A Combination Theorem for Metric Bundles

arXiv:0912.2715v2 [math.GT] 27 Apr 2010 A COMBINATION THEOREM FOR METRIC BUNDLES MAHAN MJ AND PRANAB SARDAR Abstract. We define metric bundles which provide a purely topological/coarsegeometric generalization of the notion of Trees of metric spaces a la BestvinaFeighn in the case that the inclusions of the edge spaces into the vertex spaces are uniform (coarsely surjective) quasi-isometries. We prove the existence of q(uasi)-i(sometric) sections in this generality. Then we prove a combination theorem for metric bundles (including exact sequences of groups) that establishes sufficient conditions (particularly flaring) under which the metric bundles are hyperbolic. We also show that in typical situations, flaring is also a necessary condition. Contents 1. Introduction 1.1. Metric Bundles 1.2. Hyperbolic metric spaces 1.3. Trees of hyperbolic and relatively hyperbolic metric spaces 2. QI Sections 2.1. Existence of qi sections 2.2. Nearby qi sections 3. Construction of Hyperbolic Ladders 3.1. Hyperbolicity of ladders: Special case 3.2. Hyperbolicity of ladders: General case 4. The Combination Theorem 4.1. Proof of Main Theorem 5. Consequences and Applications 5.1. Sections, Retracts and Cannon-Thurston maps 5.2. Hyperbolicity of base and flaring 5.3. Necessity of Flaring 5.4. An Example References 1 4 5 7 11 11 15 16 17 22 24 24 26 26 27 29 32 33 1. Introduction In this paper we introduce the notion of metric bundles which provide a purely topological/coarse-geometric generalization of the notion of Trees of metric spaces a la Bestvina-Feighn [BF92] in the case that the inclusions of the edge spaces into the Research of first author partially supported by a CEFIPRA Indo-French Research grant. The second author is partly supported by a CSIR Junior Research Fellowship. This paper is part of PS’s PhD thesis written under the supervision of MM. . 1 2 MAHAN MJ AND PRANAB SARDAR vertex spaces are uniform (coarsely surjective) quasi-isometries. We generalize the base space from a tree to an arbitrary hyperbolic metric space. In [FM02], Farb and Mosher introduced the notion of metric fibrations which was used by Hamenstadt to give a combination theorem in [Ham05] (see also [KL08]). Metric fibrations can be thought of as metric bundles (in our terminology) equipped with a foliation by totally geodesic sections of the base space. We first prove the following Proposition which ensures the existence of q(uasi)-i(sometric) sections in the general context of metric bundles, generalizing (and giving a different proof of) a Lemma of Mosher [Mos96] in the context of exact sequences of groups. Proposition 2.10 : Existence of qi sections:Suppose p : X → B is a metric bundle, with the following properties: ′ (1) Each of the fibers Xb , b ∈ B is a δ -hyperbolic metric space with respect to the induced path metric from X. (2) The barycenter map φb : ∂ 3 Xb → Xb is uniformly coarsely surjective for all b ∈ B, i.e. ∃N > 0 such that Xb is contained in the N -neighbourhood of the image of the map φb for all b ∈ B. ′ Then through each point of X there is a K = K(δ , N )-qi section. Proposition 2.10 provides a context for developing a ‘coarse theory of bundles’ and proving the following combination theorem, which is the main theorem of this paper. Theorem 4.2 Suppose p : X → B is a metric bundle with the following properties: (1) B is a δ-hyperbolic metric space. ′ (2) Each of the fibers Xb , b ∈ B is a δ -hyperbolic metric space with respect to the induced path metric from X and the barycenter maps ∂ 3 Xb → Xb are (uniformly) coarsely surjective. (3) The metric bundle satisfies a flaring condition. Then X is a hyperbolic metric space. Theorem 4.2 generalizes Hamenstadt’s combination theorem [Ham05] and is a first step towards proving a combination Theorem for more general complexes of spaces (cf. Problem 90 of [Kap08], part of the Geometric Group Theory Problem list). A word about the proof ahead of time. Proposition 2.10 ensures the existence of q(uasi)-i(sometric) sections through points of X. We use the notion of flaring from Bestvina-Feighn [BF92] and a criterion for hyperbolicity introduced by Hamenstadt in [Ham07] to construct certain path families and use them to prove hyperbolicity. Another crucial ingredient is a ‘ladder-construction’ due to Mitra [Mit98b], which may be regarded as an analogue of the hallways of Bestvina-Feighn [BF92]. Theorem 4.2 generalizes the main combination Theorem of [Ham05] by a) Removing the hypothesis of properness of the base space B (a hypothesis that is crucial in [Ham05] to ensure compactness of the boundary of the base space and hence allow the arguments in [Ham05] to work). This generalization is relevant for two reasons. First, underlying trees in trees of spaces are frequently non-proper. Secondly, curve complexes of surfaces are mostly non-proper metric spaces and occur as natural base spaces for metric bundles. See [LMS08] by Leininger-MjSchleimer for a closely related example in this context. b) Removing the hypothesis on existence of totally geodesic sections in [Ham05] A COMBINATION THEOREM FOR METRIC BUNDLES 3 altogether. Proposition 2.10 ensures the existence of q(uasi)-i(sometric) sections under mild technical assumptions. Remark 1. In fact, Theorem 4.2, in the special case that the base is a tree, gives a different proof of the Bestvina-Feighn [BF92] result when the inclusions of edge spaces-to-vertex spaces are quasi-isometries. As an application of Theorem 4.2 we obtain a rather plentiful supply of examples from the following, where the base space need not be a tree (as in all previously known examples). Proposition 5.12 Let (K, dK ) be a hyperbolic metric space satisfying the following, There exists C > 0 such that for any two points u, v ∈ K, there exists a bi-infinite C-quasigeodesic γ ⊂ K with dK (u, γ) ≤ C and dK (v, γ) ≤ C. Let j : K → (T eich(S), dT ) be a quasi-isometric embedding such that E ◦ j : K → (T eich(S), de ) is also a quasi-isometric embedding. Let U (S, K) be the pull-back bundle under j of the universal curve over T eich(S) equipped with the natural path metric. Then U^ (S, K) is a hyperbolic metric space. where U^ (S, K) denotes the metric bundle obtained by taking fiberwise universal covers. Theorem 4.2 also provides the following combination Theorem whenever we have an exact sequence with hyperbolic quotient and kernel, giving a converse to Mosher’s theorem [Mos96]. Theorem 5.1 Suppose that the short exact sequence of finitely generated groups 1 → K → G → Q → 1. satisfies a flaring condition and that K, Q are word hyperbolic and K is nonelementary. Then G is hyperbolic. The next Proposition links the flaring condition with hyperbolicity of the base. Proposition 5.6 Consider the short exact sequence of finitely generated groups 1 → K → G → Q → 1. such that K is non-elementary word hyperbolic but Q is not hyperbolic. Then the short exact sequence cannot satisfy a flaring condition. We also prove an analogue of Proposition 5.6 for relatively hyperbolic groups and use it to generalize a Theorem of Mosher [Mos96] as follows. Proposition 5.8 Suppose we have a short exact sequence of finitely generated groups p 1 → (K, K1 ) → (G, NG (K1 )) → (Q, Q1 ) → 1 with K (strongly) hyperbolic relative to the cusp subgroup K1 such that G preserves cusps. Suppose further that G is (strongly) hyperbolic relative to NG (K1 ). Then Q is hyperbolic. Finally we show the necessity of flaring. Proposition 5.10 Let P : X → B be a metric bundle such that 1) X is hyperbolic 2) There exists δ0 such that each fiber Fz = p−1 (z) ⊂ X equipped with the path metric is δ0 -hyperbolic. Then the metric bundle satisfies a flaring condition. In particular, any exact sequence of finitely generated groups 1 → N → G → Q → 1 with N and G hyperbolic, satisfies a flaring condition. 4 MAHAN MJ AND PRANAB SARDAR Precise definitions of terms appearing in Theorem 4.2 are given in the subsection that follows. Acknowledgements We would like to thank Panos Papasoglu for explaining the proof of the last statement of Theorem 5.5 to us. This paper also owes an intellectual debt to Hamenstadt’s paper [Ham05], which inspired us to find a combination Theorem in the generality described here. 1.1. Metric Bundles. Let X, Y be metric spaces and let k ≥ 1, ǫ ≥ 0. Recall [Gro85] [Gd90] that a map φ : X → Y is said to be a (k, ǫ)-quasi-isometric embedding if ∀x1 , x2 ∈ X one has −ǫ + d(x1 , x2 )/k ≤ d(φ(x1 ), φ(x2 )) ≤ ǫ + k.d(x1 , x2 ). A (k, k)-quasi-isometric embedding will simply be referred to as a k-quasi-isometric embedding. A map φ : X → Y is said to be a (k, ǫ)-quasi-isometry ( resp. k-quasi-isometry) if it is a (k, ǫ)-quasi-isometric embedding (resp. k-quasi-isometric embedding) and if there is a constant D > 0 such that ∀ y ∈ Y , ∃ x ∈ X with d(φ(x), y) ≤ D. A k-quasi-geodesic in a metric space X is simply a k-quasi-isometric embedding γ : I → X, where I ⊆ R is an interval. Definition 1.1. Suppose (X, d) and (B, dB ) are geodesic metric spaces. We say that X is a metric bundle over B if there is a continuous surjective map p : X → B such that the following conditions hold: 1) For each point z ∈ B, Xz := p−1 (z) is a geodesic metric space, with the induced path metric dz . The inclusion maps i : (Xz , dz ) → X are uniformly proper embeddings, i.e. there is a function f : R → R such that for all z ∈ B and x, y ∈ Xz we have dz (x, y) ≤ f (d(i(x), i(y))). 2) For z1 , z2 ∈ B and xi ∈ p−1 (zi ) = Xzi , i = 1, 2 we have dB (p(x1 ), p(x2 )) ≤ d(x1 , x2 ). 3) Suppose z1 , z2 ∈ B, d(z1 , z2 ) ≤ 1 and let γ be a geodesic in B joining them. (i) Then for any point x ∈ Xz , z ∈ γ, there is a path in p−1 (γ) of length at most c joining x to both Xz1 and Xz2 . (ii) Let φ : Xz1 → Xz2 , be any map such that ∀x1 ∈ Xz1 there is a path of length at most c in p−1 (γ) joining x1 to φ(x1 ). Then φ is a K-quasi-isometry, for some constant K, independent of γ. The spaces Xz , z ∈ B will be referred to as the horizontal spaces of the metric bundle and distance of two points in Xz will be referred to as their horizontal distance. A geodesic in Xz will be called a horizontal geodesic. Moreover, we shall refer to the constants c, K and the function f as the parameters of the metric bundle. Definition 1.2. Let X be a metric bundle over B and let k ≥ 1. Then X1 ⊆ X is said to be a k- qi section (of B) if there is a k-quasi-isometric embedding s : B → X such that p ◦ s = I and X1 = Im(s). If X1 is a k-qi section and x ∈ X1 then we say that X1 is a k-qi section through x. Let γ : I → B be a geodesic, where I ⊆ R is an interval. By a k- qi lift of γ in X, we mean a k-quasi isometric embedding γ̃ : I → X such that p ◦ γ̃ = γ. Suppose X1 ⊆ X is k- qi section and γ : I → B is a geodesic. By the lift of γ in X1 we mean the k- qi lift of γ whose image is contained in X1 . A COMBINATION THEOREM FOR METRIC BUNDLES 5 Definition 1.3. Suppose p : X → B is a metric bundle. We say that it satisfies a flaring condition if for all k ≥ 1 we can find Mk ≥ 1, λk > 1 and nk > 0 such that the following holds: Let γ : [−nk , nk ] → B be a geodesic and let γ˜1 and γ˜2 be two k-quasi-isometric lifts of γ in X. If dγ(0) (γ˜1 (0), γ˜2 (0)) ≥ Mk . then we have λk .dγ(0) (γ˜1 (0), γ˜2 (0)) ≤ max{dγ(nk ) (γ˜1 (nk ), γ˜2 (nk )), dγ(−nk ) (γ˜1 (−nk ), γ˜2 (−nk ))} Lemma 1.4. Suppose X is a metric bundle over B with parameters as in the Definition 1.1 and b1 , b2 ∈ B with d(b1 , b2 ) ≤ 1. Then there is a function g : R+ → R+ such that for all c1 ≥ c the following holds: Let φ : Xb1 → Xb2 be any map such that ∀x1 ∈ X1 , d(x1 , φ(x1 )) ≤ c1 . Then φ is a g(c1 )-quasi-isometry. Proof: For all c1 ≥ c we can always find such a map by the condition 3 of metric bundles. Moreover, for c1 = c we know that such a map, say φ0 , exists which is a K-quasi-isometry. Let φ1 be a map defined for c1 > c. Then it is clear that for all x ∈ Xb2 , d(φ0 (x), φ1 (x)) ≤ c + c1 . Thus we have db2 (φ0 (x), φ1 (x)) ≤ f (c + c1 ). Thus choosing g(c1 ) to be K + 2.f (c + c1 ) we are through. ✷ As an immediate corollary we have the following result. Corollary 1.5. For all k ≥ 1 there is a function µk : R≥0 → R≥1 such that the following holds: Suppose γ ⊆ B is a geodesic joining b1 , b2 ∈ B, and γ˜1 , γ˜2 are two k-qi lifts of γ in X which join the pairs of points (x1 , x2 ) and (y1 , y2 ) respectively, so that p(xi ) = p(yi ) = bi , i = 1, 2. If dB (b1 , b2 ) ≤ N then db1 (x1 , y1 ) ≤ µk (N ).max{db2 (x2 , y2 ), 1} In the rest of the paper, we will say that metric bundles satisfy bounded flaring condition only to refer to this corollary. 1.2. Hyperbolic metric spaces. We assume that the reader is familiar with the basic facts and definitions about the hyperbolic metric spaces [Gro85], [Gd90], [ABC+ 91]. In this subsection we collect together some of them for later use. Convention: We shall use suffixes for constants to indicate the Lemmas or Propositions where they first appear. Definition 1.6. Suppose ∆x1 x2 x3 ⊂ X is a geodesic triangle, where [xi , xj ] denote the geodesic joining xi , xj and let δ ≥ 0. (1) For all i 6= j 6= k, let ck ∈ [xi , xj ] be such that d(xi , cj ) = d(xi , ck ). The points ci will be called internal points of ∆x1 x2 x3 . Note that, for all i 6= j 6= k, d(xi , cj ) = 21 {d(xi , xj ) + d(xi , xk ) − d(xj , xk )}. (2) Diameter of the set {c1 , c2 , c3 } will be referred to as the insize of the triangle ∆x1 x2 x3 . (3) We say that the triangle ∆x1 x2 x3 is δ-slim if each of the sides is contained in the δ-neighbourhood of the union of the remaining two sides. (4) We say the triangle ∆x1 x2 x3 is δ-thin if for all i 6= j 6= k and p ∈ [xi , cj ], q ∈ [xi , ck ] with d(p, xi ) = d(q, xi ) one has d(p, q) ≤ δ. Definition 1.7. Gromov inner product: Let X be any metric space and let x, y, z ∈ X. Then the Gromov inner product of y, z with respect to x, denoted (y.z)x , is defined to be the number 12 {d(x, y) + d(x, z) − d(y, z)}. 6 MAHAN MJ AND PRANAB SARDAR Definition 1.8. A geodesic metric space X is said to be δ-hyperbolic, for some δ ≥ 0, if all the triangles in X are δ-slim. Lemma 1.9. (See Proposition 2.1,[ABC+ 91]) Suppose X is a δ-hyperbolic metric space. Then the following hold: (1) All the triangles in X have insize at most 4δ. (2) All the triangles in X are 6δ-thin. Definition 1.10. Let X be a geodesic metric space and let A ⊆ X. For K > 0, we say that A is K-quasiconvex in X if any geodesic with end points in A is contained in a K-neighbourhood of A. A subset A ⊂ X is said to be quasi-convex if it is K-quasi-convex for some K. Lemma 1.11. [Gd90] Stability of quasigeodesics: Suppose Y is a δ-hyperbolic metric space. Then for all k ≥ 1, the Hausdorff distance between a geodesic and a k-quasi-geodesic joining the same pair of end points is less than equal to a constant D1.11 depending on δ and k. Definition 1.12. Local quasi-geodesics: Let X be a metric space and K ≥ 1, ǫ ≥ 0, L > 0 be three real numbers. A map f : I → X, where I ⊂ R is an interval, is said to be a (K, ǫ, L)-local quasi-geodesic if for all s, t ∈ I, with |s − t| ≤ L, one has −ǫ + (1/K).|s − t| ≤ d(f (s), f (t)) ≤ ǫ + K.|s − t|. For the following important lemma we refer to(See Theorem 1.4, Chapter 3,[CDA90]; or Theorem 21, Chapter 5, [Gd90]). Lemma 1.13. Local quasi-geodesic vs global quasi-geodesic: Suppose X is a δ-hyperbolic metric space and K ≥ 1, ǫ ≥ 0 be two real numbers. Then there are constants L = L(δ, K, ǫ) and λ = λ(δ, K, ǫ) such that any (K, ǫ, L)-local quasigeodesic in X is a λ-quasi-geodesic. ′ Lemma 1.14. Suppose Y , Y are δ-hyperbolic metric spaces and y1 , y2 , y3 ∈ Y . Then the following hold. 1) There is a point y ∈ Y such that y ∈ N4δ ([yi , yj ]), for any geodesic [yi , yj ] joining yi , yj , i 6= j . We call such a point a barycenter of the triangle △y1 y2 y3 . ′ 2) Suppose φ : Y → Y is a (k, ǫ)-quasi-isometric embedding. If y is a barycentre ′ ′ ′ of △y1 y2 y3 ⊆ Y and y ∈ Y is a barycentre of △φ(y1 )φ(y2 )φ(y3 ) ⊆ Y then we ′ have d(y , φ(y)) ≤ D1.14 for some constant D1.14 depending only on δ, k and ǫ. Proof: By Lemma 1.9 the internal points of △y1 y2 y3 are barycenters of △y1 y2 y3 . This proves the part (1) of the lemma. ′ ′ For (2) let ci be the internal points of △φ(y1 )φ(y2 )φ(y3 ). Suppose z ∈ Y is contained in a D-neighbourhood of each of the sides of △φ(y1 )φ(y2 )φ(y3 ). Let pi ∈ [φ(yj ), φ(yk )], i 6= j 6= k be such that d(pi , z) ≤ D, 1 ≤ i, j, k ≤ 3; then d(pi , pj ) ≤ 2D. Now, a straightforward calculation shows that d(ci , pi ) ≤ 6D and thus d(z, ci ) ≤ 7D for all i, 1 ≤ i ≤ 3. Now, since φ is a (k, ǫ)-q.i. embedding, by Lemma 1.11, it follows that there is a constant D1 depending on k, ǫ, δ such that φ(y) is contained in a D1 -neighbourhood of each of the sides of △φ(y1 )φ(y2 )φ(y3 ). The lemma follows from these observations. ✷ Lemma 1.15. Suppose X is a δ-hyperbolic metric space and V ⊆ U are K-quasiconvex subsets of X. Let x ∈ X and let x1 , x2 be nearest point projections of x on ′ U and V respectively. Then for every point x1 ∈ U the union, of the geodesics, A COMBINATION THEOREM FOR METRIC BUNDLES 7 ′ [x, x1 ] ∪ [x1 , x1 ] is a (3 + 2K)-quasi-geodesic. Moreover, if x3 is a nearest point projection of x1 on V , then there is a constant D1.15 , depending only on δ and K, such that d(x2 , x3 ) ≤ D1.15 . ′ Proof: It easily follows that [x, x1 ] ∪ [x1 , x1 ] is a (3, 2K)-quasi-geodesic, whence it is a (3 + 2K)-quasi-geodesic. Thus, [x, x1 ] ∪ [x1 , x2 ] is a (3 + 2K)-quasi-geodesic. Now, by Lemma 1.11, there is a point x4 ∈ [x, x2 ] with d(x1 , x4 ) ≤ D = D1.11 (δ, 3 + 2K). Similarly, there is ′ ′ a point x3 ∈ [x1 , x2 ] such that d(x3 , x3 ) ≤ D. Using the δ-slimness of the triangle ′′ ′ ′′ x1 x2 x4 , now we find a point x3 ∈ [x2 , x4 ] such that d(x3 , x3 ) ≤ D + δ. Thus it ′′ ′′ follows that d(x3 , x3 ) ≤ 2D + δ. Hence, x2 being a nearest point projection of x3 , ′′ ′′ ′′ we have d(x2 , x3 ) ≤ 2D + δ. Thus d(x2 , x3 ) ≤ d(x2 , x3 ) + d(x3 , x3 ) ≤ 4D + 2δ. This completes the proof of the lemma. ✷ The following Lemma due to Mitra [Mit98b] says that quasi-isometries and nearest point projections ‘almost commute’. Lemma 1.16. (Lemma 3.5 of [Mit98b]) Suppose φ : X → Y is a k-quasi isometric embedding of δ-hyperbolic metric spaces. Let x, y, z ∈ X and let γ be a geodesic in X joining x, y. Let u be a nearest point projection of z on γ. Then there is constant D1.16 = D1.16 (δ, k) such that if v is a nearest point projection of φ(z) onto a geodesic joining φ(x) and φ(y), then d(v, φ(u)) ≤ D1.16 . To prove our theorem, the following characterization of hyperbolicity turns out to be very useful. Lemma 1.17. (Proposition 3.5 of [Ham07]) Suppose X is geodesic metric space and there is a collection of rectifiable curves {c(x, y) : x, y ∈ X}, one for each pair of points x, y ∈ X and constants D1 , D2 ≥ 1 with the following properties: Let x, y, z ∈ X. (1) If d(x, y) ≤ D1 then length of the curve c(x, y) is less than equal to D2 . ′ ′ ′ ′ (2) If x , y ∈ c(x, y) then the Hausdorff distance of c(x , y ) and the segment ′ ′ of c(x, y) between x and y is S bounded by D2 . (3) We have c(x, y) ⊆ ND2 (c(x, z) c(y, z)). Then X is δ1.17 = δ1.17 (D1 , D2 )-hyperbolic and each of the curves c(x, y) is a K1.17 = K1.17 (D1 , D2 )-quasi-geodesic in X. ✷ 1.3. Trees of hyperbolic and relatively hyperbolic metric spaces. We refer to [Far98] for details of definitions and terminology regarding relative hyperbolicity. We refer to Mj-Reeves[MR08] for all the definitions and results of this subsection. Let (X, d) be a path metric space. A collection of closed subsets H = {Hα } of X will be said to be uniformly separated if there exists ǫ > 0 such that d(H1 , H2 ) := inf {d(x1 , x2 ) : xi ∈ Hi , i = 1, 2} ≥ ǫ for all distinct H1 , H2 ∈ H. b Then Farb [Far98] defines the electric space (or coned-off space) E(X, H) (or X for simplicity) corresponding to the pair (X, H) as a metric space which consists of X and a collection of vertices vα (one for each Hα ∈ H) such that each point of Hα is joined to (coned off at) vα by an edge of length 12 . The sets Hα shall be referred to as horosphere-like sets and the vertices vα as cone-points. A slightly modified description is given in [MR08] where we attach a metric product Hα × [0, 1] to X along Hα for each Hα ∈ H and equip Hα × {1} with the zero metric. 8 MAHAN MJ AND PRANAB SARDAR Let i : X → E(X, H) denotes the natural inclusion of spaces. Then for a path γ ⊂ X, the path i(γ) lies in E(X, H). Replacing maximal subsegments [a, b] of i(γ) lying in a particular Hα by a path that goes from a to vα and then from vα to b, and repeating this for every Hα that i(γ) meets we obtain a new path γ̂. If γ̂ is an electric geodesic (resp. P -quasigeodesic), γ is called a relative geodesic (resp. relative P -quasigeodesic). We shall usually be concerned with the case that γ is an ambient geodesic/quasigeodesic without backtracking, i.e. paths which do not return to an Hα after leaving it. Definition 1.18. [Far98] Relative P -quasigeodesics in (X, H) are said to satisfy bounded region penetration if, for any two relative P -quasigeodesics without backtracking β, γ, joining x, y ∈ X, there exists B = B(P ) such that Similar Intersection Patterns 1: if precisely one of {β, γ} meets a horospherelike set Hα , then the length (measured in the intrinsic path-metric on Hα ) from the first (entry) point to the last (exit) point (of the relevant path) is at most B. Similar Intersection Patterns 2: if both {β, γ} meet some Hα then the length (measured in the intrinsic path-metric on Hα ) from the entry point of β to that of γ is at most B; similarly for exit points. X is strongly hyperbolic relative to the collection H if b is hyperbolic a) E(X, H) = X b) Relative quasigeodesics satisfy the bounded region penetration property. Definition 1.19. (Bestvina-Feighn [BF92]) P : X → T is said to be a tree of geodesic metric spaces satisfying the q(uasi) i(sometrically) embedded condition if the geodesic metric space (X, d) admits a map P : X → T onto a simplicial tree T , such that there exist ǫ and K > 0 satisfying the following: 1) For all vertices v ∈ T , Xv = P −1 (v) ⊂ X with the induced path metric dXv is a geodesic metric space Xv . Further, the inclusions iv : Xv → X are uniformly proper, i.e. for all M > 0, v ∈ T and x, y ∈ Xv , there exists N > 0 such that d(iv (x), iv (y)) ≤ M implies dXv (x, y) ≤ N . 2) Let e be an edge of T with initial and final vertices v1 and v2 respectively. Let Xe be the pre-image under P of the mid-point of e. There exist continuous maps fe : Xe ×[0, 1] → X, such that fe |Xe ×(0,1) is an isometry onto the pre-image of the interior of e equipped with the path metric. Further, fe is fiber-preserving, i.e. projection to the second co-ordinate in Xe ×[0, 1] corresponds via fe to projection to the tree P : X → T . 3) fe |Xe ×{0} and fe |Xe ×{1} are (K, ǫ)-quasi-isometric embeddings into Xv1 and Xv2 respectively. fe |Xe ×{0} and fe |Xe ×{1} will occasionally be referred to as fe,v1 and fe,v2 respectively. (This condition ensures that the maps fe,vi do not collapse very large sets, i.e. they are coarsely Lipschitz.) A tree of spaces as in Definition 1.19 above is said to be a tree of hyperbolic metric spaces, if there exists δ > 0 such that Xv , Xe are all δ-hyperbolic for all vertices v and edges e of T . Definition 1.20. A tree P : X → T of geodesic metric spaces is said to be a tree of relatively hyperbolic metric spaces if in addition to the conditions of Definition 1.19, we have the following: 4) Each vertex (or edge) space Xv (or Xe ) is strongly hyperbolic relative to a collection Hv (or He ) A COMBINATION THEOREM FOR METRIC BUNDLES 9 −1 (Hvi ,α ), 5) the maps fe,vi above (i = 1, 2) are strictly type-preserving, i.e. fe,v i i = 1, 2 (for any Hvi ,α ∈ Hvi ) is either empty or some He,β ∈ He . Also, for all He,β ∈ He , there exists v and Hv,α , such that fe,v (He,β ) ⊂ Hv,α ). 6) There exists δ > 0 such that each E(Xv , Hv ) is δ-hyperbolic. 7) Condition (5) above ensures that we obtain an induced tree (the same tree T ) of coned-off, or electric spaces (see below). We demand further that the induced maps of the coned-off edge spaces into the coned-off vertex spaces fd e,vi : E(Xe , He ) → E(Xvi , Hvi ) (i = 1, 2) are uniform quasi-isometries. This is called the qi-preserving electrocution condition dv and de will denote path metrics on Xv and Xe respectively. iv , ie will denote inclusion of Xv , Xe respectively into X. For a tree of relatively hyperbolic spaces, the sets Hvα and Heα will be referred to as horosphere-like vertex sets and edge sets respectively. When all the horosphere like vertex sets and the edge sets are coned off, the resulting tree of coned-off spaces will be called the induced tree of coned-off b spaces and will be denoted as X. b Definition 1.21. The cone locus of the induced tree (T) of coned-off spaces, X, is the graph whose vertex set V consists of horosphere like vertex sets and the edge set E consists of the horosphere like edge sets where the incidence relations is the obvious one. The connected components of the cone-locus will be called a maximal cone-subtree. The collection of maximal cone-subtrees will be denoted by T and elements of T will be denoted as Tα . For each maximal cone-subtree Tα , we define the maximal cone-subtree of horosphere-like spaces to be the tree of metric spaces whose vertex and edge spaces are the horosphere like vertex and edge sets Hvα , Heα , v ∈ V(Tα ), e ∈ E(Tα ), along with the restrictions of the maps fe ’s to Heα × [0, 1]. The collection of Cα ’s will be denoted as C. Definition 1.22. (Bestvina-Feighn [BF92]) A disk f : [−m, m]×I → X is a hallway of length 2m if it satisfies: 1) f −1 (∪Xv : v ∈ T ) = {−m, · · · , m}×I 2) f maps i×I to a geodesic in Xv for some vertex space. 3) f is transverse, relative to condition (1), to ∪e Xe . Definition 1.23. (Bestvina-Feighn [BF92]) A hallway is ρ-thin if d(f (i, t), f (i + 1, t)) ≤ ρ for all i, t. A hallway is λ-hyperbolic if λl(f ({0} × I)) ≤ max{l(f ({−m} × I)), l(f ({m} × I))} A hallway is essential if the edge path in T resulting from projecting X onto T does not backtrack (and is therefore a geodesic segment in the tree T ). Hallway flaring condition: The tree of spaces, X, is said to satisfy the hallways flare condition if there are numbers λ > 1 and m ≥ 1 such that for all ρ there is a constant H(ρ) such that any ρ-thin essential hallway of length 2m and girth at least H is λ-hyperbolic. Definition 1.24. An essential hallway of length 2m is cone-bounded if f (i × ∂I) lies in the cone-locus for i = {−m, · · · , m}. 10 MAHAN MJ AND PRANAB SARDAR The tree of spaces, X, is said to satisfy the cone-bounded hallways strictly flaring condition condition if there are numbers λ > 1 and m ≥ 1 such that any cone-bounded hallway of length 2m is λ-hyperbolic. The following theorem is one of the main results of [MR08]. Theorem 1.25. (Mj-Reeves [MR08]) Let X be a tree (T ) of strongly relatively hyperbolic spaces satisfying (1) the qi-embedded condition (2) the strictly type-preserving condition (3) the qi-preserving electrocution condition (4) the induced tree of coned-off spaces satisfies the hallways flare condition Then X is weakly hyperbolic relative to the family C of maximal cone-subtrees of horosphere-like spaces. Definition 1.26. Partial Electrocution Let (X, H, G, L) be an ordered quadruple such that the following holds: (1) X is (strongly) hyperbolic relative to a collection of subsets Hα , thought of as horospheres (and not horoballs). (2) For each Hα there is a uniform large-scale retraction ga lpha : Hα → Lα to some (uniformly) δ-hyperbolic metric space Lα , i.e. there exist δ, K, ǫ > 0 such that for all Hα there exists a δ-hyperbolic Lα and a map gα : Hα → Lα with dLα (gα (x), gα (y)) ≤ KdHα (x, y) + ǫ for all x, y ∈ Hα . Further, we denote the collection of such gα ’s as G. The partially electrocuted space or partially coned off space corresponding to (X, H, G, L) is obtained from X by gluing in the (metric) mapping cylinders for the maps gα : Hα → Lα . Lemma 1.27. [MR08] (X, dpel ) is a hyperbolic metric space and the sets Lα are uniformly quasiconvex. We end this subsection with a Proposition a special case of which is due to Hamenstadt [Ham05] (where the tree is taken to be Z). We give a different proof below as our proof applies in a more general context. We need the following definition to state the proposition. Definition 1.28. Suppose Y is a δ-hyperbolic metric space and U1 , U2 are two quasi-convex subsets. Let ǫ > 0, D > 0. 1) We say that U1 , U2 are ǫ-separated if inf {d(y1 , y2 ) : yi ∈ Ui , i = 1, 2} ≥ ǫ. 2) We say that U1 , U2 are D-cobounded if any nearest point projection of U1 on U2 has diameter at most D and vice versa. Proposition 1.29. Suppose Y is a tree of δ-hyperbolic metric spaces satisfying the K-q.i. embedding condition such that the images of the edge spaces in the vertex spaces are ǫ-separated and mutually D-cobounded. Then Y is a δ1.29 -hyperbolic metric space and all the vertex spaces and edge spaces are k1.29 -quasi-convex in Y , where the constants δ1.29 , k1.29 depend on δ, K, ǫ, and D. Proof: First of all we note that by [Bow97] the vertex spaces are strongly relatively hyperbolic with respect to the images of edge spaces. Hence Y can be thought of as a tree of relatively hyperbolic metric spaces whose horosphere like edge sets and vertex sets are respectively the whole of the edge spaces and the A COMBINATION THEOREM FOR METRIC BUNDLES 11 image of the edge spaces in the vertex spaces. It is clear that all the conditions of Theorem 1.25 are satisfied in this case. Thus Y is strongly relatively hyperbolic with respect to the edge spaces. Now the edge spaces are uniformly hyperbolic with respect to the induced length metric from Y . Hence, by lemma 1.27, we see that when the maps gα are taken to be identity maps of the edge spaces, the partially electrocuted space is hyperbolic. This space is clearly quasi-isometric to Y . Hence the result. ✷ 2. QI Sections 2.1. Existence of qi sections. Definition 2.1. Sequential Boundary(See Chapter 4,[ABC+ 91]) Let X be a δ hyperbolic metric space. A sequence of points {xn } in X is said to converge to infinity, written xn → ∞, if for some (and hence for any) point p ∈ X, limm,n→∞ (xm .xn )p = ∞. Now we define an equivalence relation on the set of all sequences in X converging to infinity, by setting {xn } ∼ {yn } iff limn→∞ (xn .yn )p = ∞. The set of all equivalence classes {[{xn }] : xn → ∞} will be denoted by ∂X and be referred to as the sequential boundary of X. Suppose {xn } is a sequence of points in X and xn → ∞. We shall write xn → ξ ∈ ∂X to mean that ξ = [{xn }]. Lemma 2.2. Suppose f : X → Y is a (K, ǫ)-quasi isometric embedding of δhyperbolic metric spaces. Then we have an induced map ∂X → ∂Y , denoted by ∂(f ), such that the following hold. (1) If IX : X → X is the identity map then the ∂(IX ) is the identity map on the sequential boundary of X. (2) If f : X → Y and g : Y → Z are two (K, ǫ) quasi isometric embeddings then ∂(g ◦ f ) = ∂(g) ◦ ∂(f ) (3) If f, g : X → Y are two (K, ǫ) quasi isometric embeddings such that supx∈X d(f (x), g(x)) < ∞ then ∂(f ) = ∂(g). (4) If f : X → Y is a quasi-isometry then ∂(f ) : ∂X → ∂Y is a bijection. Proof: Suppose f : X → Y is a (K, ǫ)-quasi-isometric embedding and ξ = [{xn }] ∈ ∂X. Then we claim that f (xn ) → ∞ and setting ∂(f )(ξ) := [{f (xn )}] gives a well defined map ∂(f ) : ∂X → ∂Y . This can be shown as follows. Suppose ′ ′ {xn } ∼ {xn } in X and fix p ∈ X; we need to show that limn→∞ (f (xn ).f (xn ))p = ′ ′ ∞. Let q ∈ [p, xn ], r ∈ [p, xn ] be such that d(p, q) = d(p, r) = (xn .xn )p , and ′ ′ ′ ′ ′ let q ∈ [f (p), f (xn )], r ∈ [f (p), f (xn )] be such that d(f (p), q ) = d(f (p), r ) = ′ ′ (f (xn ).f (xn ))f (p) . By the proof of Lemma 1.14(1), we know that q and q are ′ ′ barycenters of the triangles ∆pxn xn ⊂ X and ∆f (p)f (xn )f (xn ⊂ Y respectively. ′ Thus, we have a constant D1.14 depending only on δ, K, ǫ such that d(f (q), q ) ≤ ′ ′ D1.14 . Hence (f (xn ).f (xn ))f (p) = d(f (p), q ) ≥ d(f (p), f (q)) − D1.14 . Now, using ′ the fact that φ is a (K, ǫ) quasi-isometric embedding, we have f (xn ).f (xn ))f (p) ≥ ′ ′ 1 K (xn , xn )p − ǫ − D1.14 . Thus limn→∞ (f (xn ).f (xn ))f (p) = ∞. The assertions (1), (2), (3) follow immediately from this definition of the induced map on the boundary, and (4) follows from (3) along with the standard ′ fact that for every quasi-isometry f : X → Y there is a quasi-isometry f : Y → X, 12 MAHAN MJ AND PRANAB SARDAR ′ called the quasi-isometric inverse of f , such that supx∈X dX (x, f ◦ f (x)) < ∞ and ′ supy∈Y dY (y, f ◦ f (y)) < ∞. ✷ The following lemma is a straightforward consequence of the stability of quasigeodesics(Lemma 1.11) in hyperbolic metric spaces. Lemma 2.3. Let X be a δ-hyperbolic metric space and let γ : [0, ∞) → X be a (K, ǫ)-quasi geodesic ray. Let {tn } be any sequence of non-negative real numbers tending to ∞; then γ(tn ) → ∞ and the point of ∂X represented by {γ(tn )} is independent of the sequence {tn }. The point of ∂X determined by a quasi-geodesic γ will be denoted by γ(∞). Lemma 2.4. Suppose X is a δ-hyperbolic metric space. Then there is a constant K = K2.4 (δ) such that the following hold: (1) Given any point ξ ∈ ∂X and p ∈ X there is a K-quasi-geodesic ray γ : [0, ∞) → X of X with γ(0) = p and γ(∞) = ξ. (2) Given two points ξ1 6= ξ2 ∈ ∂X there is a K-quasi-geodesic line α : R → X with α(−∞) = ξ1 and α(∞) = ξ2 . Terminology: Any quasi-geodesic ray as in (1) of the above lemma will be referred to as a quasi-geodesic ray joining the points p and ξ. Similarly any quasigeodesic as in (2) of the above lemma will be referred to as a quasi-geodesic line joining the points ξ1 and ξ2 . Proof of Lemma 2.4: (1). We shall inductively construct a sequence of points {pn } and finally show that ∪[pn , pn+1 ] is a uniform quasi-geodesic. Suppose xn → ξ, xn ∈ X, for all n. Fix N ≥ 1 and let p0 = p. Since xn → ∞ we can find a positive integer n1 ∈ N such that (xi .xj )p0 ≥ N for all i, j ≥ n1 . Let [p0 , xn1 ] be a geodesic joining p0 and xn1 . Choose p1 ∈ [p0 , xn1 ] such that d(p0 , p1 ) = N . Now suppose pl has been constructed. To construct pl+1 , let nl+1 ≥ max{nk : 1 ≤ k ≤ l} be an integer such that (xi .xj )pl ≥ (l + 1)N for all i, j ≥ nl+1 . Choose pl+1 ∈ [pl , xnl+1 ] such that d(pl , pl+1 ) = (l + 1)N . Now let αN be the arc length parametrization of the concatenation of the geodesic segments [pi , pi+1 ], i ∈ Z≥0 . Claim: For N > 7δ + 1, αN is a uniform quasi-geodesic. First we show that [pi , pi+1 ] ∪ [pi+1 , pi+2 ] is a uniform quasi-geodesic for each i. Let n > ni+2 . Join pi with xn . Since (xn .xni+1 )pi ≥ (i + 1)N and triangles in X are 6δ-thin by Lemma 1.9(2), we can find a point qi ∈ [pi , xn ] such that d(pi , qi ) = (i + 1)N and d(pi+1 , qi ) ≤ 6δ. Similarly there is a point qi+1 ∈ [pi+1 , xn ] such that d(pi+1 , qi+1 ) = (i + 2)N and d(pi+2 , qi+1 ) ≤ 6δ. Now consider the triangle ∆pi pi+1 xn . We know that the point qi+1 ∈ [pi+1 , xn ] is contained in a δ-neighbourhood of [pi , pi+1 ] ∪ [pi+1 , xn ]. In fact, we intend to show that there is point ri ∈ [qi , xn ] ⊂ [pi , xn ] such that d(ri , qi+1 ) ≤ δ. Suppose ri ∈ [pi , qi ]∪[pi , pi+1 ] then, there is a point si ∈ [pi , pi+1 ] such that d(ri , si ) ≤ 6δ; therefore, we have d(qi+1 , si ) ≤ 7δ. Now, d(pi+1 , qi+1 ) ≤ d(pi+1 , si ) + d(si , qi+1 ) ≤ d(pi , pi+1 ) + 7δ gives (i + 2)N ≤ (i + 1)N + 7δ. This is clearly false, for any integer i ≥ 0, if N > 7δ + 1. Now, it follows that [pi , pi+1 ] ∪ [pi+1 , pi+2 ] is a (1, 42δ)-quasi-geodesic if N > 7δ + 1. Therefore, for N sufficiently large, αN is a uniform quasi-geodesic by Lemma 1.13. Now we show that γ(∞) = ξ. For this, we just need to check that {pn } ∼ {xn }. Again, to show this, it is enough to check that {pk } ∼ {xnk−1 }, i.e. limk→∞ (pk .xnk−1 )p = A COMBINATION THEOREM FOR METRIC BUNDLES 13 k−1 ∞. By the above proof we know that [pk−1 , xnk−1 ] ∪ (∪i=1 [pi−1 , pi ]) is a uniform quasi-geodesic. Thus, by stability of quasi-geodesics (Lemma 1.11), we can find a constant D depending only on δ such that there is a point u ∈ [p, xnk−1 ] with d(pk−1 , u) ≤ D; similarly there is a point v ∈ [p, pk ] such that d(pk−1 , v) ≤ D. Now, we have d(u, v) ≤ 2D and (pk .xnk−1 )p ≥ (u.v)p ≥ d(p, u) − d(u, v) ≥ d(p, pk−1 ) − d(u, pk−1 ) − d(u, v) ≥ d(p, pk−1 ) − 3D. As limk→∞ d(p, pk ) = ∞, we have limk→∞ (pk .xnk−1 )p = ∞. (2). Using the proof of (1) first we construct two uniform quasi-geodesics, (say K1 -quasi-geodesics), γ1 , γ2 joining a point p to ξ1 and ξ2 respectively. Claim : ∃N0 > 0 such that (x.y)p ≤ N0 for all x ∈ γ1 and y ∈ γ2 . Otherwise for all n ∈ N, we can find xn ∈ γ1 , yn ∈ γ2 such that xn → γ1 (∞) = ξ1 and yn → γ2 (∞) = ξ2 and (xn .yn )p ≥ n for all n which contradicts the fact that ξ1 6= ξ2 . ′ ′ Let N1 = sup{(x.y)p : x ∈ γ1 , y ∈ γ2 }. Fix 0 < ǫ < N1 and let x ∈ γ1 , y ∈ γ2 ′ ′ ′ ′ be such that N1 − ǫ < (x .y )p . Let u ∈ [p, x ], v ∈ [p, y ] be such that d(p, u) = ′ ′ ′′ d(p, v) = (x .y )p . By stability of quasi-geodesics (Lemma 1.11) we can find x ∈ ′ ′′ ′ ′′ ′′ [p, x ] ⊂ γ1 and y ∈ [p, y ] ⊂ γ2 such that d(u, x ) ≤ D1.11 (δ, K1 ), and d(v, y ) ≤ ′′ ′′ D1.11 (δ, K1 ). Thus we have d(x , y ) ≤ 2.D1.11 (δ, K1 ). Now it is easy to see ′′ ′′ ′′ ′′ ′′ that [x , ∞) ∪ [x , y ] ∪ [y , ∞) is a uniform quasi-geodesic, where [x , ∞) ⊂ γ1 , ′′ [y , ∞) ⊂ γ2 . ✷ Using stability of quasi-geodesics (Lemma 1.11) the proofs of the following lemma and corollary are standard (see Lemma 1.15, Chapter III.H, [BH99]). Lemma 2.5. Asymptotic rays are uniformly close Suppose X is a δ hyperbolic metric space and γ1 , γ2 : [0, ∞) → X are two asymptotic k-quasi-geodesic rays. Then there is constant D depending only on δ and k such that γ1 (t) ∈ ND (Im(γ2 )) and γ2 (t) ∈ ND (Im(γ1 )) ≤ D for all t ≥ T , for some T ≥ 0. Corollary 2.6. Suppose X is a δ-hyperbolic metric space and let γ1 , γ2 be two K-quasi-geodesic lines in X joining the same pair of points ξ1 , ξ2 ∈ ∂X. Then the Hausdorff distance of γ1 and γ2 is at most D2.6 = D2.6 (δ, K). Lemma 1.11 and Lemma 2.5 combined with the proof of Lemma 1.14, immediately imply the following result. Lemma 2.7. Suppose X is a δ-hyperbolic metric space. Then we have the following: (1) Let ∆ξ1 ξ2 ξ3 be a k-quasi-geodesic ideal triangle in X i.e. a union of three k-quasigeodesic lines in X joining the pairs of points (ξi , ξj ). Let us denote the quasi geodesic lines by [ξi , ξj ]. Then there is a point x ∈ X such that x ∈ ND ([ξi , ξj ]) for each i 6= j, for some constant D = D2.7 (δ, k). ′ ′ (2) Given k ≥ 1 and D ≥ 0 there is a constant L = L2.7 (δ, k, D ) such that ′ ′ if x, x are two points each of which is contained within a D - neighbourhood of each of the sides of the ideal k-quasi-geodesic triangle ∆ξ1 ξ2 ξ3 then ′ d(x, x ) ≤ L. ′ If a point x ∈ X is contained in the D -neighbourhood of each of sides of an ′ ideal quasi-geodesic triangle ∆ξ1 ξ2 ξ3 , then x will be called a D -barycenter. A D2.7 -barycenter will be simply referred to as a barycenter. Now, Corollary 2.6 along with the proof of Lemma 1.14(2) gives the following Lemma. 14 MAHAN MJ AND PRANAB SARDAR Lemma 2.8. Suppose f : X → Y is a K1 -quasi isometric embedding of two δhyperbolic metric spaces and let ∆ξ1 ξ2 ξ3 be a K2 -quasi-geodesic ideal triangle in ′ ′ X. Given D ≥ 0 there is a constant D = D(δ, K1 , K2 , D ) such that if x ∈ X is a ′ D -barycenter of ∆ξ1 ξ2 ξ3 then f (x) ∈ Y is a D-barycenter of the K2 -quasi-geodesic ideal triangle ∆∂(f )(ξ1 )∂(f )(ξ2 )∂(f )(ξ3 ). The barycenter map Suppose X is a δ-hyperbolic metric space such that ∂X has more than three points. Let us denote the set of all distinct triplets of points in ∂X by ∂ 3 X. Now, given ξ = (ξ1 , ξ2 , ξ3 ) ∈ ∂ 3 X by lemma 2.4 we can construct a K2.4 (δ)-quasi-geodesic ideal triangle, say ∆1 , with vertices ξi , i = 1, 2, 3. Then, by lemma 2.7(2) there is a coarsely well defined barycenter of ∆1 . Suppose bξ is a barycenter of ∆1 . Henceforth, we sha11 refer to it simply as a barycenter of the triplet (ξ1 , ξ2 , ξ3 ). For a different set of choices of the K2.4 (δ)-quasi-geodesic lines joining the pairs ′ (ξi , ξj ), suppose we obtain a new ideal triangle ∆2 , and suppose bξ is a barycenter of (ξ1 , ξ2 , ξ3 ) defined with respect to ∆2 . Then by the stability of quasi-geodesic lines ′ (Lemma 2.6), bξ is a D1 := (D2.7 (δ) + D2.6 (δ, K2.4 (δ))-barycenter of the triangle ′ ∆1 . Hence, by lemma 2.7, d(bξ , bξ ) ≤ L2.7 (δ, K2.4 (δ), D1 ). In other words, we have the following result. Lemma 2.9. Suppose X is a δ-hyperbolic metric space. Then there is a constant D2.9 = D2.9 (δ) such that following holds. ′ ′ Let ξ = (ξ1 , ξ2 , ξ3 ) ∈ ∂ 3 X. If bξ and bξ are two barycenters of ξ, then d(bξ , bξ ) ≤ D2.9 We shall say that a map f : (U, dU ) → (V, dV ) satisfying properties P1 , · · · , Pk is coarsely unique if there exists C > 0 such that for any two maps f1 , f2 : (U, dU ) → (V, dV ) satisfying properties P1 , · · · , Pk , and any u ∈ U , dV (f1 (u), f2 (u)) ≤ C. Thus, from Lemma 2.9 we have a coarsely unique map φ : ∂ 3 X → X, ξ 7→ bξ . Any such map will be referred to as the barycenter map. Now we are ready to state the main result of this subsection. Proposition 2.10. Existence of qi sections: Suppose p : X → B is a metric bundle with parameters as in the definition 1.1, with the following properties: ′ (1) Each of the fiber Xb , b ∈ B is a δ -hyperbolic metric space with respect to the induced path metric from X. (2) The barycenter map φb : ∂ 3 Xb → Xb is uniformly coarsely surjective for all b ∈ B, i.e. ∃N > 0 such that Xb is contained in the N -neighbourhood of the image of the map φb for all b ∈ B. ′ Then through each point of X there is a K = K(δ , N )-qi section. Proof : First let us suppose that v ∈ B and x ∈ Xv is contained in the image of the barycenter map. Choose a point ξ = (ξ1 , ξ2 , ξ3 ) ∈ ∂ 3 Xv such that φv (ξ) = x. Now note that given any geodesic γ ⊂ B starting from v, there is a c-qi lift of γ starting from y, for all y ∈ Xv . Thus for any w ∈ B, w 6= v, we have a map fvw : Xv → Xw which is coarsely uniquely determined, by the definition of the metric bundle, and that this map is a quasi-isometry where the parameters of quasi-isometry depend on d(v, w). Let us denote the induced map between the spaces of distinct triplets of points of the boundaries by ∂ 3 fvw . Now we make the following observations: A COMBINATION THEOREM FOR METRIC BUNDLES 15 1. First of all, ∂fvw and hence ∂ 3 fvw does not depend on the on the choices involved in the definition of fvw , by lemma 2.2(3). Thus we have a coarsely unique map s = sξ,x : B → X where s(v) = x, and s(w) = φw ((∂ 3 fvw (ξ))), for all w 6= v. 2. Writing ξw = ∂ 3 fvw (ξ), note that for any w, z ∈ B, ∂ 3 fwz (ξw ) = ξz . This follows from the fact that by definition of the maps fvz , fwz , fvw we have sup{dz (fvz (x), fwz ◦ fvw (x)) : x ∈ Xv } ≤ c.{d(v, z) + d(w, z) + d(v, w) + 3c} Now we may apply lemma 2.2(3). 3. Lastly, if d(w, z) ≤ 1 then d(s(w), s(z)) ≤ C for some constant C. This can be shown as follows. If d(w, z) ≤ 1 then we know that each point of Xw is within a distance c of Xz by the definition of metric bundle. The map fwz is simply sending a point xw ∈ Xw to xz ∈ Xz such that d(xw , xz ) ≤ c. This map is a g(c)- quasi-isometry by Lemma 1.4. Let ξw = (β1 , β2 , β3 ) and ∂ 3 fwz (ξw ) = ξz = (η1 , η2 , η3 ). Choose K2.4 (δ)-quasigeodesic ideal triangles ∆w and ∆z respectively in Xw and Xz , with vertices ξi ’s and ηi ’s, and suppose s(w) and s(z) are D2.7 (δ)barycenters of these triangles. Now the map fwz takes the ideal triangle ∆w into an ideal K1 -quasigeodesic triangles with vertices ηi ’s, where K1 can be chosen to be K1 = K2.4 (δ).g(c) + g(c), and fwz (s(w)) is a D1 := {D2.7 (δ).g(c) + g(c)}barycenter of the new triangle. Thus by Lemma 2.6, fwz (s(w)) is a D2 -barycenter of the triangle ∆z , where D2 = D2.6 (δ, K1 ) + D1 . Hence by Lemma 2.7 we have d(s(z), fwz (s(w))) ≤ L2.7 (δ, K2.4 (δ), D2 ). Since d(s(w), fwz (s(w))) ≤ c we have d(s(w), s(z)) ≤ C := c + L2.7 (δ, K2.4 (δ), D2 ). Now we are ready to show that s is a uniform qi section. For w, z ∈ B we have d(s(w), s(z)) ≥ d(w, z) by the definition of metric bundle. Also from the observations (3) above, we have d(s(w), s(z)) ≤ C.d(w, z) + C. If x ∈ Xv is not in the image of φ, we can choose x1 ∈ Xv such that d(x, x1 ) ≤ N and x1 ∈ Im(φ). Now construct as above a C-q.i. section s = sξ,x1 , and define a ′ ′ ′ new section s by setting s (b) = s(b) for all b ∈ B, b 6= v and s (v) = x. Clearly this is a (N + C)-qi section passing through x. Thus we can take K = N + C to finish the proof the Proposition.✷ As an immediate consequence, we have the following qi-section Theorem due to Mosher [Mos96]. Theorem 2.11. (Mosher [Mos96]) Let us consider the short exact sequence of finitely generated groups 1 → A → G → Q → 1. such that A is non-elementary word hyperbolic. Then there exists a q(uasi)-i(sometric) section σ : Q → G. Hence, if G is hyperbolic, then so is Q. 2.2. Nearby qi sections. Definition 2.12. Suppose X is metric bundle over B. For two sections X1 , X2 and A > 0 we define UA (X1 , X2 ) to be the set {b ∈ B : db (X1 ∩ Xb , X2 ∩ Xb ) ≤ A}. The following lemma is proved in [Ham05] in the context of metric fibrations. The proof is almost the same, nevertheless we include it for the sake of completeness. Suppose X is a metric bundle over B with parameters as in Definition 1.1; suppose X satisfies the flaring condition and the symbols Mk , λk , nk have the same meaning as in Definition 1.3 and µk as in Corollary 1.5. For the sake of the convenience of exposition we will write λ for λc1 , n for nc1 and µ for µc1 for the proof of the lemma below. 16 MAHAN MJ AND PRANAB SARDAR Lemma 2.13. Suppose X1 , X2 are two c1 -q.i. sections of B in X. Let A ≥ Mc1 . (1) Let γ : [t0 , t1 ] → B be a geodesic such that γ(t1 ) ∈ UA but for all t ∈ [t0 , t1 ), γ(t) 6∈ UA . Then the length of γ is bounded by a constant D2.13 depending on c1 and the ratio dγ(t0 ) (X1 ∩ Xγ(t0 ) , X2 ∩ Xγ(t0 ) )/A. (2) UA is K2.13 -quasi-convex in B, for some constant K2.13 depending on c1 but independent of A. Proof: (1) Let t1 − t0 = n.L + ǫ where L ∈ Z≥0 and 0 ≤ ǫ < n. Without loss of generality we may assume that L ≥ 3. Let F : [t0 , t1 ] → R be the function t 7→ dγ(t) (X1 ∩Xγ(t) , X2 ∩Xγ(t) ). Now consider the sequence of numbers F (t0 +n.i), i = 1, · · · , L. Since F (t0 + n.i) ≥ Mc1 , for all i ∈ [1, L − 1] we have by the flaring condition, λ.F (t0 + n.i) ≤ max{F (t0 + n.(i − 1)), F (t0 + n.(i + 1))} Suppose F (t0 +n) > F (t0 ) then F (t0 +n.(i+1)) ≥ λ.F (t0 +n.i) for all i ∈ [1, L−1]. Thus it follows that F (t0 + n.L)) ≥ λL−1 .F (t0 ). Now using boundedness of the flaring condition (Corollary 1.5) we have, F (t0 + n.L) ≤ µc1 (n). max{F (t1 ), 1}. Putting all these together and using the fact that F (t1 ) ≤ A and F (t0 ) > A, we have L − 1 < logµ(n)/logλ. Since L is an integer, we have L ≤ logµ(n)/logλ. Hence, F (t0 + n) > F (t0 ) implies l(γ) ≤ max{3n, n(L + 1)} ≤ max{3n, n + n.logµ(n)/logλ} Now, suppose F (t0 ) ≥ F (t0 + n) and let k ≤ L be the largest integer such that F (t0 ) ≥ F (t0 + n) ≥ · · · ≥ F (t0 + k.n). Applying the flaring condition we get F (t0 + (i − 1).n) ≥ λ.F (t0 + i.n) for all i ∈ [1, k − 1]. Thus F (t0 ) ≥ λk−1 .F (t0 + (k − 1).n) > λk−1 .A. Hence k ≤ {logF (t0 ) − logA}/logλ. Again by the first part of the proof we know that l(γ|[t0 +k.n,t1 ] ) ≤ max{3n, n + n.logµ(n)/logλ}. Hence, in any case, we have l(γ) ≤ n.{logF (t0 ) − logA}/logλ + max{3n, n + n.logµ(n)/logλ} Therefore, choosing D2.13 = D2.13 (c1 , F (t0 )/A) to be the right hand side of the above inequality we are done. (2) Suppose γ : [t0 , t1 ] → B is a geodesic joining two points of UA , such that for all t ∈ (t0 , t1 ), γ(t) 6∈ UA . Without loss of generality, we may assume that t1 − t0 > n. Let t2 = t0 + n. Then by the boundedness of the flaring condition we have F (t2 ) ≤ µ(n).F (t0 ) ≤ µ(n).A. Again by the first part of the lemma l(γ|[t2 ,t1 ] ) ≤ D2.13 (c1 , F (t2 )/A). Thus l(γ) ≤ n + D2.13 (c1 , µ(n)). Hence, taking K2.13 (c1 ) = n + D2.13 (c1 , µ(n)) completes the proof of the lemma.✷ 3. Construction of Hyperbolic Ladders Notations and conventions: For the rest of the paper we fix the following notations. For us p : X → B will be a fixed metric bundle. The symbols f, c; λk , Mk will bear the same meaning as in the definitions 1.1 and 1.3, and g, µk will be as in Lemma 1.4 and Corollary 1.5 respectively. We shall assume that B is δ′ hyperbolic and each of the horizontal space Xb , b ∈ B if δ -hyperbolic. We assume that the barycenter maps ∂ 3 Xb → Xb are (uniformly) coarsely surjective. Thus by the Proposition 2.10 we know that the metric bundle admits uniform qi sections through each point of X. We shall assume that through each point of X there is a A COMBINATION THEOREM FOR METRIC BUNDLES 17 K0 qi section. Lastly, often the dependence on these constants and functions will not be explicitly stated if it is clear from context. We use the term ladder below due to a similar ladder construction of Mitra in [Mit98b]. The term girth is taken from Bestvina-Feighn[BF92]. Definition 3.1. Suppose X1 and X2 are two c1 -q.i. sections of B in X. For each b ∈ B, join the points X1 ∩ Xb , X2 ∩ Xb by a geodesic in Xb . We denote the union of these geodesics by C(X1 , X2 ), and call it a ladder formed by the sections X1 and X2 . Since the Hausdorff distance between any pair of ladders determined by the sections X1 , X2 is uniformly bounded, in general, C(X1 , X2 ) will refer to any one of them, and abusing notation we refer to it as the ladder determined by X1 , X2 . This is justified since we are concerned only about the large scale properties of the spaces. Definition 3.2. Suppose X1 and X2 are two c1 -q.i. sections of B in X. We define dh (X1 , X2 ) to be the quantity inf{db (Xb ∩ X1 , Xb ∩ X2 ) : b ∈ B} and call it the girth of the ladder C(X1 , X2 ). Lemma 3.3. Suppose X1 and X2 are two c1 -q.i. sections of B in X. Then through each point x ∈ C(X1 , X2 ) there is C3.3 = C3.3 (c1 )-q.i. section contained in C(X1 , X2 ). ′ Proof: We already know that there is a K0 -q.i. section, say X , through x in ′′ ′′ X. Now define a new section X as follows: X ∩ Xb is a nearest point projection ′ of X ∩ Xb into the horizontal geodesic C(X1 , X2 ) ∩ Xb . This defines a set theoretic section. We need to check that this is indeed a q.i. section. For this it is enough ′′ to check that ∀ b1 , b2 ∈ B, with d(b1 , b2 ) ≤ 1, the distance between Xb1 ∩ X and ′′ Xb1 ∩ X is small. This follows immediately from lemma 1.4, and lemma 1.16. In ′ ′ ′ ′ fact, we can take C3.3 = c + D1.16 (δ , g(c )), where c = 2.max{K0 , c1 }. ✷ Notation: We fix the following conventions and notation to be followed in the rest i i is the i-th (c1 ), i = 1, 2, 3 where C3.3 of this section. Fix c1 ≥ K0 . Let ci+1 = C3.3 ′ iterate of the function C3.3 . Note that if X is a k-q.i. section, k ≤ c4 , then it is also a c4 -q.i. section, since the map p : X → B is distance decreasing. We shall write M for Mc4 so that, for any geodesic γ : [−nc4 , nc4 ] → B, and any two c4 -qi lifts of γ, with initial horizontal distance at least M , flares exponentially. We shall denote the function µc4 simply by µ. 3.1. Hyperbolicity of ladders: Special case. This subsection is devoted to proving the hyperbolicity of small girth ladders. Let X1 , X2 be two c1 -q.i. sections in X and let A0 > 0 be such that dh (X1 , X2 ) ≤ A0 and UA0 (X1 , X2 ) 6= ∅. Then we have the following Proposition 3.4. (1) C(X1 , X2 ) is δ3.4 -hyperbolic with the induced path metric from X, and X1 ,X2 are K3.4 -quasi-convex in C(X1 , X2 ), where the constants δ3.4 and K3.4 depend on c1 , A0 . (2)If dh (X1 , X2 ) ≥ Mc1 , then there is a constant D = D3.4 depending on c1 , A0 , such that X1 , X2 are D-cobounded in C(X1 , X2 ). The proof of this proposition is rather long. Therefore, we shall break it up into several lemmas. The idea of the proof is that we define a set of curves c(x, y) for 18 MAHAN MJ AND PRANAB SARDAR each pair of points x, y ∈ C(X1 , X2 ) and check that they satisfy the three properties of lemma 1.17. ′ ′′ Let A = max{A0 , M }. In the course of the proof whenever X , X are two c4 -q.i. ′ ′′ ′ ′′ sections, we write U (X , X ) for UA (X , X ). Definition of curve family: Let x, y ∈ C(X1 , X2 ). By Lemma 3.3 we can choose two c2 -q.i. sections X3 and X4 through x and y respectively, in C(X1 , X2 ). By lemma 2.13, U (X3 , X4 ) ⊆ B is a K2.13 (c2 )-quasi-convex subset. Join p(x) to U (X3 , X4 ) by a shortest geodesic γx,y in B; let its end point be bx,y ; let γ̃x,y be the lift of γx,y in X3 , with end point sx,y . Let tx,y be the lift of bx,y in X4 . We note that dbx,y (tx,y , sx,y ) ≤ A. Now let βx,y be a geodesic in B joining p(y) and bx,y , and let β̃x,y be the lift of βx,y in X4 . We define c(x, y) to be the union of the three curve segments: γ̃x,y , β̃x,y and the segment of Xbx,y ∩ C(X1 , X2 ) between tx,y and sx,y . We see that there is an asymmetry in the definition of c(x, y) and a number of choices are involved. However, for each unordered pair {x, y} make the choices once for all and choose either c(x, y) or c(y, x) as the curve joining the points x, y. Let us denote this set of curves by S. (See figure below.) Figure 1:Path families: Special case Lemma 3.5. Suppose x, y ∈ C(X1 , X2 ) with d(x, y) = D1 > 0. Then the length of the curve c(x, y) ∈ S is at most D3.5 , for a constant D3.5 depending on A0 , c1 and D1 . Proof: Let ỹ be the lift of p(x) in X4 . By criterion 2 of metric bundles we have d(p(y), p(ỹ)) ≤ D1 . Thus we have d(y, ỹ) ≤ c2 .D1 + c2 , so that d(ỹ, x) ≤ c2 .D1 + D1 + c2 . Let b = p(x). Then db (ỹ, x) ≤ f (c2 .D1 + D1 + c2 ), by criterion 1 of metric bundles. Now by lemma 2.13, we have ′ D1 := d(p(x), bx,y ) ≤ D2.13 (c2 , ′ f (c2 .D1 + D1 + c2 ) ) A Thus d(p(y), bx,y ) ≤ D1 + D1 . Now it follows that the lengths of the curves γ̃x,y ′ and β̃x,y are bounded in terms of c2 , D1 , and D1 . This proves the lemma. ✷ We next show that the curve family is coarsely well-defined, i.e. ambiguities in the definition of the curves can be ignored. More precisely, the curves corresponding A COMBINATION THEOREM FOR METRIC BUNDLES 19 to the different choices, joining the same pair of points, are at a uniformly bounded Hausdorff distance. ′ Suppose X3 , X3 are two k-qi sections in C(X1 , X2 ) containing x and X4 is a k-qi section containing y, where k ≤ c4 . Consider the two curves joining x, y ′ using X3 , X4 and X3 , X4 respectively, defined in the same manner as in case of ′ c(x, y). We shall denote these two curves by c(x, y) and c (x, y) respectively. Let ′ ′ ′ V := U (X1 , X2 ), W := U (X3 , X4 ) and W := U (X3 , X4 ). Then V ⊂ W , V ⊂ W . ′ Join p(x) to V by a shortest geodesic γ in B and let γ̃, γ̃ be the lifts of γ in X3 ′ ′ and X3 respectively. Similarly join p(x) to W, W respectively by shortest geodesics ′ ′ ′ γx,y and γx,y and call their lifts, respectively in X3 and X3 , γ̃x,y and γ̃x,y . Let sx,y , ′ ′ ′ sx,y be the end points of γ̃x,y , γ̃x,y respectively, and let bx,y , bx,y be the end points ′ of γx,y and γx,y . Now, first of all, we observe the following. ′ Lemma 3.6. d(bx,y , bx,y ) is bounded by a constant D3.6 = D3.6 (k). ′ ′ Proof: Since V ⊂ W ∩ W and V, W, W are K2.13 (k)-convex subsets of B, by ′ joining each of bx,y , bx,y with the end point of γ we get two (3, 2K2.13(k))-quasigeodesics connecting the initial and the end points of γ. In particular these are (3 + 2K2.13 (k))-quasi-geodesics. Since B is hyperbolic, by lemma 1.11 we can find ′ ′ ′ b, b ∈ γ, such that d(bx,y , b) ≤ D3 , d(bx,y , b ) ≤ D3 , where D3 := D1.11 (δ, 3 + ′ ′ 2K2.13(k)). If b ∈ [p(x), b ] ⊂ γ then clearly bx,y ∈ N2.D3 +δ (γx,y ). Otherwise, ′ ′ b ∈ [p(x), b], so that bx,y ∈ N2.D3 +δ (γx,y ). So without loss of generality, let us ′ assume that b ∈ [p(x), b ]. ′ Since the end points of γ are in U (X3 , X3 ) which is a K2.13 (k)-quasi-convex set in B by Lemma 2.13, by the bounded flaring condition we know that for all point b2 ∈ γ, ′ db2 (X3 ∩ Xb2 , X3 ∩ Xb2 ) ≤ A.µ(K2.13 (k)). In particular, one has ′ ⇒ db (X3 ∩ Xb , X3 ∩ Xb ) ′ db (X3 ∩ Xb , X4 ∩ Xb ) ≤ A.µ(K2.13 (k)) ′ ≤ db (X3 ∩ Xb , X3 ∩ Xb ) + db (X3 ∩ Xb , X4 ∩ Xb ) ≤ A.µ(K2.13 (k)) + A.µ(D3 ) ′ ′ ′ ′ ′ Now, we know [p(x), b ] ⊂ Nδ+D3 (γx,y ). Let b1 ∈ γx,y be such that d(b, b1 ) ≤ δ+D3 . Then ′ ′ db′ (X3 ∩ Xb′ , X4 ∩ Xb′ ) ≤ µ(δ + D3 ).max{db (X4 ∩ Xb1 , X3 ∩ Xb1 ), 1} 1 1 1 ′ ⇒ db′ (X3 ∩ Xb′ , X4 ∩ Xb′ ) ≤ A.µ(δ + D3 ){µ(D3 ) + µ(K2.13 (k))} 1 1 1 ′ Denoting the right hand side of the inequality by D , we have, by Lemma 2.13 that ′ ′ ′ d(b1 , bx,y ) ≤ D2.13 (k, D /A) ′ ′ ′ ′ Since, d(bx,y , bx,y ) ≤ d(bx,y , b) + d(b, b1 ) + d(b1 , bx,y ), finally, we have ′ ′ ′ d(bx,y , bx,y ) ≤ D3 + (δ + D3 ) + D2.13 (k, D /A) = δ + 2D3 + D2.13 (k, D /A) The lemma is now proved. ✷ ′ Lemma 3.7. The Hausdorff distance of c(x, y) and c (x, y) is bounded by a constant D3.7 depending on A and k. 20 MAHAN MJ AND PRANAB SARDAR ′ Proof: Step 1: By the above lemma we have d(bx,y , bx,y ) ≤ D3.6 (k). Hence, ′ by the δ-hyperbolicity of B, we have Hd(βx,y , βx,y ) ≤ δ + D3.6 (k). Since X4 is a k-q.i. section, we have ′ Hd(β̃x,y , β̃x,y ) ≤ k + k.(δ + D3.6 (k)) Step 2: Similarly, ′ ′ Hd([sx,y , tx,y ], [sx,y , tx,y ]) ≤ A + k + k.D3.6 (k) ′ ′ where [sx,y , tx,y ], [sx,y , tx,y ] are the horizontal geodesic segments of c(x, y) and ′ c (x, y) respectively, each of length at most A. ′ ′′ Step 3: Now we calculate the Hausdorff distance of γ̃x,y and γ̃x,y . Let γ̃x,y be ′ ′ ′′ the lift of γx,y in X3 . Then, as in the step 1, we have Hd(γ̃x,y , γ̃x,y ) ≤ k + k.(δ + ′ D3.6 (k)). Since γx,y joins two points p(x) and bx,y of U (X3 , X3 ), by Lemma 2.13 ′ and the bounded flaring condition, for all point b2 ∈ γ, db2 (X3 ∩ Xb2 , X3 ∩ Xb2 ) ≤ A.µ(K2.13 (k)). Since there is a point b ∈ γ such that d(b, bx,y ) ≤ D3 , using the boundedness of the flaring condition, we have ′ dbx,y (X3 ∩ Xb2 , X3 ∩ Xb2 ) ≤ ≤ ′ µ(D3 ).max{db (X3 ∩ Xb , X3 ∩ Xb ), 1} A.µ(D3 ).µ(K2.13 (k)) ′ Let A1 = A.µ(D3 ).µ(K2.13 (k)). Now we note that γx,y joins two points of UA1 (X3 , X3 ). Therefore, by lemma 2.13 and the bounded flaring condition, we have ∀b1 ∈ γx,y , ′′ ′ db1 (X3 ∩Xb1 , X3 ∩Xb1 ) ≤ µ(K2.13 (k)).A1 . Hence Hd(γ̃x,y , γ̃x,y ) ≤ µ(K2.13 (k)).A1 . Now we have ′ Hd(γ̃x,y , γ̃x,y ) ′′ ′ ′′ ≤ Hd(γ̃x,y , γ̃x,y ) + Hd(γ̃x,y , γ̃x,y ) ≤ µ(K2.13 (k)).A1 + k.(δ + D3.6 (k)) + k Finally, since ′ Hd(c(x, y), c (x, y)) ′ ′ ′ ′ ≤ max{Hd(β̃x,y , β̃x,y ), Hd([sx,y , tx,y ], [sx,y , tx,y ]), Hd(γ̃x,y , γ̃x,y )} the proof of the lemma follows. ✷ Lemma 3.8. Hd(c(x, y), c(y, x)) is bounded by a constant D3.8 depending on k and A, where c(x, y), c(y, x) are defined using two k-q.i sections X3 , X4 such that x ∈ X3 and y ∈ X4 . Proof: Let α be a geodesic in B joining bx,y and by,x . Since α joins two points of U (X3 , X4 ), by Lemma 2.13 and the bounded flaring condition we have ∀b ∈ α, db (Xb ∩ X3 , Xb ∩ X4 ) ≤ µ(K2.13 (k)).A Also note that since U (X3 , X4 ) is a K2.13 (k)-quasi-convex set in B, γx,y ∪ α is a (3+2K2.13(k))-quasi-geodesic by Lemma 1.15. Thus the Hd(γx,y ∪α, [p(x), by,x ]) ≤ D1.11 (δ, 3 + 2K2.13(k)), by lemma 1.11, where [p(x), by,x ] is any geodesic in B joining the points p(x) and by,x . Similarly we have Hd(γy,x ∪ α, [p(y), bx,y ]) ≤ D1.11 (δ, 3 + 2K2.13(k)). Now it follows that Hd(c(x, y), c(y, x)) ≤ µ(K2.13(k)).A + 2k + (δ + D1.11 (δ, 3 + 2K2.13(k))).k ✷ A COMBINATION THEOREM FOR METRIC BUNDLES 21 Corollary 3.9. Let x, y ∈ C(X1 , X2 ) and k ≤ c4 . Then the Hausdorff distance between any two curves joining x, y defined in the same way as that of c(x, y) using k-qi sections passing through x, y, is at most D3.9 for some constant D3.9 depending on k and A0 . Proof: Choosing D3.9 = 2.D3.7 + 3.D3.8 we are done. ✷ Lemma 3.10. Suppose k ≤ c4 , and X3 , X4 , X5 are k-q.i. sections in C(X1 , X2 ) such that z ∈ X5 , y ∈ X4 ⊂ C(X1 , X5 ) and x ∈ X3 ⊂ C(X1 , X4 ). Then the triangle formed by the curves c(x, y), c(y, z), c(x, z), defined using the pairs X3 , X4 ; X4 , X5 and X3 , X5 respectively, is D3.10 = D3.10 (k, A)-slim. Proof: We have U (X3 , X5 ) ⊂ U (X4 , X5 ) ∩ U (X3 , X4 ) and we know, by lemma 2.13, that all of these three sets are K2.13 (k)-quasi-convex in B. Case 1: Suppose x, y are in the same horizontal space and dp(x) (x, y) ≤ A. Thus we have p(x) ∈ U (X3 , X4 ). Since γx,z ends in U (X3 , X5 ) ⊂ U (X3 , X4 ), it joins two points of U (X3 , X4 ). Hence, by Lemma 2.13 and Corollary 1.5, we have for all ′ b ∈ γx,z , db′ (X3 ∩ Xb′ , X4 ∩ Xb′ ) ≤ A.µ(K2.13 (k)). Now we show that d(bx,z , by,z ) is small. Recall that bx,z ∈ U (X3 , X5 ) ⊂ U (X4 , X5 ) and by,z ∈ U (X4 , X5 ). Thus γy,z ∪ [by,z , bx,z ] is a (3 + 2K2.13 (k))-quasi-geodesic in B, by Lemma 1.15. Hence, there is a point b2 ∈ γx,z , such that d(by,z , b2 ) ≤ D1.11 (δ, 3 + 2K2.13 (k)), by lemma 1.11. Since dby,z (Xby,z ∩ X4 , Xby,z ∩ X5 ) ≤ A, we have by the boundedness of the flaring condition, db2 (Xb2 ∩ X4 , Xb2 ∩ X5 ) ≤ A.µ(D1.11 (δ, 3 + 2K2.13(k))). Therefore, db2 (Xb2 ∩ X3 , Xb2 ∩ X5 ) = db2 (Xb2 ∩ X3 , Xb2 ∩ X4 ) + db2 (Xb2 ∩ X4 , Xb2 ∩ X5 ) ≤ A.{µ(K2.13 (k)) + µ(D1.11 (δ, 3 + 2K2.13(k)))}. Now, by the lemma 2.13 we have d(b2 , bx,z ) ≤ D2.13(k, µ(K2.13 (k)) + µ(D1.11 (δ, 3 + 2K2.13(k)))) Hence d(bx,z , by,z ) ≤ D1.11 (δ, 3 + 2K2.13 (k)) + D2.13 (k, µ(K2.13 (k)) + µ(D1.11 (δ, 3))) Now it follows by arguments similar to that of lemma 3.7 that Hd(c(x, z), c(y, z)) ≤ A.µ(K2.13 (k)) + k + k.(δ + d(bx,z , by,z )) ′ Let us denote the expression on the right hand side by D3.10 . Therefore, the proof ′ of the lemma follows, in this case, if we choose D3.10 = A + D3.10 . Case 2: Now, we consider the general case. For the rest of the proof we shall assume that all the curves of the form c(u, v), where u, v are points in X3 ∪X4 ∪X5 , are defined using these sections only, unless otherwise specified. Now, we first show that Hd(c(x, z), γ̃x,y ∪ c(sx,y , z)) is bounded by a constant depending on k and A. Let b̄ be a nearest point projection of bx,y on U (X3 , X5 ). Then we know by lemma 1.15 that d(b̄, bx,z ) ≤ D1.15 = D1.15 (δ, K2.13(k)). Let γ2 be a geodesic joining bx,y to b̄ and let γ˜2 be a lift of γ2 in X3 . Note that γx,y ∪ γ2 is a (3 + 2K2.13 (k))quasi geodesic in B. Thus the Hausdorff distance of γx,z and γx,y ∪ γ2 is at most δ + D1.15 + D1.11 (δ, 3 + 2K2.13 (k)). Hence, clearly the Hausdorff distance of γ̃x,y ∪c(sx,y , z) and c(x, z) is at most k+k{δ+D1.15 +D1.11 (δ, 3+2K2.13(k))}+A = D1 , say. ′ Again by case 1, we know that Hd(c(sx,y , z), c(tx,y , z)) ≤ D3.10 . Thus Hd(c(x, z), γ̃x,y ∪ ′ [sx,y , tx,y ] ∪ c(tx,y , z)) is at most A + D1 + D3.10 . Also, if we define the curves c(z, tx,y ), c(z, y) with respect to the sections X4 , X5 taking γz,tx,y = γz,y , clearly 22 MAHAN MJ AND PRANAB SARDAR the triangle formed by the curves c(z, tx,y ), c(z, y) and β̃y,tx,y is (k +k.δ)-slim. Thus by corollary 3.9, the triangle formed by the curves β̃y,tx,y , c(tx,y , z) and c(y, z) is D2 ′ slim where D2 = k+k.δ+2.D3.9 . Thus we can take D3.10 to be A+D1 +D3.10 +D2 , in any case. ✷ Proof of Proposition 3.4 We verify that the set S of curves defined earlier in this section satisfies the properties of Lemma 1.17, with D1 = c1 and D2 = max{D3.5 , D3.10 (c4 , A) + 2.D3.9 (c4 )}. Proof of property 1: This is the content of Lemma 3.5. ′ ′ Proof of property 2: Suppose x, y ∈ C(X1 , X2 ). Now if x , y ∈ c(x, y) ′ ′ then clearly the segment of c(x, y) between x , y , say c(x, y)|[x′ ,y′ ] , is a possible ′ ′ candidate for the definition of c(x , y ). Thus by corollary 3.9 the Hausdorff distance ′ ′ of c(x, y)|[x′ ,y′ ] and c(x , y ) is bounded by D3.9 ≤ D2 . Proof of property 3: Let x, y, z ∈ C(X1 , X2 ). Then using lemma 3.3 we may assume, without loss of generality, that x, y, z are contained in three c4 -qi sections X3 , X4 , X5 respectively, where X4 ⊆ C(X1 , X5 ), X3 ⊆ C(X1 , X4 ). Now, the triangle formed by the curves c(x, y), c(y, z), c(x, z) defined using these sections is D3.10 (c4 , A)-slim by lemma 3.10. Hence, by the corollary 3.9, in any case, the triangle formed by the curves c(x, y), c(y, z), c(x, z) is {D3.10(c4 , A) + 2.D3.9 (c4 )}slim. Also since X1 , X2 are c1 -quasi-isometric images of the geodesic metric space B in the hyperbolic metric space C(X1 , X2 ), they are uniformly quasi-convex in C(X1 , X2 ). This completes this proof of the first part of the proposition. For the second part of the proposition note that if A0 = dh (X1 , X2 ) ≥ Mc1 , then by Lemma 2.13, the diameter of U (X1 , X2 ) ⊆ B is bounded by D2.13 (c1 , A/A0 ). Thus, for any point x ∈ X1 and y ∈ X2 the quasi-geodesic c(x, y) passes through the bounded set p−1 (U (X1 , X2 )) ∩ C(X1 , X2 ). Since C(X1 , X2 ) has been proven to be hyperbolic, stability of quasi-geodesics (Lemma 1.11) completes the proof. ✷ 3.2. Hyperbolicity of ladders: General case. Lemma 3.11. There is a function D3.11 : R+ → R+ such that the following holds: Suppose I, J are intervals in R and φ : I → J is a k-quasi-isometric embedding. Let x1 , x2 , x3 ∈ I, x1 ≤ x2 ≤ x3 and suppose φ(x1 ) belongs to the interval with end points φ(x2 ), φ(x3 ). Then x2 − x1 ≤ D3.11 (k). P roof : Without loss of generality, we may assume that φ(x2 ) ≤ φ(x1 ) ≤ φ(x3 ). Let x4 = inf {y ∈ [x2 , x3 ] : φ(y) ≥ φ(x1 )}. ′ ′ If x2 = x4 then ∃x ∈ [x2 , x2 + 1] ∩ [x2 , x3 ] such that φ(x ) ≥ φ(x1 ). Now ′ ′ x − x2 ≤ 1 implies |φ(x ) − φ(x2 )| ≤ 2k, since φ is a k-q.i. embedding. Therefore, it follows that φ(x1 ) − φ(x2 ) ≤ 2k. Thus we have x2 − x1 ≤ 3k 2 . ′ ′′ ′ If x2 < x4 we choose x ∈ [x2 , x4 ) and x ∈ [x4 , x3 ] such that x4 − x ≤ 1 and ′′ ′′ ′′ ′ ′ ′′ x − x4 ≤ 1 with φ(x ) ≥ φ(x1 ). Now x − x ≤ 2 implies |φ(x − φ(x )| ≤ 3k. ′′ ′ ′ Thus φ(x ) − φ(x1 ) ≤ 3k, since φ(x ) < φ(x1 ) by the choice of x . Hence x2 − x1 ≤ ′′ x − x1 ≤ 4k 2 . Therefore, in any case, we may choose D3.11 (k) = 4k 2 . ✷ Proposition 3.12. Suppose X1 , X2 are two c1 -q.i. sections in X. Then C(X1 , X2 ) is a δ3.12 = δ3.12 (c1 )- hyperbolic metric space with respect to the induced path metric from X. A COMBINATION THEOREM FOR METRIC BUNDLES 23 2 Proof: Let c2 = C3.3 (c1 ), c3 = C3.3 (c1 ), as in the proof of the Proposition 3.4. Let A = Mc3 + D3.11 (2g(c3 )). The idea of the proof is that we break the ladder C(X1 , X2 ) into small girth ladders and then apply Proposition 1.29. Step 1 : Defining small girth ladders Fix a horizontal geodesic I = Xb0 ∩ C(X1 , X2 ). I has two end points respectively in X1 , and X2 . Choose a parametrization α : [0, l] → I, by arc length so that α(0) ∈ X1 and α(l) ∈ X2 . Next we inductively construct a finite sequence of real numbers 0 = s0 < s1 < · · · < sm = l, for some m ∈ N and a sequence of c2 -q.i. ′ sections Xi , passing through α(si ) for each i and contained in C(X1 , X2 ). Let ′ ′ X0 = X1 ; suppose si has been found, si < l and Xi has been constructed. If ′ ′ dh (Xi , X2 ) ≤ A then define si+1 = l, Xi+1 = X2 and the construction is over. Otherwise, consider the set ′ ′ ′ Si+1 = {t ∈ [si , l] : ∃ a c2 − q.i. section X through α(t) with dh (X , Xi ) ≤ A} ′ Let ui+1 = sup S. If ∃t ∈ S ∪ {ui+1 } such that there is a c2 -q.i. section X through ′ ′ ′ ′ α(t) with dh (X , Xi ) = A, define si+1 = t and Xi+1 = X . Otherwise define ′ si+1 = min{l, ui+1 + 1} and let Xi+1 be any c2 -qi section through α(si+1 ). The ′ construction of these sections stops at the m-th step if dh (Xm−1 , X2 ) ≤ A, so that ′ we must have Xm = X2 and sm = l. It follows from the above construction of the ′ ′ ′ sections Xi that for each i, 1 ≤ i ≤ m − 1, we have dh (Xi−1 , Xi ) ≥ A and in case ′ ′ ′′ dh (Xi , Xi+1 ) > A, there is a section Xi through a point α(ti ), ti ∈ [si , si+1 ] with ′′ ′ dh (Xj , Xi ) ≤ A, j = i, i + 1. Step 2 : Small girth ladders form a decomposition of C(X1 , X2 ) ′ ′ ′ ′ In this step, we will show that C(X1 , X2 ) = ∪ni=0 C(Xi , Xi+1 ); moreover C(Xi−1 , Xi )∩ ′ ′ ′ C(Xi , Xi+1 ) = Xi . ′ ′ Clearly, the first assertion follows from the second. Since C(X1 , Xi )∩C(Xi , X2 ) = ′ ′ ′ Xi , it is enough to show that Xi+1 ⊆ C(Xi , X2 ), for all i, 1 ≤ i ≤ m − 2. Let ′ ′ us consider the triplets of points (X1 ∩ Xb , Xi ∩ Xb , Xi+1 ∩ Xb ), b ∈ B. They are ′ contained in the geodesic Xb ∩ C(X1 , X2 ). For b = b0 we know that Xi ∩ Xb ∈ ′ ′ ′ ′ [X1 ∩ Xb , Xi+1 ∩ Xb ]. Now if Xi+1 6⊆ C(Xi , X2 ), then for some point b ∈ B we ′ ′ must have Xi+1 ∩Xb′ ∈ [X1 ∩Xb′ , Xi ∩Xb′ ]. Therefore there are two distinct points ′ ′ b1 , b2 ∈ B, d(b1 , b2 ) ≤ 1 for which we have Xi ∩ Xb1 ∈ [X1 ∩ Xb1 , Xi+1 ∩ Xb1 ] but ′ ′ ′ ′ Xi+1 ∩Xb2 ∈ [X1 ∩Xb2 , Xi ∩Xb2 ]. We know Xi , Xi+1 are c2 -quasi-isometric sections, ′ ′ and X1 is a c1 -quasi-isometric section. Thus we have d(Xi ∩ Xb1 , Xi ∩ Xb2 ) ≤ 2c2 , ′ ′ d(Xi+1 ∩ Xb1 , Xi+1 ∩ Xb2 ) ≤ 2c2 and d(X1 ∩ Xb1 , X1 ∩ Xb2 ) ≤ 2c1 ≤ 2c2 . Now by Lemma 3.3, the choice of c3 and Lemma 1.4, we have a g(2c3 )-quasi-isometric ′ embedding [X1 ∩ Xb1 , Xi+1 ∩ Xb1 ] → [X1 ∩ Xb2 , X2 ∩ Xb2 ] which sends each of the points Xj ∩ Xb1 to Xj ∩ Xb2 , j = 1, i, i + 1. Now, applying the Lemma 3.11 we get ′ ′ d(Xi ∩ Xb1 , Xi+1 ∩ Xb1 ) ≤ D3.11 (g(2c3 )) ′ By the choice of the constant A, and the definition of Xi ’s this gives rise to a contradiction and completes the proof. Step 3 : Small girth ladders are uniformly hyperbolic ′ ′ Now we show that there are constants δ1 , k1 and D such that each C(Xi , Xi+1 ) is 24 MAHAN MJ AND PRANAB SARDAR ′ ′ δ1 -hyperbolic and Xi , Xi+1 are k1 -quasi-convex in it for each i, 0 ≤ i ≤ n. Also the ′ ′ ′ ′ sets Xi , Xi+1 are mutually D-cobounded in C(Xi , Xi+1 ), for i 6= m − 1. ′ ′ ′ ′ Since Xi , Xi+1 are c2 -q.i. section in X, they are c2 -q.i. section in C(Xi , Xi+1 ) ′ ′ ′ ′ too. Hence, they are D1.11 (δ1 , c2 )-quasiconvex in C(Xi , Xi+1 ) if C(Xi , Xi+1 ) is δ1 -hyperbolic. ′ ′ ′ ′ If dh (Xi , Xi+1 ) ≤ A then, by Lemma 3.4, each of C(Xi , Xi+1 ) is δ3.4 (c2 , A)′ ′ ′ ′ hyperbolic and Xi , Xi+1 are mutually D3.4 (c2 , A)-cobounded provided dh (Xi , Xi+1 ) = A. ′ ′ ′ Suppose dh (Xi , Xi+1 ) > A. Recall that Xj passes through α(sj ), j = i, i + 1. ′′ In this case, we can find ti ∈ [si , si+1 ] such that there is a c2 -q.i. section Xi in ′ ′′ C(X1 , X2 ), passing through α(ti ), so that dh (Xj , Xi ) ≤ A, j = i, i + 1. Now, as ′′ in the proof Lemma 3.3 we project each point of Xi into the horizontal geodesics ′ ′ ′ of C(Xi , Xi+1 ) and get a c3 -q.i. section Yi through α(ti ). Note that we still have ′ ′ ′ ′ ′ ′ dh (Xj , Yi ) ≤ A for j = i, i + 1. By Lemma 3.4, C(Xi , Yi ), and C(Xi+1 , Yi ) ′ ′ ′ ′ ′ are both δ3.4 (c3 , A)-hyperbolic. Also we have C(Xi , Yi ) ∩ C(Xi+1 , Yi ) = Yi ′ ′ which is a c3 -quasi-isometric image of B in X and hence in both of C(Xi , Yi ) ′ ′ ′ ′ ′ and C(Xi+1 , Yi ). Thus Yi is D1.11 (δ3.4 (c3 , A), c3 )-quasi-convex in C(Xj , Yi ), j = ′ ′ i, i+1. Thus as a trivial application of Proposition 1.29 we obtain that C(Xi , Xi+1 ) is δ1.29 (δ3.4 (c3 , A), D1.11 (δ3.4 (c3 , A), c3 ))-hyperbolic. Therefore, we can take δ1 = max{δ3.4 (c2 , A), δ1.29 (δ3.4 (c3 , A), D1.11 (δ3.4 (c3 , A), c3 ))}. ′ ′ ′ ′ We next show that Xi , Xi+1 are mutually cobounded in C(Xi , Xi+1 ) in this case too. This breaks up into two cases: ′ ′ If U (Xj , Yi ), j = i, i + 1 are well separated in B then the lift of a geodesic in B to ′ ′ ′ ′ ′ Yi is the (coarsely) unique shortest geodesic joining Xi , Xi+1 in C(Xi , Xi+1 ). This follows from the fact that these sets are uniformly quasiconvex in B, by Lemma 2.13 and the definition of the curves c(x, y) which are uniform quasi-geodesics by Proposition 3.4 and Lemma 1.17. ′ ′ ′ ′ On the other hand if U (Xj , Yi ), j = i, i+1 are not well separated then dh (Xi , Xi+1 ) ′ ′ is small, say A1 ≥ A. Hence the by Proposition 3.4, Xi , Xi+1 are D3.4 (c2 , A1 )cobounded. This proves the assertions of step 3. Finally using Proposition 1.29 we see that C(X1 , X2 ) is δ1.29 (δ1 , k1 )-hyperbolic. ✷ 4. The Combination Theorem 4.1. Proof of Main Theorem. Suppose X1 ,X2 and X3 are three c1 -q.i. sections of B in X. In each of the fibers Xb , construct a geodesic triangle with vertices X1 ∩ Xb , X2 ∩ Xb and X3 ∩ Xb by joining each pair of these points with a geodesic in Xb . Let us denote the union of all these triangles by C(X1 , X2 , X3 ). Then we have the following Proposition 4.1. C(X1 , X2 , X3 ) is δ4.1 = δ4.1 (c1 )- hyperbolic with the induced path metric from X and each of C(Xi , Xj ), i 6= j is K4.1 = K4.1 (c1 )-quasi-convex in C(X1 , X2 , X3 ). Proof: For b1 , b2 ∈ B with d(b1 , b2 ) ≤ 1, we have a g(2c1 )-quasi-isometry Xb1 → Xb2 by Lemma 1.4, which sends Xi ∩ Xb1 to Xi ∩ Xb2 for i = 1, 2, 3. Therefore, by Lemma 1.14, choosing the barycenters of the geodesic triangles C(X1 , X2 , X3 ) ∩ Xb A COMBINATION THEOREM FOR METRIC BUNDLES ′ 25 ′ in each of the fiber spaces Xb , b ∈ B we have a c1 -q.i. section of B in X where c1 := ′ ′ 2c1 + D1.14 (δ , g(2c1 ), g(2c1 )). Let us call this section X4 . Let K1 := max{c1 , c1 }. Let Zi := C(Xi , X4 ), ∀i = 1, 2, 3 and let Z := Z1 ∪ Z2 ∪ Z3 . We first show that Z is a hyperbolic metric space with the induced path metric from X and Zi ’s are quasi-convex in Z. By proposition 3.12 each of these subspaces Zi ⊂ Z is δ3.12 (K1 )-hyperbolic; since Xi , X4 are K1 -quasi-isometric image of B, by Lemma 1.11 they are D1.11 (δ3.12 (K1 ), K1 )-quasi-convex subsets of Zi , i = 1, 2, 3. Also note that Zij := Zi ∩ Zj , 1 ≤ i 6= j ≤ 3 can be easily shown to be contained in ′ the horizontal 8δ -neighbourhood of X4 . Thus Zij is K2 -quasi-convex in Zi and ′ Zj , where K2 := D1.11 (δ3.12 (K1 ), K1 ) + 16δ . Thus using Proposition 1.29, we get that Zi ∪ Zj is δij := δ1.29 (δ3.12 (K1 ), K2 )-hyperbolic and Zi , Zj are kij := k1.29 (δ3.12 (K1 ), K2 )- quasi-convex subsets of Zi ∪ Zj . Applying Proposition 1.29 ′ again to Z3 and Z1 ∪ Z2 , it follows that Z1 ∪ Z2 ∪ Z3 is δ4.1 -hyperbolic, and each ′ ′ of the spaces Zi is k4.1 -quasi-convex in Z1 ∪ Z2 ∪ Z3 , for some constants δ4.1 and ′ k4.1 depending on c1 . ′ Next, we note that since Xb is δ -hyperbolic, by the definition of barycenters (see Lemma 1.14), the Hausdorff distance of C(Xi , Xj ) ∩ Xb and (C(Xi , X4 ) ∪ ′ C(Xj , X4 )) ∩ Xb , i 6= j, 1 ≤ i, j ≤ 3 in Xb is at most 5.δ . Now given a geodesic triangle in C(X1 , X2 , X3 ), taking a nearest point projection of each of its points ′ on Z1 ∪ Z2 ∪ Z3 in each of the horizontal spaces we get a 1 + 10.δ -quasi-geodesic ′ ′ ′ triangle in Z1 ∪ Z2 ∪ Z3 which is, therefore, δ4.1 + 2.D1.11 (δ4.1 , 1 + 10.δ )-slim. ′ ′ ′ ′ Thus we can take δ4.1 to be 10.δ + δ4.1 + 2.D1.11(δ4.1 , 1 + 10.δ ). Similarly, we ′ ′ ′ ′ can take K4.1 to be D1.11 (δ4.1 , 1 + 10.δ ) + k4.1 + 5δ . ✷ We are finally in a position to assemble the proof of the main combination Theorem. Theorem 4.2. Suppose p : X → B is a metric bundle with the following properties: (1) B is a δ-hyperbolic metric space. ′ (2) Each of the fiber Xb , b ∈ B is a δ -hyperbolic metric space with respect to the induced path metric from X. (3) The barycenter maps ∂ 3 Xb → Xb , b ∈ B are (uniformly) coarsely surjective. (4) The metric bundle satisfies a flaring condition. Then X is a hyperbolic metric space. Proof: We define a set of curves joining each pair of points x, y ∈ X and check that they satisfy the properties of lemma 1.17 to complete the proof. Definition of the curves: For each pair of points x, y in X, choose, once for all, two K0 -q.i. sections X1 , X2 passing through x and y respectively. Now define c(x, y) to be a geodesic in C(X1 , X2 ) joining x,y. Proof of property 1: This follows from Lemma 3.5. Proof of property 2: Clearly it suffices to show that for any x, y ∈ X, given ′ two sections X3 , X3 passing through x and a section X4 passing through y, the two possible candidates for the curve c(x, y) defined using the sections X3 , X4 and ′ X3 , X4 respectively, are at a uniformly bounded Hausdorff distance. Now by Propo′ sition 4.1 we know that C(X3 , X3 , X4 ) is δ4.1 (K0 )-hyperbolic and its subspaces ′ C(X3 , X4 ), C(X3 , X4 ) are K4.1 (K0 )-quasi-convex. Hence the curves in considera′ ′ ′ tion are K -quasi-geodesics in C(X3 , X3 , X4 ) for some constant K depending on 26 MAHAN MJ AND PRANAB SARDAR δ4.1 and K4.1 . Since the curves join the same pair of points x, y, their Hausdorff ′ distance is at most 2.D1.11(δ4.1 , K ). Proof of property 3: Given x, y, z ∈ X choose three K0 -q.i. sections X3 , X4 , X5 ′ ′ ′ containing x, y, z ∈ X respectively and define the curves c (x, y), c (x, z) and c (y, z) using these sections in the same way as the curves c(x, y)’s are defined. Since these ′ curves are K -quasi-geodesics, as observed in the proof of property 2, by lemma ′ 1.11 the triangle formed by these curves is {δ4.1 + D1.11 (δ4.1 , K )}-slim. Now by the proof of the property 2 we know that the Hausdorff distance of any two curves joining same pair of end points, defined in the same manner as c(x, y)’s is ′ at most 4.D1.11 (δ4.1 , K ). Thus the triangle formed by the curves c(x, y), c(y, z) ′ and c(x, z) is {δ4.1 + 5.D1.11 (δ4.1 , K )}-slim. ✷ 5. Consequences and Applications 5.1. Sections, Retracts and Cannon-Thurston maps. We shall say that an exact sequence of finitely generated groups 1 → K → G → Q → 1 satisfies bounded flaring if the associated metric bundle of Cayley graphs does. An immediate consequence of Theorem 4.2 coupled with the existence of qi-sections from Theorem 2.11 is the following converse to Mosher’s Theorem 2.11. Theorem 5.1. Suppose that the short exact sequence of finitely generated groups 1 → K → G → Q → 1. satisfies a flaring condition such that K, Q are word hyperbolic and K is nonelementary. Then G is hyperbolic. Theorem 2.11 was generalized by Pal [Pal08] as follows. Theorem 5.2. (Pal [Pal08]) Suppose we have a short exact sequence of finitely generated groups p 1 → K → G → Q → 1, with K strongly hyperbolic relative to a subgroup K1 such that G preserves cusps, i.e. for all g ∈ G there exists k ∈ K with gK1 g −1 = kK1 k −1 . Then there exists a (k, ǫ) quasi-isometric section s : Q → G for some constants k ≥ 1,ǫ ≥ 0. Next, suppose we have a short exact sequence of pairs of finitely generated groups p 1 → (K, K1 ) → (G, NG (K1 )) → (Q, Q1 ) → 1 with K (strongly) hyperbolic relative to the cusp subgroup K1 . If G preserves cusps, then Q1 = Q and there is a quasi-isometric section s : Q → NG (K1 ) satisfying 1 dQ (q, q ′ ) − ǫ ≤ dNG (K1 ) (s(q), s(q ′ )) ≤ RdQ (q, q ′ ) + ǫ R where q, q ′ ∈ Q and R ≥ 1,ǫ ≥ 0 are constants. Further, if G is weakly hyperbolic relative to K1 , then Q is hyperbolic. The setup of Theorem 5.2 naturally gives an associated metric bundle P : X → Q b of spaces, where Q is the quotient group and fibers are the coned off spaces K obtained by electrocuting copies of K1 in K. We now use another construction of Mitra [Mit98a]. Let X1 , X2 be two qisections of a metric bundle p : X → B, where each fiber (but not necessarily the base B) is uniformly δ-hyperbolic. Let C(X1 , X2 ) be the associated ladder. By A COMBINATION THEOREM FOR METRIC BUNDLES 27 construction, the intersection C(X1 , X2 ) ∩ Xb of the ladder with a fiber Xb is a geodesic λb ⊂ Xb in the path metric space (Xb , db ). Define a projection πb : Xb → λb to be the nearest point projection of Xb onto the geodesic λb in the metric db and ΠX1 ,X2 : X → C(X1 , X2 ) by ΠX1 ,X2 (x) = πb (x)∀x ∈ Xb . The main technical Theorem of [Mit98a] states Theorem 5.3. [Mit98a] For X, B, X1 , X2 , p as above, ΠX1 ,X2 is a coarse Lipschitz retract, i.e. there exists C ≥ 1 such that ∀x, y ∈ X, d(ΠX1 ,X2 (x), ΠX1 ,X2 (y)) ≤ Cd(x, y) + C. Theorem 5.3 is proven in [Mit98a] in the context of an exact sequence of finitely generated groups 1 → N → G → Q → 1, with N hyperbolic; but all that the proof requires is the existence of qi sections (which is automatic in the context of groups by the qi section Theorem 2.11 of Mosher above). As in [Mit98a], the existence of a qi-section through each point of X guarantees, via Theorem 5.3, the existence of a continuous extension to the boundary (also called a Cannon-Thurston map [CT07] [CT85]) provided X is hyperbolic. The proof is identical to that in [Mit98a] and we refer the reader to [Mit98a]. Combining this fact with Theorem 4.2 we have the following. Theorem 5.4. Suppose p : X → B is a metric bundle with the following properties: (1) B is a δ-hyperbolic metric space. ′ (2) Each of the fiber Xb , b ∈ B is a δ -hyperbolic metric space with respect to the induced path metric from X. (3) The barycenter maps ∂ 3 Xb → Xb , b ∈ B are (uniformly) coarsely surjective. (4) The metric bundle satisfies a flaring condition. Then the inclusion ib : Xb → X extends continuously to a map from the Gromov cb → X. b compactifications î : X 5.2. Hyperbolicity of base and flaring. In our main combination theorem 4.2 flaring was a sufficient condition. In this subsection and the next we investigate its necessity. This problem ties up with that of hyperbolicity of the base space B in a metric bundle p : X → B. We study it with special attention to hyperbolic and relatively hyperbolic groups as in Theorems 2.11 and Theorem 5.2. A Theorem of Papasoglu ([Pap95], Lemma 3.8 of [Pap05]) states the following. Theorem 5.5. (Papasoglu [Pap95], Lemma 3.8 of [Pap05]) Let G be a finitely generated group and let Γ be the Cayley graph of G with respect to to a finite generating set. If there is an ǫ such that geodesic bigons in Γ are ǫ-thin then G is hyperbolic. Similarly, let X be a geodesic metric space such that for every K there exists C such that K-quasigeodesic bigons are C-thin, then X is hyperbolic. In fact there is some (universal) constant C > 0 such that if G is finitely generated and non-hyperbolic, then ∀R > 0 there is some R′ > R and a (C, C) q.i. embedding of a Euclidean circle of radius R′ in Γ. We now look at short exact sequences of finitely generated groups 1 → K → G → Q → 1. 28 MAHAN MJ AND PRANAB SARDAR such that K is non-elementary word hyperbolic, but Q is not hyperbolic. By Theorem 5.5, Q contains (C, C) qi embeddings of Euclidean circles of arbitrarily large radius. Now, given any l, A0 construct a (C, C) qi embedding τl of a Euclidean circle σ of circumference > 4l in Q. Construct two qi sections s1 , s2 of Q into G by Theorem 2.11 such that dh (s1 ◦ τl (σ), s2 ◦ τl (σ)) > A0 . Let q ∈ σ be such that the length dq (s1 ◦ τl (q), s2 ◦ τl (q)) in the fiber Fq over q is maximal. Let the two arcs of length l in τl starting at q (in opposite directions) end at q1 , q2 . Let q1 qq2 denote the union of these arcs. Then the two quasigeodesics s1 ◦ τl (q1 qq2 ), s2 ◦ τl (q1 qq2 ) violate flaring as the fiber length achieves a maximum at the midpoint q. We have thus shown the following. Proposition 5.6. Consider the short exact sequence of finitely generated groups 1 → K → G → Q → 1. such that K is non-elementary word hyperbolic but Q is not hyperbolic. Then the short exact sequence cannot satisfy a flaring condition. We next turn to the relatively hyperbolic situation described in Theorem 5.2 with Q non-hyperbolic, i.e. we assume that K is (strongly) hyperbolic relative to K1 . We have an analogue of Proposition 5.6 in this situation too. The proof is the same as that of the above Proposition. The existence of qi sections in this case, follows from Theorem 5.2. Lemma 5.7. Suppose we have a short exact sequence of finitely generated groups p 1 → (K, K1 ) → (G, NG (K1 )) → (Q, Q1 ) → 1 with K strongly hyperbolic relative to the cusp subgroup K1 and G preserves cusps, but Q is not hyperbolic. Let P : X → Q be the associated metric bundle of spaces, b obtained by where Q is the quotient group and fibers Fq are the coned off spaces K electrocuting copies of K1 in K. Then X does not satisfy flaring. Now suppose we have an exact sequence of groups as in the above Proposition with the additional condition that G is (strongly) hyperbolic relative to NG (K1 ). We claim that in this case Q is hyperbolic. Suppose Q is not hyperbolic. Let X be the Cayley graph of G and let B the Cayley graph of Q. Then the quotient map G → Q gives rise to a metric bundle p : X → B. This metric bundle admits uniform qi sections through each point of X by Theorem 5.2. Also note that B is not a hyperbolic metric space. Thus, by Theorem 5.5 of Papasoglu, above, we can construct (C, C)-qi embeddings of large Euclidean circles in B. Claim: Lifts of each of these circles in a k-q.i. section of the metric bundle p : X → B is contained in a D = D(k)-neighbourhood of a coset of NG (K1 ). Proof of the claim: Let τ1 be a (C, C)-q.i. embedding in Q of an Euclidean circle σ and τ2 : Q → G be a k-qi section. Let τ = τ2 ◦ τ1 . Then τ is clearly a k1 := (kC + k)-q.i. section. Let u, v be a pair of antipodal points of the circle and x = τ (u), y = τ (v). Join x, y by a geodesic γ in X h . Since G is strongly relatively hyperbolic w.r.t. NG (K1 ), any pair of adjacent vertices in γ which lie in different cosets of NG (K1 ) are contained in a D = D(δ, k1 )-neighbourhood of the images under τ of each of the semi-circular arcs bounded by u, v, where we assume that ′ X h is δ-hyperbolic. Suppose (z, z ) is such a pair of points. Thus we can find two points, say q1 , q2 , on the images under τ of the two halves of the circle σ such that d(z, qi ) ≤ D, so that d(q1 , q2 ) ≤ 2D. Choose ui ∈ σ, i = 1, 2 such that τ (ui ) = qi . A COMBINATION THEOREM FOR METRIC BUNDLES 29 Without loss of generality, we may assume that u is contained on a geodesic joining u1 , u2 in the circle σ. Thus it follows that distance of x from one of q1 , q2 is at most D1 for some constant depending only on k1 and D. Thus d(z, x) ≤ D+D1 . Thus the pair of ’antipodal’ points x, y is contained in the (D + D1 )-neighbourhood of a coset of NG (K1 ). Suppose H1 = g1 NG (K1 ) be such that x, y ∈ ND+D1 (H1 ). If possible, suppose the image of τ is not contained in the (D + D1 )-neighbourhood of H1 . ′ ′ Then we can find two pairs of antipodal points (v1 , v2 ) and (v1 , v2 ) on the Euclidean circle such that τ (v1 ), τ (v2 ) are contained in the (D + D1 )-neighbourhood of H1 ′ ′ and τ (v1 ), τ (v2 ) are contained in the (D + D1 )-neighbourhood of H2 = g2 NG (K1 ), ′ where H1 6= H2 and d(vi , vi ) ≤ 1. Since G is strongly relatively hyperbolic w.r.t NG (K1 ), H1h and H2h are mutually D0 -cobounded for some constant D0 . Let wi ∈ ′ H2 , i = 1, 2, be such that d(wi , τ (vi )) ≤ D + D1 . Let w̄i be the projections of wi on H1 . Then clearly dh (w¯1 , w¯2 ) ≥ dh (τ (v1 ), τ (v2 )) − 4(D + D1 + k1 ). Thus dh (τ (v1 ), τ (v2 )) ≤ 4(D + D1 + k1 ) + D0 . This fails to hold if we choose circles of sufficiently large radius, since X is properly embedded in X h . Thus, the image under τ of all large circles are contained in a uniformly bounded neighbourhood of a coset of NG (K1 ). This proves the claim. ✷ Now, let Y1 , Y2 be two k-qi sections of a large Euclidean circle σ in B, such that Y1 and Y2 lie uniformly close to two distinct cosets of NG (K1 ). Let C(Y1 , Y2 ) be the union ∪q∈σ λq λq is a horizontal geodesic in the Xq joining Y1 ∩ Xq to Y2 ∩ Xq . ′ Suppose b, b are antipodal points on σ. As in the proof of Lemma 3.3, we know that through each point of λb passes a uniform q.i. section of σ in C(Y1 , Y2 ); any such qi section is uniformly close to a coset of NG (K1 ). Since Y1 , Y2 are close to ′ ′ distinct cosets of NG (K1 ), we can find two qi section Y1 , Y2 of σ passing through z1 , z2 ∈ λb with d(z1 , z2 ) ≤ 1 and are close to two distinct cosets of NG (K1 ). ′ ′ ′ ′ ′ ′ Suppose Y1 , Y2 intersect λb′ in z1 and z2 respectively. If d(z1 , z2 ) is large, in the ′ ′ same way as before, we can construct two qi sections Y3 , Y4 of σ in C(Y1 , Y2 ) close to two distinct cosets of NG (K1 ) such that d(Y3 ∩ λb′ , Y4 ∩ λb′ ) = 1. Thus we have two qi sections of long subarcs of σ that start and end close by in X but lie close to distinct cosets of NG (K1 ). Since the circle σ can be chosen to be arbitrarily large, this violates strong relative hyperbolicity of X with respect to the cosets of NG (K1 ). Hence we have shown the following. Proposition 5.8. Suppose we have a short exact sequence of finitely generated groups p 1 → (K, K1 ) → (G, NG (K1 )) → (Q, Q1 ) → 1 with K (strongly) hyperbolic relative to the cusp subgroup K1 such that G preserves cusps. Suppose further that G is (strongly) hyperbolic relative to NG (K1 ). Then Q is hyperbolic. 5.3. Necessity of Flaring. We are finally in a position to prove that flaring is a necessary condition for hyperbolicity. The crucial ingredient is the following Lemma which is a specialization to our context of the fact that geodesics in a hyperbolic space diverge exponentially. (See Proposition 2.4 and the proof of Theorem 4.11 in [Mit97]). Suppose p : X → B is a metric bundle with parameters as in definition 1.1, X is a hyperbolic metric space and that each of the fiber Xz , z ∈ b is δ0 -hyperbolic. Suppose γ : [−L, L] → B is a geodesic and α, β are two K-qi lifts of γ. Similar to 30 MAHAN MJ AND PRANAB SARDAR the constructions of ladders, we define Y to be the union of horizontal geodesics [α(t), β(t)] ⊂ Xt , t ∈ [−L, L], and refer to it as the ladder formed by α and β. Lemma 5.9. Let γ : [−L, L] → X be a geodesic as above. Given K ≥ 1, D ≥ 0 there exist b = b(K, D) > 1, A = A(K, D) > 0 and C = C(K, D) > 0 such that the following holds: If α, β are two K-qi sections of γ with d(α(0), β(0)) ≤ D and there exists T ∈ [0, L] with d(α(T ), β(T )) ≥ C then any path joining α(T + t) to β(T + t) and lying outside t the union of the T +t−1 2K -balls around α(0), β(0) has length greater than Ab for all t ≥ 0 such that T + t =∈ [0, L]. In particular, the horizontal distance of α(T + t) to β(T + t) is greater than Abt for all t ≥ 0 such that T + t ∈ [0, L]. Now, we use Lemma 5.9 to show that the ladder Y flares in at least one direction of γ in some special cases. We start the proof with the following important note. For each geodesic γ ⊂ B and x ∈ X there is a 2c-q.i. section of γ through x. This follows in a straightforward way from the definition of metric bundles. Now, as in the proof of Lemma 3.3, it follows that through every point of the ladder Y there is a F (K)-q.i. section of γ contained in Y , for some function F depending on K (and δ0 ). Flaring of ladders in special cases: Let K1 ≥ 1. Suppose γ : [−L, L] → B is a geodesic and let α : [−L, L] → X and β : [−L, L] → X be two K1 -qi lifts of γ, which form the ladder Y . Let η : [0, M ] → Xγ(0) be the geodesic Y ∩ Xγ(0) . We shall assume L to be sufficiently large for the following arguments to go through. Let K2 = F (K1 ) and K3 = F (K2 ). Case 1: We show that if M ≥ D1 , for some constant D1 depending on K1 then the ladder satisfies flaring, where λK1 = 4 and nK1 (see Definition 1.3) depends on M , K1 . Let [a, b] ⊂ [−L, L]. First we make the following note: Since X is a hyperbolic space and α : [a, b] → X and β : [a, b] → X are two K1 -quasi-geodesics, there are constants C = C(d1 , d2 , K1 ) and D = D(K1 ) such that if [t − C, t + C] ⊂ [a, b] for some t ∈ [a, b] then d(α(t), β(t)) ≤ D, where d1 = d(α(a), β(a)) and d2 = d(α(b), β(b)). This follows easily from stability of quasi-geodesics and slimness of triangles in ′ X. Let D1 = C5.9 (K1 , D). Since the horizontal spaces in X are uniformly properly embedded in X there is a D1 such that for all v ∈ B and x, y ∈ Xv if dv (x, y) ≥ D1 ′ then d(x, y) ≥ D1 . Now, let us assume that dγ(0) (α(0), β(0)) = M ≥ D1 . Let D2 = C5.9 (K1 , M ) and let C1 := 1 + 2.C(M, D2 , K1 ). If d(α(C1 ), β(C1 )) ≥ D3 then for all t ≥ 0 the length of the horizontal geodesic joining α(C1 + t) to β(C1 + t) is greater than equal to A1 .bt1 for some A1 , b1 depending on K1 and D2 , by Lemma 5.9. Choose t1 > 0 such that for all t ≥ t1 , A1 .bt11 ≥ 4.M . Otherwise, suppose d(α(C1 ), β(C1 )) < D2 . In this case, by the observation made in the beginning, d(α( C12−1 ), β( C12−1 )) ≤ D. By the choice of the constants D, D2 we can again apply Lemma 5.9 so that for all t ≥ 0 the length of a horizontal geodesic Y ∩ Xγ(−t) is greater than equal to A2 .bt2 , where the constants A2 , b2 depends on K1 and D. Choose t2 > 0 such that for all t ≥ t2 , A2 .bt2 ≥ 4.M . Now let n1 = max{C1 +t1 , t2 }. Thus we have max{dγ(−t) (α(−t), β(−t)), dγ(t) (α(t), β(t))} ≥ 4M for all t ≥ n1 = n1 (M, K1 ) if M ≥ D1 = D1 (K1 ). Case 2: In this case, we fix l > 0 and suppose that for any s ∈ [0, M − 2] there is a K2 - q.i. section α1 of γ in Y through η(s) such that d(α(t), α1 (t)) ≤ l for some t ∈ [−L, L]. (Without loss of generality, we shall assume that t ∈ [−L, 0].) A COMBINATION THEOREM FOR METRIC BUNDLES 31 Let C3 = C5.9 (l, K3 ), A3 = A(l, K3 ), b3 = b(l, K3 ). First, we note the following easy consequence of lemma 1.4, Corollary 1.5 and the definition of metric bundles. ′ ′′ Suppose we have two K3 -qi sections α , α : [−L, L] → X of γ. For all m > 0 we can ′ ′ ′ ′′ ′ ′ ′′ find D4 = D4 (m) > 0 such that if dγ(0) (α (0), α (0)) ≥ D4 then d(α (t), α (t) ≥ C3 for all t ∈ [−m, 0]. Let m0 := min{m ∈ N : A3 .bm 3 ≥ D3.11 (g(2K3 ))}, where g refers to the function ′ defined in the Lemma 1.4, and D4 := D4 (m0 ). Let M − 2 = N.D4 + r where N ∈ N and 0 ≤ r < D4 . Now construct a K2 -qi section β1 in the ladder Y such that dγ(0) (β(0), β1 (0)) = r + 2 and d(α(t), β1 (t)) ≤ l for some t ∈ [−L, 0]. We now break the subladder of Y , formed by α and β1 , by K3 -q.i. sections α0 = α, α1 , · · · , αN = β1 over γ such that dγ(0) (αi (0), αi+1 (0)) = D4 . We have d(αi (t), αi+1 (t)) ≤ l. Thus by the choice of the constant D4 , dγ(t) (αi (t), αi+1 (t)) ≥ D3.11 (g(2K3 )).bt3 for all t ≥ 0 and ∪[αi (t), αi+1 (t)] is a partition of the horizontal geodesic segment [α(t), β1 (t)] ⊂ Y ∩ Xγ(t) . Therefore, we can choose n2 = n2 (K1 , l) such that for all t ≥ n2 , max{dγ(t) (α(t), β(t)), dγ(t) (α(t), β(t))} ≥ 4.M if M − 2 ≥ D4 = D4 (K1 , l). Flaring of general ladders: In the general case, first of all, we break the ladder Y into subladders of special types as described above (see Figure 2). Then a simple argument will finish the proof. Let us assume that Y is made of the Kq.i. sections α, β of a geodesic γ : [−L, L] → X. Let K1 = F (K), K2 = F (K1 ), K3 = F (K2 ) and l = D3.11 (g(2.K3 )). Let MK := max{D1 (K1 ), 2 + D4 (K1 , l)}, and nK := max{n1 (K1 , D1 (K1 )), n2 (K1 , l)} where the functions D1 , D4 , n1 , n2 are as in the proof of the flaring for the special ladders. Now we inductively construct K1 -qi sections α0 = α, α1 , · · · , αi = β in Y to decompose it into some special subladders. Let us assume that M ≥ MK . As observed earlier, we can construct a K1 -qi section α1 through η(MK ). Now, suppose α1 , · · · , αj has been constructed through the points η(s1 ), · · · , η(sj ) respectively. If dγ(0) (αj (0), β(0)) ≤ MK define αj+1 = β. Otherwise, if there is a K1 -qi section through η(MK + sj ) in the ladder formed by αj and β define it to be αj+1 . If neither of that happens then consider the following set: {t ≥ sj + MK : any K1 -qi section through η(t) enters the ladder formed by αj and α }. Let tj be the supremum of this set. Define αj+1 to be a K1 -qi section through sj+1 := tj + 1, in the ladder formed by αj and β that does not enter the ladder formed by αj and α. Figure 2:Flaring subladders 32 MAHAN MJ AND PRANAB SARDAR Clearly for each j, αj and αj+1 form a special ladder and hence it must flare. Thus η can be expressed as the disjoint union of subsegments that flare to the left and the union of the subsegments that flare to the right respectively. Length of one of these must be at least half of the length of η. Thus we have shown the following: For each K ≥ 1 we can find n = nK > 0, D = DK > 0 such that the following holds. Suppose γ : [−n, n] → B be a geodesic and let α, β : [−n, n] → X be two K-qi lifts of γ in X. If dγ(0) (α(0), β(0)) ≥ D then max{dγ(−n) (α(−n), β(−n)), dγ(n) (α(n), β(n))} ≥ 2.dγ(0) (α(0), β(0)) In other words, we have proved the following proposition. Proposition 5.10. Let P : X → B be a metric bundle such that 1. X is hyperbolic 2. There exist δ0 such that each of the fiber Xz is δ0 -hyperbolic equipped with the induced path metric from X. Then the metric bundle satisfies a flaring condition. In particular, any exact sequence of finitely generated groups 1 → N → G → Q → 1 with N, G hyperbolic satisfies a flaring condition. The last statement follows from theorem 2.11 which guarantees the existence of qi sections. 5.4. An Example. Let (T eich(S), dT ) be the Teichmuller space of S equipped with the Teichmuller metric dT . Teichmuller space can also be equipped with an electric metric de by electrocuting the cusps. Note (as per work of Masur-Minsky [MM99], see also [Mj09]) that (T eich(S), de ) is quasi-isometric to the curve complex CC(S). Let E : (T eich(S), dT ) → (T eich(S), de ) be the identity map from the Teichmuller space of S equipped with the Teichmuller metric dT to the Teichmuller space of S equipped with the electric metric de . We shall need the following Theorem due to Hamenstadt [Ham09] which used an idea of Mosher [Mos03] in its proof. Theorem 5.11. Hamenstadt [Ham09]: For every L > 1 there exists D > 0 such that the following holds. Let f : R → (T eich(S), dT ) be a Teichmuller L-quasigeodesic such that E ◦ f : R → (T eich(S), dE ) is also an L-quasigeodesic. Then for all a, b ∈ R there is a Teichmller geodesic ηab joining f (a), f (b) ∈ T eich(S) such that the Hausdorff distance dH (f ([a, b]), ηab ) ≤ D0 . We are now in a position to prove a rather general combination Proposition for metric bundles over quasiconvex subsets of CC(S). Proposition 5.12. Let (K, dK ) be a hyperbolic metric space satisfying the following, There exists C > 0 such that for any two points u, v ∈ K, there exists a bi-infinite C-quasigeodesic γ ⊂ K with dK (u, γ) ≤ C and dK (v, γ) ≤ C. Let j : K → (T eich(S), dT ) be a quasi-isometric embedding such that E ◦ j : K → (T eich(S), de ) is also a quasi-isometric embedding. Let U (S, K) be the pull-back bundle under j of the universal curve over T eich(S) equipped with the natural path metric. Then U^ (S, K) is a hyperbolic metric space. where U^ (S, K) denotes the metric bundle obtained by taking fiberwise universal covers. A COMBINATION THEOREM FOR METRIC BUNDLES 33 Proof: Clearly, U^ (S, K) is a metric bundle over K. Hence, by Theorem 4.2 it suffices to prove flaring. Let Sx denote the fiber of U (S, K) over x ∈ i(K). Let [a, b] be a geodesic segment of sufficiently large length in K. By the hypothesis on K, there exists a bi-infinite geodesic passing within a bounded neighborhood of [a, b]. Hence without loss of generality, we may assume that a, b lie on a bi-infinite geodesic in K. By Theorem 5.11 we may assume that j(a), j(b) lie in a uniformly bounded neighborhood of a fat Teichmuller geodesic ηab whose end-points in ∂T eich(S) are two singular foliations F+ , F− . Let ds be the singular Euclidean metric on S induced by the pair of singular foliations F+ , F− . The rest of the argument follows an argument of Mosher [Mos97]. 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