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Geometry and dynamics in Gromov hyperbolic metric spaces : with an emphasis on non-proper settings / Tushar Das, David Simmons and Mariusz Urbanski.

By: Contributor(s): Material type: TextTextSeries: Mathematical surveys and monographs ; v 218.Publication details: Providence : American Mathematical Society, ©2017.Description: xxxv, 281 pages : illustrations ; 27 cmISBN:
  • 9781470434656 (alk. paper)
Subject(s): DDC classification:
  • 510MS 23 Am512
Contents:
1. Introduction and overview -- Part 1. Preliminaries. 2. Algebraic hyperbolic spaces -- 3. R-trees, CAT( -1) spaces, and Gromov hyperbolic metric spaces -- 4. More about the geometry of hyperbolic metric spaces -- 5. Discreteness -- 6. Classification of isometries and semigroups -- 7. Limit sets -- Part 2. The Bishop-Jones theorem. 8. The modified Poincare exponent -- 9. Generalization of the Bishop-Jones theorem -- Part 3. Examples. 10. Schottky products -- 11. Parabolic groups -- 12.Geometrically finite and convex-cobounded groups -- 13. Counterexamples -- 14. R-trees and their isometry groups -- Part 4. Patterson-Sullivan theory. 15. Conformal and quasiconformal measures -- 16. Patterson-Sullivan theorem for groups of divergence type -- 17. Quasiconformal measures of geometrically finite groups.
Summary: Presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behaviour not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory.
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Includes bibliographical references and index.

1. Introduction and overview --
Part 1. Preliminaries.
2. Algebraic hyperbolic spaces --
3. R-trees, CAT( -1) spaces, and Gromov hyperbolic metric spaces --
4. More about the geometry of hyperbolic metric spaces --
5. Discreteness --
6. Classification of isometries and semigroups --
7. Limit sets --
Part 2. The Bishop-Jones theorem.
8. The modified Poincare exponent --
9. Generalization of the Bishop-Jones theorem --
Part 3. Examples.
10. Schottky products --
11. Parabolic groups --
12.Geometrically finite and convex-cobounded groups --
13. Counterexamples --
14. R-trees and their isometry groups --
Part 4. Patterson-Sullivan theory.
15. Conformal and quasiconformal measures --
16. Patterson-Sullivan theorem for groups of divergence type --
17. Quasiconformal measures of geometrically finite groups.

Presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behaviour not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory.

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