorem 5.5 (Theorem 4.8 suffices in the case where the attaching˜ are quasi isometries), there is E5.5 ≥ D5.4 and a E5.5 -telescopic chain P such maps of X 26

that for i = 0, 1 we have dH (gi , P) ≤ C5.5 . Hence dH (g0 , g1 ) ≤ 2C5.5 and Theorem 6.2 is proved.  The following lemma was first indicated to the author by I. Kapovich: Lemma 6.3. [14] 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. Theorem 6.2 together with Lemma 6.3 imply Theorem 6.1.



7. Strong relative hyperbolicity Let G be a group with finite generating set S and associated Cayley graph ΓS (G), and let H = {H1 , · · · , } be a family of subgroups Hi of G. Let ΓH S (G) be the coned Cayley graph (see Definition 2.1). For u ≥ 1, v ≥ 0 let gˆ be a (u, v)-quasi geodesic path in ΓH ˆ in ΓS (G) S (G). A trace g of g is obtained by substituting each subpath of gˆ not in ΓS (G) by a subpath of ΓS (G) with same endpoints, which is a geodesic for the metric induced by ΓS (G) on the corresponding left Hi -class. We say that g (or gˆ) backtracks if g reenters a left Hi -class that it left before. Definition 7.1. [12] With the notations above: The coned Cayley graph ΓH S (G) satisfies the Bounded-Coset Penetration property (BCP) if and only if for any u ≥ 1 and v ≥ 0 there exists C ≥ 0 such that, for any two (u, v)-quasi geodesics gˆ0 , gˆ1 of ΓH S (G) with traces g0 , g1 in ΓS (G), which have the same initial point, which have terminal points at most 1-apart in ΓS (G) and which do not backtrack, the following two properties are satisfied: (a) if both g0 and g1 intersects a same left Hi -class then their first intersection points with this class are C-close in ΓS (G), (b) if g0 intersects a left Hi -class that g1 does not intersect, then the length in ΓS (G) of the subpath of g0 between its entrance and exit-points in this class is smaller than C. Farb’s notion of strong relative hyperbolicity is that the group G is strongly hyperbolic relative to H if and only if ΓH S (G) is hyperbolic and satisfies the BCP. This definition is equivalent to the definition given in the introduction, which relied upon Bowditch’s approach (see [5]). Let (G, He , Hv ) be a graph of relatively hyperbolic groups (recall that G = (Γ, {Ge }, {Gv }, {ıe })). We associate to G a graph of metric spaces (Γ, {Xe }, {Xv }, {e }) which satisfies the following properties: • Each edge- and vertex space of Γ is a PL 2-complex with fundamental group the corresponding edge- or vertex-group of G. • The 1-skeleton of the above 2-complex is the rose whose petals are identified to the generators of this edge- or vertex-group. • The attaching-maps e are continuous PL-maps which induce the corresponding monomorphisms ıe between the fundamental groups of the edge- and vertex-spaces. e π, T ). Then we consider a geometric realization of Γ and its universal covering (X, e is the Cayley graph of the corresponding vertex- or The 1-skeleton of each stratum of X edge-group. We then consider the space obtained • by putting a cone over each left H-class (H a parabolic subgroup in the edge- or vertex-group) of this Cayley graph, 27

• by adding a “triangular” 2-cell for each pair of elements g, gh in same left H-class, with h a generator of H, the boundary of the 2-cell being a geodesic reading h between g and gh and the two exceptional edges incident to g and gh, • by extending the attaching maps over these coned Cayley graphs in a π(G)equivariant way. By definition of a graph of relatively hyperbolic groups, the tree of spaces that we b π, T ). eventually get is a tree of qi-embedded Gromov hyperbolic spaces, denoted by (X, b π, T ), and its geometric realization, will be called Such a tree of qi-embedded spaces (X, a relative (G, He , Hv )-tree of metric spaces. The attaching-maps from the edge-spaces to the vertex-spaces are denoted by b e . Observe that in the setting of strong relative hyperbolicity, if parabolic subgroups are almost malnormal and infinite then these maps are uniquely defined over the exceptional vertices. This allows us to give the following Definition 7.2. Let (G, Hv , He ) be a fine graph of strongly relatively hyperbolic groups. b T , π) be some relative (G, Hv , He )-tree of spaces (see the construction above). Let (X, b is a maximal set S of exceptional vertices in X b such that An exceptional orbit in X 0 0 0 (v, v ) ∈ S × S if and only if b [π(v),π(v0 )] (v) = v , where [π(v), π(v )] denotes the unique reduced edge-path from v to v 0 and b [π(v),π(v0 )] is the composition of the associated attachingmaps. Definition 7.3. Let (G, Hv , He ) be a fine graph of strongly relatively hyperbolic groups. b T , π) satisfies the strong hallways-flare propA relative (G, Hv , He )-tree of spaces (X, erty if it satisfies the hallways-flare property and for any m ≥ 0, there is t ≥ 0 such that the vertical width of any region where two exceptional orbits remain at horizontal distance smaller than m one from each other is smaller than t. b satisfies The second condition in the above definition is needed for the BCP. Since X the hallways-flare property, it suffices in fact that the existence of t be satisfied for a constant m greater than the constant of hyperbolicity M . Theorem 7.4. Let G be a fine graph of strongly relatively hyperbolic groups. If some relative (G, Hv , He )-tree of spaces satisfies the strong hallways-flare property, then the fundamental group of G is strongly hyperbolic relative to a family composed of • exactly one representative from each non-periodic parabolic orbit, • the free extensions of exactly one representative from each periodic parabolic orbit. We begin the proof with the Lemma 7.5. With the assumptions and notations of Theorem 7.4: There exists C ≥ 0 such that any exceptional orbit is a discrete subset of a C-vertical tree. Proof. This lemma is a consequence of the following two observations: • there are finitely many parabolic subgroups preserved up to conjugacy, • the free groups which fix these subgroups up to conjugacy are finitely generated. Let us prove this last assertion. We consider the universal covering of our graph of groups. If no composition of conjugacies and morphisms eıe over a geodesic p of T starting at some vertex v is defined on a parabolic subgroup H in Hv then the same property holds over any geodesic of T properly containing p. It follows that there is a constant C > 0, only depending on the number of parabolic subgroups and of vertices of G, such that for any vertex v of T , for any parabolic subgroup in Hv , if no composition of conjugacies and morphisms eıe over a geodesic of length C starting at v is defined on H then H does not 28

belong to a periodic parabolic orbit. This readily implies that the free subgroup fixing (a conjugate of) a parabolic subgroup up to conjugacy is finitely generated. It follows from the above two observations that there are only finitely many conjugation elements. Let m be the maximum of their word-lengths. Then m + 32 ( 21 for going from b to X plus 1 for going through a left H-class coned in X) b gives an exceptional vertex of X the announced constant.  The following lemma is a straightforward consequence of the strong hallways-flare property: Lemma 7.6. With the assumptions and notations of Theorem 7.4: There exists C ≥ 0 b then the such that, if v, v 0 are any two exceptional vertices in a same stratum of X 0 exceptional orbits of v and v are connected by a diagonal (see Definition 4.7) of horizontal length greater or equal to 1, the endpoints of which are exponentially separated in all the directions outside a region whose vertical width is smaller than C. Lemma 7.7. With the assumptions and notations of Theorem 7.4: For any a ≥ 1 and b between two b ≥ 0 there exists C ≥ 0 such that if g, g 0 are two (a, b)-quasi geodesics of X 0 0 exceptional orbits L1 , L2 then g, g admit decompositions g = g1 g2 g3 and g = g10 g20 g30 with the following properties: g1 ⊂ NXCb (L1 ), g10 ⊂ NXCb (L1 ), g3 ⊂ NXCb (L2 ), g30 ⊂ NXCb (L2 ) and 0 0 0 H dH b (g, g ) ≤ C. b (g2 , g2 ) ≤ C. If g and g have the same endpoints then dX X Proof. This is an easy consequence of Theorem 5.5. For simplicity assume that the b are quasi isometries so that Theorem 4.8 can be applied. The C7.5 attaching-maps of X vertical trees through the given two exceptional orbits bound a C7.5 -corridor. Both g and g 0 are approximated by two chains G and G0 which only possibly differ by their first and last maximal vertical segments in L1 and L2 . These last vertical segments 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 orbits. As written before, the extension to the general case where there is not a corridor, but only a generalized corridor, between the two exceptional orbits, is easily dealt with by using Theorem 5.5 instead of Theorem 4.8.  Definition 7.8. Let (G, Hv , He ) be a fine graph of strongly relatively hyperbolic groups. b is the metric space obtained from a relative (G, Hv , He )-tree An exceptional space C(X) b by putting a cone over each exceptional orbit. of spaces X Lemma 7.9 below stresses the importance of this exceptional space. Lemma 7.9. With the assumptions and notations of Theorem 7.4: The exceptional space b is hyperbolic and satisfies the BCP with respect to the exceptional orbits if and only C(X) if the fundamental group of G is strongly hyperbolic relative to a family of subgroups as given by Theorem 7.4. Remark 7.10. Let G be a finitely generated group which is strongly hyperbolique relative to H = {H1 , H2 }. Let α ∈ Aut(G, H) such that α(H1 ) is a conjugate of H2 and α(H2 ) is a conjugate of H1 . Consider the graph of groups G with one vertex and one edge (a b be a relative (G, H, H)-tree of spaces. Then in C(X) b loop) associated to G oα Z. Let X cones are put above the left Hi -classes, and their exceptional vertices all belong to a same exceptional orbit. However only one of the two subgroups H1 , H2 appears in the subgroups of the relative part described by Theorem 7.4 because otherwise the condition of almost malnormality would be violated. 29

b according to the parabolic Proof of Lemma 7.9. Let Y be the space obtained by coning X subgroups described in Theorem 7.4. The essential difference between Y and the coned b of Definition 7.8 is the following one: space C(X) b a horizontal cone is first put over all the left-classes for the parabolic subgroups In C(X) in the edge and vertex groups; then a “vertical cone” is put over all the vertices which belong to a same exceptional orbit. In Y, a cone is put on the left-classes of exactly one subgroup from each finite orbit, and of exactly one free extension of subgroup in each periodic orbit. b and Y, there is exactly one exceptional vertex for each Observe that in both C(X) exceptional orbit. One thus has a natural one-to-one correspondence, denoted by B, beb and those of Y. Assume that there is a horizontal tween the exceptional vertices of C(X) b over two exceptional vertices x, y in a same stratum of X. ˜ It belongs to cone in C(X) an exceptional orbit and we denote by v(gH) the exceptional vertex associated to this leaf. Consider the exceptional vertex B(v(gH)) of Y. Assume that x, y do not belong to the cone with vertex B(v(gH)). Then there are two points x0 , y 0 in another stratum which are at bounded telescopic distance from x and y and belong to this cone. This is straightforward if v(gH) is the vertex of the cone over a finite exceptional orbit. Otherwise this comes from the finite generation of the free groups which preserve the parabolic subgroups up to conjugacy and from the fact that there is an upper-bound on the length of the conjugacy elements. b whose restriction to X b is the identity-map and There is a natural map j : Y → C(X) which maps each exceptional vertex v(gH) of Y to the exceptional vertex B −1 (v(gH)) of b The observation of the previous paragraph readily implies the following assertion: C(X). b (with possible different if g is a quasi geodesic of Y, then j(g) is a quasi geodesic of C(X) b constants of quasi geodesicity) and traces in X of g and j(g) are Hausdorff-close. The lemma follows.  b follows from the quasi convexRemark 7.11. The hyperbolicity of the coned space C(X) ity of the exceptional orbits implied by Lemma 7.5 and from the arguments developed for proving Proposition 1 of [33]. However we re-prove it when listing below the arguments for checking the BCP. Lemma 7.12. With the assumptions and notations of Theorem 7.4: For any v ≥ D5.4 , for any a ≥ 1 and b, r ≥ 0 there exists C ≥ 0 such that if g1 , g2 are two (a, b)-quasi b the terminal points of which are at most 1-apart in X, b and with same geodesics of C(X), b if C is a generalized v-corridor whose vertical boundaries pass through initial point in X, b satisfy gbi ⊂ N r (C) for i = 1, 2 then the endpoints of g1 , if traces gbi ’s of the gi ’s in X b X

dH b (g1 , g2 ) ≤ C. Furthermore, if g1 and g2 do not backtrack then they satisfy the two C(X) conditions required by the BCP with a constant D depending on v, a, b, r. We emphasize that this proposition is false if one only requires a bound on the distance b from the gi ’s to C. in C(X) Proof. For simplicity we assume that C is a corridor, the adaptation to generalized corridors is straightforward. We consider the horizontal quasi projections on C of the maximal b From Lemma 4.12, these projections are (C4.12 , C4.12 )subsets of g1 , g2 which belong to X. quasi geodesics. From Lemmas 7.5, 7.6 on the one hand and Lemma 3.18 on the other hand, there is K, depending on r and C4.6 (C7.5 ), such that the horizontal quasi projections of the exceptional orbits are K-vertical trees, for which there exists a constant L playing the rˆole of the constant t7.3 . It is equivalent to prove the announced properties for 30

the bigon g1 , g2 with respect to the exceptional orbits than to prove them for the above projections on C. If g1 , g2 go through the same exceptional orbits, then their horizontal quasi projections on C satisfy the same property with respect to the horizontal quasi projections of the exceptional orbits. From Lemma 7.7, the “bigon” obtained by projection to the generalized corridor is thin. Moreover the points where the projections of g1 and g2 penetrate a given exceptional orbit are close, because either they are close to the diagonal preceding this exceptional orbit, or they leave a same exceptional orbit: in this last case we are done by the existence of the constant L above (the analog on the corridor of the constant t7.3 ). Let us now assume that g1 enters in an exceptional leaf S but g2 does not. Of course this also holds for the respective projections on C. We then distinguish three cases: First case: 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 Theorem 4.8. It remains before in a bounded horizontal neighborhood of the exceptional orbit, 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 orbit is bounded above by a constant depending on a, b and r. Second case: the exit point of g1 is followed by another exceptional orbit. Thanks to the existence of the constant L and Lemma 7.6, we can follow the same arguments as above, appealing to Proposition 4.9 rather than directly Theorem 4.8. We leave the reader work out details and computations. Third case: the exit point of g1 is followed by a horizontal geodesic with horizontal length bounded above by the constant of the first case. In this case, this horizontal geodesic ends at the vertical boundary of C. The entrance-point of g1 in S is close to a point in g2 . Since g2 is a (a, b)-quasi geodesic and g2 does not pass through S, it cannot happen that the passage of g1 though S is a long passage at small horizontal distance from the considered vertical boundary. Thus, if it is a long passage then there is a stratum which is closest to the entrance-point of g1 in S and where the horizontal distance between S and the considered vertical boundary is smaller than the critical constant. From Proposition 4.9, g2 lies in a bounded neighborhood of S until reaching this stratum. Once again, this gives an upper-bound on the vertical length of S. The proof of Lemma 7.12 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 7.13. With the assumptions of Lemma 7.12: For any v ≥ D5.4 , for any a ≥ 1 and b ≥ 0 there exist C ≥ 1 and D > 0 such that, if x0 , x1 , · · · , xn are consecutive points in some exceptional orbit L, which lie outside the horizontal D-neighborhood of a generalized v-corridor C, and if the vertical distance between the strata of x0 and xn is b with both endpoints greater than C, then no non-backtracking (a, b)-quasi geodesic of C(X) in the horizontal D-neighborhood of C contains the cone over {x0 , xn }. See proof in subsection 9.7. b Proof of Theorem 7.4. Let g, g 0 be two non-backtracking (a, b)-quasi geodesics of C(X) b We assume with same initial point, and with terminal points at most 1-apart in X. b are quasi isometries, the adaptation to the for simplicity that the attaching-maps of X general case is easy. There is a corridor C (in the whole generality only a generalized corridor) the vertical boundaries of which pass through the initial and terminal points of g. 31

Let p be a passage of g (resp. of g 0 ) through the cone over a subset S of an exceptional b From Proposition 7.13, substituting p by orbit outside the D7.13 -neighborhood of C in X. b with S yields a non-backtracking (κ(a, b), κ0 (a, b))-quasi geodesics h (resp. h0 ) of C(X), 0 H κ(a, b) = C7.13 ∗ C7.5 ∗ a and κ (a, b) = C7.13 ∗ C7.5 ∗ (b + 1), such that dC(X) b (g, h) ≤ 1 (resp.

0 0 dH b (g , h ) ≤ 1). We can thus assume that all passages like p have been suppressed in C(X) h and h0 as above. By Proposition 4.10, the subsets of h and h0 between two exceptional orbits are contained in the horizontal C4.10 -neighborhood of a corridor between these orbits. Thus h b From Lemma 7.12, and h0 are contained in the D7.13 + C4.10 -neighborhood of C in X. 0 0 h, h satisfy the BCP. The conclusion for g, g follows. The proof of the hyperbolicity follows the same scheme. If g, g 0 form a (a, b)-quasi b one first substitutes it by a non-backtracking (a, b)-quasi geodesic geodesic bigon of C(X), 0 0 H H 0 bigon g0 , g0 with dC(X) b (g , g0 ) ≤ b. The line of the arguments thereafter b (g, g0 ) ≤ b, dC(X) is the same than above: at the end, Lemma 7.12 gives the thinness of the quasi geodesic bigons instead of the BCP. As in Section 6, the hyperbolicity follows from Lemma 6.3. 

7.1. From Theorems 6.1 and 7.4 to the results of Section 2. Proof of Theorem 2.17. Let (G, He , Hv ) be a graph of weakly relatively hyperbolic groups with the exponential separation property (recall that G = (Γ, {Ge }, {Gv }, {ıe })). For avoiding the reader unnecessary go-and-back in the text, we recall here the construction of what has been called a (G, He , Hv )-relative tree of spaces, which will allow us to appeal to Theorem 6.1. We associate to G a graph of metric spaces (Γ, {Xe }, {Xv }, {e }) which satisfies the following properties: • Each edge- and vertex space of Γ is a PL 2-complex with fundamental group the corresponding edge- or vertex-group of G. • The 1-skeleton of the above 2-complex is the rose whose petals are identified to the generators of this edge- or vertex-group. • The attaching-maps e are continuous PL-maps which induce the corresponding monomorphisms ıe between the fundamental groups of the edge- and vertex-spaces. e π, T ). The Then we consider a geometric realization of Γ and its universal covering (X, fundamental group of G acts isometrically, cocompactly and properly discontinuously on e equipped with the telescopic metric. The 1-skeleton of each stratum of X e is the Cayley X graph of the corresponding vertex- or edge-group. We then consider the space obtained • by putting a cone over each left H-class (H a parabolic subgroup in the edge- or vertex-group) of this Cayley graph, • by adding a “triangular” 2-cell for each pair of elements g, gh in same left H-class, with h a generator of H, the boundary of the 2-cell being a geodesic reading h between g and gh and the two exceptional edges incident to g and gh, • by extending the attaching maps over these coned Cayley graphs in a π(G)equivariant way. By definition of a graph of weakly relatively hyperbolic groups, the tree of spaces that we eventually get is a tree of qi-embedded Gromov hyperbolic spaces. The hallways-flare property for the constructed geometric realization is a straightforward consequence of the exponential separation property of G, of which it is only a reformulation in a slightly different setting. Theorem 6.1 then gives that π(G) is weakly hyperbolic relative to the family composed of all the parabolic subgroups of the vertex-groups and of all the images 32

ıe (He ) of the families of parabolic subgroups of the edge-groups. By definition of a graph of weakly relatively hyperbolic groups, for each parabolic subgroup H of an edge-group Ge , ıe (H) is conjugate to a subgroup of a parabolic subgroup of a vertex-group. It readily follows that one can remove all the subgroups in the subfamily ıe (He ) from the previous family and still get the weak relative hyperbolicity of π(G). Theorem 2.17 follows.  Proof of Theorem 2.10, weak relative hyperbolicity case. With a slight abuse of terminology, we consider Fr as a subgroup of Aut(G, H) generated by the automorphisms αi ’s. The group G o Fr is the fundamental group of the graph of groups which has G as unique vertex group Gv and G as the r edge-groups Gei (the ei ’s are loops with initial and terminal vertex v). The attaching endomorphisms ıei : Gei ,→ Gv are the automorphisms αi : Gei ,→ Gv , are the identity. Since the αi ’s are relative automorphisms whereas the ıe−1 i of (G, H), each one induces a quasi isometry from (Gei , H) to (Gv , H). We so got a graph of weakly relatively hyperbolic groups. Since Fr is a uniform free group of relatively hyperbolic automorphisms, it satisfies the exponential separation property. Hence the weak relative hyperbolicity case of Theorem 2.10 is then a corollary of Theorem 2.17.  Proof of Theorem 2.21. As for the proof of Theorem 2.17 we consider a (G, He , Hv )relative tree of spaces. Item (a) of Theorem 2.21 is the property of exponential separation required by Theorem 2.17. Thus the hallways-flare property is satisfied. The second condition required by the strong hallways flare property in Definition 7.3 is an obvious consequence of Item (b) of Theorem 2.21. Therefore the assumptions required by Theorem 7.4 are satisfied and we get the strong relative hyperbolicity claimed by Theorem 2.21.  Proof of Theorem 2.10, strong relative hyperbolicity case. It suffices to check that the definition of a uniform free group of relatively hyperbolic automorphisms implies the strong hallways-flare property. The exponential separation of the vertical segments is clear but one has to prove that any two exceptional orbits also separate exponentially one from each other. Assume that this is not satisfied. Then, there is M ≥ 0 such that for any N ≥ 1, there is αw ∈ Fr with |w| ≥ N , s.t. there is a geodesic word u in (G, |.|H ) of the form h1 Hi1 h2 · · · Hik hk+1 (where hj stands for a passage of the geodesic in the Cayley graph of G whereas Hij stands for a passage of the geodesic in a left-class for Hij ) satisfying the following properties: (a) |u|H ≤ M , (b) the image under αw of any element with geodesic word HuH 0 has the form rHuH 0 s, where H, H 0 stand for passages through left-classes for the corresponding parabolic subgroups, and where the relative lengths of r and s only depend on the length of w. Here H and H 0 are the parabolic subgroups of G corresponding to the left-classes associated to the two exceptional orbits which violate, for the considered w, the strong exponential separation property. The existence of u above comes from the finiteness of the family H and from the finite generation of G: they imply together that there are only finitely many geodesic words of a given form which have relative length smaller than M . Since G is strongly hyperbolic relative to H, H is almost malnormal in G. This readily implies, by choosing elements in H and H 0 which are sufficiently long enough in (G, |.|S ), that there is an element g of the form HuH 0 .H 0 u−1 H = HuH 0 u−1 H which is not conjugate to an element of a parabolic subgroup. Furthermore g can be chosen not to be a torsion element. From Corollary 4.20 of [29], lim |g n |H = +∞. However αw (g) has the form n→+∞

33

rHuH 0 ss−1 H 0 u−1 Hr−1 = rHuH 0 u−1 Hr−1 . Thus |αw (g n )|H ≤ |g n |H + 2|r|H . Since |r|H is a constant only depending on |w|H , by choosing n sufficiently large enough we get a contradiction with the uniform hyperbolicity of Fr .  Proof of Corollary 2.12. From Theorem 2.10 and Corollary 2.11, we only have to prove that the considered automorphism α of G is hyperbolic relative to the given family of subgroups (indeed these families are quasi convex and almost malnormal - see [17] and [18]). The passage from the surface case to the torsion free one-ended hyperbolic group case is done thanks to the JSJ-decomposition theorems of [4]. We refer the reader to [17] for more precisions and concentrate on the surface case. The fundamental group of S is the fundamental group of a graph of groups G such that: • the edge groups are cyclic subgroups associated to the reduction curves and boundary components, • the vertex groups are the subgroups associated to the pseudo-Anosov components (type I vertices) and to the maximal subsurfaces with no pseudo-Anosov components (type II vertices), • the (outer) automorphism α induced by the homeomorphism preserves the graph of groups structure. We consider the universal covering of G and the associated tree of spaces. We measure the length of a geodesic in this tree of spaces as follows: • we count zero for the passages through the edge-spaces and through the type II vertex-spaces, • we measure the length of the pieces through the type I vertex-spaces by integrating against the stable and unstable measures of the invariant foliations (a boundary-component is considered to belong to both invariant foliations and so the contribution of a path in such a leaf amounts to zero). There is N ≥ 1 such that, when the total stable (resp. unstable) length of a geodesic in a type I-vertex space is two times its unstable (resp. stable) length, then it is dilated by a factor λ > 1 under N iterations of α−1 (resp. of α). In the other cases, we find N ≥ 1 such that the total length is dilated under N iterations of both α and α−1 . Similar computations have been presented in [17]. The conclusion of the relative hyperbolicity of α now comes easily since pieces with positive length, dilated either under αN or under α−N , and pieces with zero length alternate.  Proof of Corollary 2.24. Before stating a first lemma let us recall that if h is a geodesic in a Gromov hyperbolic space then Ph (.) denotes a quasi projection on h. Lemma 7.14. Let G = hSi be a finitely generated group which is strongly hyperbolic relative to a finite family of subgroups H. There exists C > 0 such that if K is a finitely generated subgroup of G satisfying the following properties: • it is almost malnormal relative to H, • it is strongly hyperbolic relative to a (possibly empty) finite family H0 the subgroups of which are conjugated to subgroups in H, • (K, |.|H0 ) is quasi isometrically embedded in (G, |.|H ), and if x, y (resp. z, t) are any two vertices in a same left-class gK (resp. hK) with g 6= h then dΓH (P[z,t] (x), P[z,t] (y)) ≤ C. S (G) Proof. In order to simplify the notations we write dH (., .) for dΓH (., .). Since ΓH S (G) S (G) is hyperbolic, there is a constant δ ≥ 0 such that the geodesic triangles of are δ-thin, 34

and 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 ΓH S (G) is δhyperbolic, 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 ΓS (G) (that is the distance associated to the given finite set of generators S of G) respectively to x0 , y0 , z0 , t0 . 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, ΓH S (G) satisfies the BCP property 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 ΓS (G) with length smaller than c1 (λ, µ, δ): indeed, since x1 , y1 , z1 , t1 were chosen to minimize the distances in ΓS (G) with respect to x0 , y0 , z0 , t0 , 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 7.14 is proved.  From Lemma 7.14, the overlapping of two distinct left Im(α)-classes is bounded above by a constant. Together with the fact that α is a relatively hyperbolic endomorphism, this implies the exponential separation property. Getting the strong version of this property is done as in the proof of the strong relative hyperbolicity case of Theorem 2.10. Corollary 2.24 now follows from Theorem 7.4.  Proof of Corollary 2.25. The fundamental group of M 3 is the fundamental group of a graph of groups G satisfying the following properties: • the vertex groups are of two kinds: there are the subgroups associated to the maximal graph-submanifolds, denoted by Gi , and the subgroups associated to the finite volume hyperbolic 3-submanifolds with cusps, denoted by Hj ; • the edge groups are Z ⊕ Z-subgroups; • two vertex groups of the first kind, Gi and Gj with i 6= j, are not adjacent. The graph G becomes a graph of strongly relatively hyperbolic groups when considering each edge-group and each vertex-group Gi strongly hyperbolic relative to itself, whereas each vertex-group Hj is considered as a group strongly hyperbolic relative to the Z ⊕ Zsubgroups of the cusps [12] (i.e. boundary components). It is in fact a fine graph of strongly relatively hyperbolic groups because the cusp subgroups are malnormal. This last property gives the following stronger assertion: if T is the universal covering of G ˜ T , π), for any v ≥ 0 there is a uniform bound M on then for any G-tree of spaces (X, ˜ T , π) the vertical length of the v-corridors. In other words, if C is a v-corridor in (X, then π(C) has diameter smaller than M in T . It follows from Theorem 7.4 that the fundamental group of G is strongly hyperbolic relative to a family of subgroups as given by Corollary 2.25.  An alternative proof is obtained by combining [12] (any hyperbolic 3-manifold with boundary tori is strongly hyperbolic relative to the boundary subgroups) and the combination theorem of [10]. 35

8. Proof of Proposition 4.9 Conventions: The constants of hyperbolicity and of quasi isometry are chosen sufficiently

large enough to satisfy the conclusions of Lemma 3.19, and also sufficiently large enough so that computations make sense. Moreover the horizontal subsets of the (a, b)-quasi geodesics considered will be assumed to be horizontal geodesics. The hyperbolicity of the strata gives, for any a ≥ 1 and b ≥ 0, a positive constant C(a, b) such that any (a, b)-quasi 0 geodesic g may be substituted by another one g 0 with dH ˜ (g, g ) ≤ C(a, b) and satisfying X this latter property. In the proofs of the various intermediate statements, when referring to a constant provided by an earlier result we will sometimes indicate between parentheses the values of some of the parameters from which it depends. Our first lemma is 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 which satisfies the hallways-flare Lemma 8.1. Let (X, property. For any a ≥ 1, b ≥ 0 and for any v ≥ D5.4 there exist C ≥ 0 and D ≥ 0 such ˜ if [x, y] ⊂ g ∩ Xα satisfies dhor (x, y) ≥ C then that, if g is a (a, b)-quasi geodesic in X, for any T -geodesic ω starting at α with |ω|T ≥ D + nt0 , n ≥ 1, we have dihor (ωx, ωy) ≥ 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-vertical segments s, s0 of vertical length n? t0 , starting at x and y and with π(s) = π(s0 ), then for any T -geodesic ω0 such that ω0 π(s) is a T -geodesic and |ω0 |T = t0 , dihor (ω0 x0 , ω0 y 0 ) ≥ λdhor (x0 , y 0 ) holds. Proof of Claim: Assume the existence of ω with |ω|T = n? t0 such that for some x0 , y 0 with x ∈ ωx0 , y ∈ ωy 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 chain between x and y. But the inequality given at the beginning of the proof tells us that the existence of such a telescopic chain is a 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-vertical segments s, s0 . The claim follows from the exponential separation of the v-vertical 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 ω0 is as in the Claim but with length N? t0 then dihor ([ω0 π(s)]x, [ω0 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, Notations: δ a fixed non negative constant, (X, w ≥ D5.4 and v ≥ C4.6 (w) two constants, λ > 1, M, t0 ≥ 1 the associated constants of hyperbolicity, λ+ , µ the associated constants of quasi isometry. Lemma 8.2. For any a ≥ 1, b ≥ 0, there exists C ≥ 0 such that if C is a generalized w-corridor with exponentially separated v-vertical segments, if g is a v-telescopic chain which is a (a, b)-quasi geodesic of (C, dvtel ), if the endpoints x, y of g both lie in a same stratum Xα , if dhor (x, y) ≥ C then, for any T -geodesic ω starting at α with |ω|T ≥ C+nt0 , n ≥ 1, and ω ∩ π(g) = {α}, we have: 36

dihor (ωx, ωy) ≥ λn dhor (x, y). Proof. Let us observe that, if [p, q] is any horizontal geodesic in g then the v-vertical trees 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 (C8.1 + t0 + µ) then for any ω as given by the current Lemma with |ω|T ≥ D8.1 + t0 , dihor (ωp0 , ωq 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 hallways-flare property, they are exponentially separated after t0 in the direction of ω, 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 a T -geodesic r with |r|T = nt0 and r ∩ ω = {α}. Therefore dhor (p, q) ≥ C8.1 + t0 . Lemma 8.1 then implies that p, q are exponentially separated in the direction of [π(p), π(p0 )] after D8.1 + 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 two v-vertical trees through 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) = D8.1 + nt0 . Let h be the horizontal geodesic in C ∩ Xβ which connects the two v-vertical trees through x and y. Assume that the endpoints of h are exponentially separated after t0 in the direction of [β, α]. Then: λn |ID |hor ≤ |h|hor ≤ λ−n (|ID |hor + |IC |hor )

(1) 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)

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 Thus the number of elements in IC is at least the integer part of 2Cte dhor (x, y) + 1. Furthermore, since g is a v-telescopic chain, the telescopic length of any subset 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 chain 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

37

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 (ωx, ωy) ≥ λdhor (x, y) for any T -geodesic ω with |ω|T = N t0 and [π(x), π(h)] ⊂ ω. Lemma 8.2 is then easily deduced.  As a consequence we have: Corollary 8.3. For any a ≥ 1, b ≥ 0 and d ≥ M , there exists C ≥ d such that if C is a generalized w-corridor with exponentially separated v-vertical segments, if g is any v-telescopic chain which is a (a, b)-quasi geodesic of (C, dvtel ), if x, y are the endpoints of two v-vertical segments s, s0 over a same edge-path in T , with π(s) ∩ π(g) = {α} and such that dihor (s, s0 ) ≤ d, then dhor (x, y) ≤ C. Remark 8.4. At this point, we would like to notice that Lemma 8.2 is similar to Lemma 6.7 of [14]. 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 8.2 is true here, in the generalized corridor, but there the constant λ should have been modified to take into account the so-called “cancellations”. Lemma 8.5. For any r ≥ 0, there exists C ≥ 0 such that if C is a generalized wcorridor with exponentially separated v-vertical segments, if x and y are the endpoints of a r-vertical segment s in C, if the intersection-point z of some v-vertical tree through y in C with the stratum Xπ(x) satisfies dhor (x, z) ≥ C, then for any T -geodesic ω with |ω|T = nt0 , n ≥ 1, and ω ∩ π(s) = {π(x)}, dihor (ωx, ωz) ≥ λn dhor (x, z). Proof. If |s|vert ≤ t0 , the existence of the constants of quasi isometry, Item (a) of Lemma 3.11, and the definition of a r-vertical 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 that dhor (x, z) ≥ C and that x and z are exponentially separated in the direction given by s. If [π(x), π(y)] = ω0 ω 0 with |ω0 |T = t0 , then dihor (ω0 x, ω0 z) ≥ λdhor (x, z). Thanks to the inequality used to define d, one easily concludes that the horizontal distance between s and the vertical tree through y increases along s when going from x to y which of course cannot happen. The conclusion follows from the hallways-flare property.  Proof of Proposition 4.9. We are given a w-corridor C, L the horizontal distance between two points x and y in C, and g a (a, b)-quasi geodesic in (C, dvtel ) from a v-vertical tree through x to a v-vertical tree through y with v ≥ C4.6 (w). We assume that the v-vertical segments in C are exponentially separated. We consider the region R with vertical width C8.3 centered at the stratum Xα with α = π(x). We decompose g in three subsets: the first one, denoted g0 , from the initial point of g until the first point z in g ∩ R, the second one, denoted g1 , from z to the last point t in g ∩ R, the third one, denoted g2 , from t to the terminal point of g. Obviously g1 can be approximated by the concatenation of two vertical segments with a horizontal geodesic in Xα (the approximation constant only depend on L, a and b). We denote by g10 the resulting set. We now consider a maximal chain in g0 which satisfies the following properties: • its endpoints lie in a same stratum Xβ , • its image under π does not intersect [α, β). From Corollary 8.3, the endpoints of such a subchain are at horizontal distance smaller than C8.3 one to each other. Thus, by substituting each such subchain by a horizontal 38

geodesic connecting its endpoints, we construct a C8.3 -vertical segment g00 . We do the same thing for g2 , so obtaining a C8.3 -vertical segment g20 . From Lemma 8.5, g 0 = g00 ∪g10 ∪ g20 lies in a bounded neighborhood of the v-vertical segments connecting its endpoints to 0 x1 and x2 . From the construction, dH tel (g, g ) ≤ aC8.3 + b + 1. The proposition follows.  9. Quasiconvexity of corridors In this section we prove Proposition 4.10, its adaptation to generalized corridors and Proposition 7.13. 9.1. Two basic lemmas. We need first a very general lemma about Gromov hyperbolic spaces. Lemma 9.1. Let (X, d) be a Gromov hyperbolic space. There exists C ≥ 0 such that for any r ≥ C there is D ≥ 0, increasing and affine in r, such that if [x, y] is a diameter of a ball Bx0 (r), if ω is any chain in X with ω ∩ Bx0 (r) = {x, y}, then |ω|d ≥ eD . This lemma is a rewriting of Lemma 1.6 of [9].



˜ be a tree of δ-hyperbolic spaces which satisfies the hallways-flare Lemma 9.2. Let X property. For any v ≥ D5.4 , there exists C ≥ 0 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 T -geodesic ω with |ω|T ≥ C3.19 + nt0 and starting at π(x), when considering the v-vertical segments over ω we have: dihor (ωx, ωy) ≥ λn dhor (x, y) ⇔ dihor (ωz, ωt) ≥ λn dhor (z, t) Proof. If A, B are two subsets of a metric space (X, d), we set ds (A, B) =

sup d(x, y). x∈A,y∈B

Let us consider any T -geodesic ω with |ω|T = t0 starting at α. From Lemma 3.11, dshor (ωx, ωz) ≤ λt+0 (2δ + µ) and dshor (ωy, ωt) ≤ λt+0 (2δ + µ). Assume dihor (ωx, ωy) ≥ λdhor (x, y) but dihor (ωz, ωt) < λdhor (z, t). We take dhor (x, y) ≥ M and dhor (z, t) ≥ M . Assume dshor (ωz, ωt) ≤ 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 (ωz, ωt) > λ1 dhor (z, t) then the lemma follows from the definition of the constant C3.19 , 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. 9.2. Approximation of quasi geodesics with bounded vertical deviation. ˜ T ) a quasi geodesic Lemma 9.3 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. 39

˜ T , π) be a tree of hyperbolic spaces. For any κ, b ≥ 0, a ≥ 1 Lemma 9.3. Let (X, ˜ with and v ≥ D5.4 there exists C ≥ 0 such that if g is any (a, b)-quasi geodesic of X diamT (π(g)) ≤ κ, if C is a generalized v-corridor whose vertical boundaries pass through the endpoints of g then g ⊂ NXC˜ (C). ˜ T , π) a tree 9.3. Stairs. Notations: The sign '1 stands for an equality up to ±1, (X, of hyperbolic spaces which satisfies the hallways-flare property, v ≥ D5.4 a constant. Definition 9.4. Let r ≥ M . A r-stair relative to a generalized v-corridor C is a vtelescopic chain S the maximal vertical segments of which have vertical length greater than C3.19 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 , PChor (ai )), (b) any two 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 )]. See Figure 8.

Figure 8. A stair Lemma 9.5. With the notations of Definition 9.4: there exist C ≥ C9.2 such that for any r ≥ C, if C is a generalized v-corridor, if S is a r-stair relative to C, if U is a generalized v-corridor between a vertical tree through the terminal point of S and a vertical boundary of C, then r+2δ S ⊂ Nhor (U). Proof. Let ai , bi ∈ S as given in Definition 9.4 and let z be a point at the intersection of the stratum Xπ(ai ) with a vertical tree through 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 , PChor (ai )) − K. Proof of Claim 1: Choose K such that eD9.1 (K) > 4δ + 1 and assume dihor ([ai , z], C) < dhor (ai , PChor (ai )) − K. Then Lemma 9.1 implies that [bi , z] descends at least until a 2δneighborhood of ai . Assume r ≥ C9.2 + 2δ. Then Lemma 9.2 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 3.18), we get z 0 at the intersection of the considered vertical tree through z with the stratum Xπ(ai+1 ) such that dihor ([ai+1 , z 0 ], C) < dhor (ai+1 , PChor (ai+1 )) − K. The repetition of these arguments show that the horizontal distance between S and the vertical tree through z does not decrease along S. This is an absurdity since z was chosen in a vertical tree through a point farther in S. The proof of Claim 1 is complete. 40

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 , PChor (bi )) − K(r). Proof of Claim 2: Let z? ∈ [bi , z] with dhor (z? , PChor (z? )) '1 max(dihor ([bi , z], C), dhor (ai , PChor (ai ))). From the δ-hyperbolicity of the strata, [bi , z? ] lies in the horizontal 2δneighborhood of [ai , bi ]. Assume dhor (bi , z? ) ≥ r and is sufficiently large enough to apply Lemma 9.2. Then there is K(r) such that, if z? satisfies dhor (z? , PChor (z? )) < dhor (bi , PChor (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 9.5 is easily deduced from the above two claims, we leave the reader work out the easy details.  Lemma 9.6. For any r ≥ C9.5 there exists C > 0 such that, if C is a generalized v-corridor, if S is a r-stair relative to C which is not contained in the vertical Cneighborhood of the stratum containing its initial point, then the terminal point of S ˜ does not belong to the r-neighborhood of C in X. 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 (2)

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

is an easy consequence of the definition of a stair and of Lemma 9.2 as soon as r ≥ C9.2 . Indeed, the initial segment of horizontal length r in [bi , PChor (bi )] lies in the horizontal 2δ-neighborhood of [bi , ai ]. The assertion then follows from Item (b) of Definition 9.4 and Lemma 9.2. The inequality (2) readily gives the announced result.  9.4. Approximation of a quasi geodesic by a stair. ˜ T ) a tree of δ-hyperbolic spaces which satisfies the hallways-flare propNotations: (X, erty, v ≥ D5.4 . Lemma 9.7. For any a ≥ 1, b ≥ 0 there exists D ≥ 0 such that for any r ≥ D there are C, E ≥ 0, where E is affine in r, such that if C is a generalized v-corridor, if the endpoints of a v-telescopic (a, b)-quasi geodesic g are in a horizontal r-neighborhood of C, if g lies in the closed complement of this horizontal neighborhood and if the maximal vertical segments in g have vertical length greater than 3(C3.19 + D8.1 ) then either g lies in the C-neighborhood of a E-stair relative to C or g is contained in the C-neighborhood of C. 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 9.7 when the horizontal length of any maximal horizontal path in g is greater than some constant (depending on a et b). The endpoints of any horizontal path h in g with horizontal length greater than C8.1 are exponentially separated under every geodesic ω of T with length D8.1 . If |h|hor ≥ C9.2 , 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 3.18 the endpoints of h are also exponentially separated in any v-corridor 41

containing h. If e(a, b) (we do not indicate the dependance on v) 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 vertical segment s. Let D be a corridor containing h1 and s. Then: (3)

2δ |h2 ∩ Nhor (D)|hor ≤ e(a, b).

Otherwise we have a contradiction with the fact that the endpoints of any subgeodesic of h2 whose length is greater than C8.1 are exponentially separated in the direction of h1 . From the inequality (3), 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 , PChor (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, PChor (x)) '1 r). Let s be the vertical 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(C3.19 + D8.1 ) ≤ |s|vert . By assumption x and PChor (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 p, q ∈ [x, PChor (x)] with dhor (p, q) ≥ max( κ1 r, M ) satisfy dhor (π(s)p, π(s)q) ≥ λn dhor (p, q). Thus the same arguments as those exposed above when working with h1 , h2 2δ show that |h ∩ Nhor ([y, PChor (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 take h1 = [x, PChor (x)] and h2 = h: the above arguments prove that the concatenation of h1 , s and h2 is e(v, a, b)-close to a e(a, b)-stair. If n is smaller than n∗ , then we substitute r by n (C +D ) λ+∗ 3.19 8.1 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 trees 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 C9.5 , M, δ and C9.2 . 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: (a) Ci is a corridor with boundary trees a v-vertical tree through xi−1 and the v-vertical boundary L2 of C, (b) xi is the first point following xi−1 along g such that dhor (xi , PChor (xi )) ≥ r. i The chain in 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 4.12, we 4.12 get a D4.12 -telescopic (C4.12 , C4.12 )-quasi geodesic of (Ci , dD tel ). We set X(a, b, r) = hor C4.9 (r, C4.12 , C4.12 ). From Proposition 4.9, PCi (gi−1,i ) is contained in the X(a, b, r)neighborhood of the concatenation of a subpath of [xi−1 , PChor (xi−1 )] with a vertical i−1 hor segment in Ci (and is followed by [PCi (xi ), xi ]). Consider in this approximation of (a subchain 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 42

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 C3.19 + D8.1 . Either we obtain a non-trivial r-stair relative to C which approximates a subchain 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 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 subchain g10 of g starting at (or near - recall that we constructed an approximation of a subchain of g) yk which lies in the r-neighborhood of Ck . This last corridor plays the rˆole of the corridor U of Lemma 9.5. We project the subchain g10 to Ck , so getting a (C4.12 , C4.12 )-quasi geodesic of this corridor. From Lemma 9.5, 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 large enough, κ Lemma 9.2 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 4.9 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.  9.5. Proof of Proposition 4.10. Let g and C be as given by this proposition. Assume that some subchain g 0 of g leaves and then reenters the horizontal D9.7 -neighborhood of C. Assume that g 0 is not contained in the telescopic C9.7 (D9.7 , a, b)-neighborhood of C. We set C9.7 := C9.7 (D9.7 , a, b) and E9.7 := E9.7 (D9.7 , a, b). Suppose for the moment that the vertical segments in g 0 have vertical length greater 0 than 3(C3.19 + D8.1 ). Then Lemma 9.7 gives G, a E9.7 -stair relative to C with dH tel (g , G) ≤ C9.7 . From Lemma 9.6, G does not leave the vertical C9.6 (E9.7 )-neighborhood of the stratum containing the initial point of G. Therefore, by setting V (a, b) = C9.6 (E9.7 )+C9.7 , g 0 does not leave the vertical V (a, b)-neighborhood of this stratum. From Lemma 9.3, g 0 lies in the telescopic C9.3 (V (a, b), a, b)-neighborhood of C. It remains to consider the case where the vertical segments in g 0 are not sufficiently large enough. Let s be a vertical segment in g with |s|vert < X := 3(C3.19 + D8.1 ). (†) Thanks to the assumption that all the attaching-maps of the tree of hyperbolic spaces are quasi isometries, s is contained in a vertical segment s0 of vertical length 43

greater than X. We modify g 0 by sliding, along s0 , a horizontal geodesic in g 0 incident to s until getting a vertical 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, we obtain a quasi geodesic as desired, and we are done. 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 9.7. This completes the proof of Proposition 4.10.  9.6. Adaptation to generalized corridors. The only problem is to get a telescopic chain with maximal vertical segments sufficiently large enough. We start from the sentence marked by a (†) in the preceding subsection. If s is not contained in a vertical segment s0 of vertical length greater than X, we obtain a vertical segment s from bi to ai+1 satisfying the following properties (we still denote by g 0 the (a0 , b0 )-quasi geodesic eventually obtained, we denote by s0 the vertical segment of g 0 ending at ai and by s1 the one starting at bi+1 ): (a) there is no vertical segment starting at ai (resp. at ai+1 ) over the edge π(s) (resp. over π(s1 )); (b) there is no vertical segment ending at bi over π(s0 ). Consider horizontal geodesics αi = [ai , PChor (ai )], βi = [bi , PChor (bi )], αi+1 = [ai+1 , hor PC (ai+1 )] and βi+1 = [bi+1 , PChor (bi+1 )]. By the δ-hyperbolicity of the strata, there 2δ 2δ (αi+1 ∪ βi+1 ). Because the strata (αi ∪ βi ) and b0i ∈ [ai+1 , bi+1 ] ∩ Nhor is a0i ∈ [ai , bi ] ∩ Nhor 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-vertical segment from a00i to b00i which is contained in a larger v-vertical segment going over π(s0 ) and π(s1 ). We modify g 0 by going from ai to a00i then to b00i and eventually end at bi+1 . The resulting chain 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 chain is in a bounded neighborhood of g 0 . Thanks to Item (B), we can modify it by enlarging the vertical segment from a00i to  b00i . The conclusion in then the same as in the preceding subsection. 9.7. Proof of Proposition 7.13. 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 orbit with respect to C depends linearly on the vertical variation of the orbit (Lemma 7.5). Thus, if a sufficiently large segment of the orbit remains outside a sufficiently large horizontal neighborhood of C, the exponential separation implies that the horizontal distance between the orbit and C exponentially increases with the vertical length of the orbit. Assume now that the exceptional orbit considered is followed by another one. The strong exponential separation gives the same consequence: this second exceptional orbit 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 9.5 and 9.6. Finally, if the exceptional orbit is followed by a quasi geodesic b then the approximation by a stair as was done before, yields the same conclusion. in X,  44

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