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Hyperbolic Trees of Spaces Michael Kapovich Pranab Sardar

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Hyperbolic Trees of Spaces Michael Kapovich Pranab Sardar Hyperbolic trees of spaces Michael Kapovich Pranab Sardar Author address: University of California Davis, 1 Shields Avenue, Davis, CA 95616 E-mail address: [email protected] Indian Institute of Science Education and Research, Mohali, Sec- tor 81, PB 140306, India E-mail address: [email protected] Contents Preface vii Chapter 1. Preliminaries on metric geometry 1 1.1. Graphs and trees 1 1.2. Coarse geometric concepts 2 1.3. Hyperbolic metric spaces 14 1.4. Combings and a characterization of hyperbolic spaces 19 1.5. Geometry of hyperbolic triangles 23 1.6. Quasiconvex subsets 26 1.7. Quasiconvex hulls 27 1.8. Projections 28 1.9. Images and preimages of quasiconvex subsets under projections 32 1.10. Modified projection 34 1.11. Projections and coarse intersections 35 1.12. Cobounded pairs of subsets 37 1.13. Ideal boundaries 40 Chapter 2. Graphs of groups and trees of metric spaces 43 2.1. Generalities 43 2.2. Trees of spaces 45 2.3. Coarse retractions 53 2.4. Trees of hyperbolic spaces 55 2.5. Flaring 58 2.6. Hyperbolicity of trees of hyperbolic spaces 68 Chapter 3. Flow-spaces, ladders and their retractions 75 3.1. Semicontinuous families of spaces 75 3.2. Ladders 78 3.3. Flow spaces 83 3.4. Retractions to bundles 92 Chapter 4. Hyperbolicity of ladders 107 4.1. Hyperbolicity of carpets 107 4.2. Hyperbolicity of carpeted ladders 108 4.3. Hyperbolicity of general ladders 116 Chapter 5. Hyperbolicity of flow-spaces 121 5.1. Ubiquity of ladders in F lk(Xu) 121 5.2. Projection of ladders 131 5.3. Hyperbolicity of tripods families 134 5.4. Hyperbolicity of flow-spaces 135 v vi CONTENTS Chapter 6. Hyperbolicity of trees of spaces: Putting everything together 139 6.1. Hyperbolicity of flow-spaces of special interval-spaces 139 6.2. Hyperbolicity of flow-spaces of general interval-spaces 152 6.3. Conclusion of the proof 153 Chapter 7. Description of geodesics 155 7.1. Inductive description 155 7.2. Characterization of vertical quasigeodesics 163 Chapter 8. Cannon-Thurston maps 165 8.1. Generalities of Cannon-Thurston maps 165 8.2. Cut-and-replace theorem 168 8.3. Part I: Consistency of points in vertex flow-spaces 171 8.4. Part II: Consistency in semispecial flow-spaces 181 8.5. Part III: The general case 188 8.6. The existence of CT-maps for subtrees of spaces 192 8.7. Fibers of CT-maps 192 8.8. Boundary flow 196 8.9. Cannon-Thurston lamination and ending laminations 200 8.10. Conical limit points in trees of hyperbolic spaces 205 8.11. Group-theoretic applications 206 Bibliography 213 List of symbols 217 Index 219 Preface The goal of this book is to understand geometry of metric spaces X which have structure of trees of hyperbolic spaces. The subject originates in the papers [BF92, BF96] of Bestvina and Feighn, where they proved a combination theorem, stating that under certain conditions such X itself is hyperbolic: Theorem. Suppose X = (π : X ! T ) is a tree of hyperbolic metric spaces, where vertex and edge-spaces are uniformly hyperbolic, incidence maps of edge spaces into vertex spaces are uniformly quasiisometric and which satisfies the hall- way flaring condition. Then X is a hyperbolic metric space. In chapter 2 we give definitions clarifying the result. Informally, the hallway flaring condition means that two K-quasiisometric sections of X over a geodesic γ in T \diverge at a uniform exponential rate" as we move along γ in one of the two directions. The original proof of this theorem was by verifying that X satisfies linear isoperimetric inequality. In the book we give a new (and longer) proof under weaker flaring assumption than the one made by Bestvina and Feighn; we name the weakened condition uniform (or, in another version, proper) flaring. Informally speaking, instead of requiring the exponential divergence of sections, we only require some rate of divergence, given by a uniform proper function of the arclength parameter of γ. We refer the reader to Theorem 2.47 for the precise statement. The main benefit of our proof is that it is done by constructing a slim combing of X: We find a family of (uniformly quasigeodesic) paths c(x; y) connecting pairs of points x; y in X, satisfying slim triangle property: Given three points x; y; z 2 X, one of the three paths c(x; y); c(y; z); c(z; x) is contained in a uniform neighbourhood of the union of the two other paths. The description of the paths c is a 6-step induction summarized in Chapter 7, starting with paths in the trees of spaces of the simplest kind that we call narrow carpets: These are metric interval-bundles over geodesics in T such that one of the interval-fibers has uniformly bounded length. The combing paths c in X are mostly concatenations of K-quasiisometric sec- tions of X over geodesics in T . Thus, we obtain (up to a uniformly bounded error) a description of geodesics in X in terms of its structure as a tree of spaces, i.e. vertex-spaces and sections. As an application of this description of geodesics, we prove (Theorem 8.47) the existence of Cannon-Thurston maps from Gromov-boundaries of subtrees of spaces Y ⊂ X to X, extending an earlier result by Mitra [Mit98c], who proved the existence of Cannon-Thurston maps for the inclusion maps of vertex-spaces into X. Mitra's proof (as well as the subsequent work of Mj and Sardar, [MS09]) was, in fact, a guideline for our description of geodesics in X. However, his description of vii viii PREFACE geodesics stopped at geodesics connecting points in the same vertex-space (step 3 of our 6-step description), leaving much of the work to be done in general. We also refer the reader to the related work of Gautero [Gau03, Gau16] and Gautero{Lusztig [GL04, GL07]. Organization of the book. In Chapter 1 we review basic facts of coarse geometry and geometry of hyperbolic spaces. While most of the material of the chapter is standard and well-known, we included it for the ease of reference in the rest of the book. In Chapter 2 we discuss definitions of the theory of trees of metric spaces, state and compare different flaring conditions in trees of spaces and formulate our main theorem. In Chapter 3 we define a certain class of subspaces Y in a tree of spaces X, called semicontinuous families. These subspaces (each of which also has structure of a tree of hyperbolic spaces Y) have the property that their intersections with vertex-spaces of X are uniformly quasiconvex and every point in Y is connected to the intersection Yu = Y \ Xu of Y with a distinguished vertex-space Xu, by a K-quasiisometric section of X over an interval in T . We prove that the subspaces Y are coarse Lipschitz retracts of X, which is a generalization of the horocyclic projections to a geodesic in the hyperbolic plane; its existence was first proven by Mitra, [Mit98c, Theorem 3.8] in the case of semicontinuous families called flow- spaces F lK (Xu). Flow-spaces and three other types of semicontinuous families (ladders, carpets and bundles) serve as key tools in our definition of combing paths c in X. Ladders are certain (semicontinuous) families of intervals over subtrees in X, where semicontinuity (informally) means that the lengths of the intervals can shrink substantially as we move away one edge from a vertex u (the center of the ladder). Bundles should be thought of as continuous families of quasiconvex subsets Qv of vertex-spaces of X with two (nonempty) vertex-spaces Qv;Qw uniformly Hausdorff- close, whenever v and w span an edge of T . Chapter 4 primarily deals with Steps 1{3 of our description of geodesics in X: We describe combing paths in carpets, carpeted ladders and general ladders and establish their hyperbolicity. Hyperbolicity of flow-spaces in proven in Chapter 5, which is technically the most difficult part of our work: We prove the slim triangle property for the combing paths c by analyzing triples of ladders (with the common center u) in the K-flow-space F lK (Xu) of a vertex-space Xu. Our last challenge is to connect by combing paths points in different vertex- flow-spaces F lK (Xu), F lK (Xv). This is done in Chapter 6. The case of points in intersecting flow-spaces F lK (Xu), F lK (Xv) is handled in section 6.1 where we primarily analyze the case of special intervals u; v , i.e. when F lK (Xu) \ Xv 6= ;. This covers Step 4 of our description of geodesicsJ K in X and is quite technical. The main trick is to introduce a certain generalization of flow-spaces of vertex- spaces and appeal to a special (and easy) case of Theorem 2.47 proven earlier, the quasiconvex amalgamation, when the tree T contains a single edge (Corollary 2.51). Once the case of special intervals is done, we complete easily Step 5 of our description of geodesics by considering points in flow-spaces F lK (XJ ) for subin- tervals J ⊂ T represented as unions of three special subintervals: For the proof we use quasiconvex amalgamation again. (A good example of such an interval J is given by a semispecial interval u; v , where the flow-spaces F lK (Xu), F lK (Xv) have nonempty intersection in XJ.) Lastly,K we conclude the 6-step description of PREFACE ix geodesics in X by appealing to the horizontal subdivision of geodesic intervals J in T , so that the consecutive subdivision vertices ui; ui+1 define pairwise uniformly cobounded flow-spaces F lK (Xui ), F lK (Xui+1 ) (their projections to the tree T are disjoint), while each interval ui; ui+1 between ui; ui+1 is a union of three special subintervals.
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