Hardcover ISBN:  9781470434656 
Product Code:  SURV/218 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470440480 
Product Code:  SURV/218.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470434656 
eBook: ISBN:  9781470440480 
Product Code:  SURV/218.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 
Hardcover ISBN:  9781470434656 
Product Code:  SURV/218 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470440480 
Product Code:  SURV/218.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470434656 
eBook ISBN:  9781470440480 
Product Code:  SURV/218.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

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 finitedimensional 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 Gromovhyperbolic spaces.

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 Poincaré exponent

Generalization of the Bishop–Jones theorem

Examples

Schottky products

Parabolic groups

Geometrically finite and convexcobounded 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


Additional Material

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
 Table of Contents
 Additional Material
 Requests
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 finitedimensional 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 Gromovhyperbolic 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 convexcobounded 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