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Book DetailsMathematical Surveys and MonographsVolume: 218; 2017; 281 ppMSC: Primary 20; 28; 37; Secondary 22
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.
ReadershipGraduate students and researchers interested in geometric group theory and Gromov-hyperbolic spaces.
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Table of Contents
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Preliminaries
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Algebraic hyperbolic spaces
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$\mathbb {R}$-trees, CAT(-1) spaces, and Gromov hyperbolic metric spaces
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More about the geometry of hyperbolic metric spaces
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Discreteness
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Classification of isometries and semigroups
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Limit sets
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The Bishop–Jones theorem
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The modified Poincaré exponent
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Generalization of the Bishop–Jones theorem
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Examples
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Schottky products
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Parabolic groups
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Geometrically finite and convex-cobounded groups
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Counterexamples
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$\mathbb {R}$-trees and their isometry groups
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Patterson–Sullivan theory
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Conformal and quasiconformal measures
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Patterson–Sullivan theorem for groups of divergence type
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Quasiconformal measures of geometrically finite groups
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Open problems
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Index of defined terms
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Additional Material
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RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
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- Additional Material
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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.
Graduate students and researchers interested in geometric group theory and Gromov-hyperbolic spaces.
-
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 Poincaré 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
