This book offers an alternative proof of the Bestvina-Feighn combination theorem for trees of hyperbolic spaces and describes uniform quasigeodesics in such spaces. As one of the applications of their description of uniform quasigeodesics, the authors prove the existence of Cannon-Thurston maps for inclusion maps of total spaces of subtrees of hyperbolic spaces and of relatively hyperbolic spaces. They also analyze the structure of Cannon-Thurston laminations in this setting. Furthermore, some group-theoretic applications of these results are discussed. This book also contains background material on coarse geometry and geometric group theory.
Bibliographic references
Includes bibliographical references and index.
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Contents
Intro
Contents
Preface
Acknowledgments
Chapter 1. Preliminaries on metric geometry
1.1. Graphs and trees
1.2. Coarse geometric concepts
1.3. Group actions
1.4. Length structures and spaces
1.5. Coarse Lipschitz maps and quasi-isometries
1.6. Coproducts, cones, and cylinders
1.7. Cones over metric spaces
1.8. Approximation of metric spaces by metric graphs
1.9. Hyperbolic metric spaces
1.10. Combings and a characterization of hyperbolic spaces
1.11. Hyperbolic cones
1.12. Geometry of hyperbolic triangles
1.13. Ideal boundaries
1.14. Quasiconvex subsets
1.15. Quasiconvex hulls
1.16. Projections
1.17. Images and pre-images of quasiconvex subsets under projections
1.18. Modified projection
1.19. Projections and coarse intersections
1.20. Quasiconvex subgroups and actions
1.21. Cobounded pairs of subsets
Chapter 2. Graphs of groups and trees of metric spaces
2.1. Generalities
2.2. Trees of spaces
2.3. Coarse retractions
2.4. Trees of hyperbolic spaces
2.5. Flaring
2.6. Hyperbolicity of trees of hyperbolic spaces
2.7. Flaring for semidirect products of groups
Chapter 3. Carpets, ladders, flow-spaces, metric bundles, and their retractions
3.1. Semicontinuous families of spaces
3.2. Ladders
3.3. Flow-spaces
3.4. Retractions to bundles
3.5. Examples
Chapter 4. Hyperbolicity of ladders
4.1. Hyperbolicity of carpets
4.2. Hyperbolicity of carpeted ladders
4.3. Hyperbolicity of general ladders
Chapter 5. Hyperbolicity of flow-spaces
5.1. Ubiquity of ladders in _{ }( ᵤ)
5.2. Projection of ladders
5.3. Hyperbolicity of tripods families
5.4. Hyperbolicity of flow-spaces
Chapter 6. Hyperbolicity of trees of spaces: Putting everything together
6.1. Hyperbolicity of flow-spaces of special interval-spaces.
6.2. Hyperbolicity of flow-spaces of general interval-spaces
6.3. Conclusion of the proof
Chapter 7. Description of geodesics
7.1. Inductive description
7.2. Characterization of vertical quasigeodesics
7.3. Visual boundary and geodesics in acylindrical trees of spaces
Chapter 8. Cannon-Thurston maps
8.1. Generalities on Cannon-Thurston maps
8.2. Cut-and-replace theorem
8.3. Part I: Consistency of points in vertex flow-spaces
8.4. Part II: Consistency in semispecial flow-spaces
8.5. Part III: Consistency in the general case
8.6. The existence of CT-maps for subtrees of spaces
8.7. Fibers of CT-maps
8.8. Boundary flows and CT laminations
8.9. Cannon-Thurston lamination and ending laminations
8.10. Conical limit points in trees of hyperbolic spaces
8.11. Group-theoretic applications
Chapter 9. Cannon-Thurston maps for relatively hyperbolic spaces
9.1. Relative hyperbolicity
9.2. Hyperbolicity of the electric space
9.3. Trees of relatively hyperbolic spaces
9.4. Cannon-Thurston maps for trees of relatively hyperbolic spaces
9.5. Cannon-Thurston laminations for trees of relatively hyperbolic spaces
Bibliography
List of symbols
Index.
ISBN
1-4704-7778-5
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