Description

Book Synopsis
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.

Table of Contents
  • Preliminaries: Algebraic hyperbolic spaces
  • $\mathbb{R}$-trees, CAT(-1) spaces, and Gromov hyperbolic metric spaces
  • More about the geometry of hyperbolic metric spaces
  • Discreteness
  • Classification of isometries and semigroups
  • Limit sets
  • The Bishop-Jones theorem: The modified Poincare exponent
  • Generalization of the Bishop-Jones theorem
  • Examples: Schottky products
  • Parabolic groups
  • Geometrically finite and convex-cobounded groups
  • Counterexamples
  • $\mathbb{R}$-trees and their isometry groups
  • Patterson-Sullivan theory: Conformal and quasiconformal measures
  • Patterson-Sullivan theorem for groups of divergence type
  • Quasiconformal measures of geometrically finite groups
  • Open problems
  • Index of defined terms
  • Bibliography

    Geometry and Dynamics in Gromov Hyperbolic Metri

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      Order before 4pm today for delivery by Thu 25 Jun 2026.

      A Hardback by Tushar Das, David Simmons, Mariusz Urbanski

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        Publisher: MP-AMM American Mathematical
        Publication Date: 6/30/2017 12:00:00 AM
        ISBN13: 9781470434656, 978-1470434656
        ISBN10: 1470434652

        Description

        Book Synopsis
        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.

        Table of Contents
        • Preliminaries: Algebraic hyperbolic spaces
        • $\mathbb{R}$-trees, CAT(-1) spaces, and Gromov hyperbolic metric spaces
        • More about the geometry of hyperbolic metric spaces
        • Discreteness
        • Classification of isometries and semigroups
        • Limit sets
        • The Bishop-Jones theorem: The modified Poincare exponent
        • Generalization of the Bishop-Jones theorem
        • Examples: Schottky products
        • Parabolic groups
        • Geometrically finite and convex-cobounded groups
        • Counterexamples
        • $\mathbb{R}$-trees and their isometry groups
        • Patterson-Sullivan theory: Conformal and quasiconformal measures
        • Patterson-Sullivan theorem for groups of divergence type
        • Quasiconformal measures of geometrically finite groups
        • Open problems
        • Index of defined terms
        • Bibliography

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