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Fuchsian Groups (Chicago Lectures in Mathematics)

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Fuchsian Groups (Chicago Lectures in Mathematics) Cover

 

Synopses & Reviews

Publisher Comments:

This introductory text provides a thoroughly modern treatment of Fuchsian groups that addresses both the classical material and recent developments in the field. A basic example of lattices in semisimple groups, Fuchsian groups have extensive connections to the theory of a single complex variable, number theory, algebraic and differential geometry, topology, Lie theory, representation theory, and group theory.

Description:

Includes bibliographical references (p. 167-168) and index.

About the Author

Svetlana Katok is associate professor of mathematics at Pennsylvania State University.

Table of Contents

Preface

1. Hyperbolic geometry

1.1. The hyperbolic metric

1.2.and#160;Geodesics

1.3. Isometrics

1.4. Hyperbolic area and the Gauss-Bonnet formula

1.5. Hyperbolic trigonometry

1.6. Comparison between hyperbolic, spherical and Euclidean trigonometry

Exercises for Chapter 1

2. Fuchsian groups

2.1. The group PSL(2,R)

2.2. Discrete and properly discontinuous groups

2.3. Algebraic properties of Fuchsian groups

2.4. Elementary groups

Exercises for Chapter 2

3. Fundamental regions

3.1. Definition of a fundamental region

3.2. The Dirichlet region

3.3. Isometric circles and the Ford fundamental region

3.4. The limit set of [ ]

3.5. Structure of a Dirichlet region

3.6. Connection with Riemann surfaces and homogeneous spaces

Exercises for Chapter 3

4. Geometry of Fuchsian groups

4.1. Geometrically finite Fuchsian groups

4.2. Cocompact Fuchsian groups

4.3. The signature of a Fuchsian group

4.4. Fuchsian groups generated by reflections

4.5. Fuchsian groups of the first kind

4.6. Finitely generated Fuchsian groups

Exercises for Chapter 4

5. Arithmetic Fuchsian groups

5.1. Definitions of arithmetic Fuchsian groups

5.2. Fuchsian groups derived from quaternion algebras

5.3. Criteria for arithmeticity

5.4. Compactness of [ ] for Fuchsian groups derived from division quaternion algebras

5.5. The modular group and its subgroups

5.6. Examples

Exercises for Chapter 5

Hints for Selected Exercises

Bibliography

Index

Product Details

ISBN:
9780226425832
Author:
Katok, Svetlana
Publisher:
University Of Chicago Press
Location:
Chicago :
Subject:
Calculus
Subject:
Group Theory
Subject:
Riemann surfaces
Subject:
Fuchsian groups.
Subject:
Mathematics-Group Theory
Edition Description:
1
Series:
Chicago Lectures in Mathematics (Paperback)
Series Volume:
8
Publication Date:
19920801
Binding:
Paperback
Grade Level:
Professional and scholarly
Language:
English
Illustrations:
30 line drawings, 1 table
Pages:
186
Dimensions:
8 x 5.25 in

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Related Subjects


Science and Mathematics » Mathematics » Algebra » General
Science and Mathematics » Mathematics » Calculus » General
Science and Mathematics » Mathematics » Group Theory

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