This book 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 behavior 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. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems.
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Sprache
Verlagsort
Zielgruppe
Maße
Höhe: 254 mm
Breite: 178 mm
Gewicht
ISBN-13
978-1-4704-3465-6 (9781470434656)
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Schweitzer Klassifikation
Tushar Das, University of Wisconsin, La Crosse, WI.
David Simmons, University of York, United Kingdom.
Mariusz Urbanski, University of North Texas, Denton, TX.
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
