Fuchsian Groups

Fuchsian Groups

by Svetlana Katok
     
 

ISBN-10: 0226425835

ISBN-13: 9780226425832

Pub. Date: 08/28/1992

Publisher: University of Chicago Press

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

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Overview

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.

Product Details

ISBN-13:
9780226425832
Publisher:
University of Chicago Press
Publication date:
08/28/1992
Series:
Chicago Lectures in Mathematics Series
Edition description:
1
Pages:
186
Product dimensions:
5.25(w) x 8.00(h) x 0.60(d)

Related Subjects

Table of Contents

Preface
1Hyperbolic geometry1
1.1The hyperbolic metric1
1.2Geodesics4
1.3Isometries8
1.4Hyperbolic area and the Gauss-Bonnet formula11
1.5Hyperbolic trigonometry15
1.6Comparison between hyperbolic, spherical and Euclidean trigonometry18
2Fuchsian groups23
2.1The group PSL(2,R)23
2.2Discrete and properly discontinuous groups26
2.3Algebraic properties of Fuchsian groups34
2.4Elementary groups37
3Fundamental regions49
3.1Definition of a fundamental region49
3.2The Dirichlet region52
3.3Isometric circles and the Ford fundamental region56
3.4The limit set of [Gamma]63
3.5Structure of a Dirichlet region67
3.6Connection with Riemann surfaces and homogeneous spaces75
4Geometry of Fuchsian groups80
4.1Geometrically finite Fuchsian groups80
4.2Cocompact Fuchsian groups84
4.3The signature of a Fuchsian group90
4.4Fuchsian groups generated by reflections99
4.5Fuchsian groups of the first kind102
4.6Finitely generated Fuchsian groups104
5Arithmetic Fuchsian groups112
5.1Definitions of arithmetic Fuchsian groups112
5.2Fuchsian groups derived from quaternion algebras113
5.3Criteria for arithmeticity120
5.4Compactness of [actual symbol not reproducible] for Fuchsian groups derived from division quaternion algebras129
5.5The modular group and its subgroups133
Hints for Selected Excercises155
Bibliography167
Index169

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