**Memoirs of the American Mathematical Society**

1995;
204 pp;
Softcover

MSC: Primary 30; 32; 20;

Print ISBN: 978-0-8218-0361-5

Product Code: MEMO/117/561

List Price: $50.00

AMS Member Price: $30.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-0140-5
Product Code: MEMO/117/561.E**

List Price: $50.00

AMS Member Price: $30.00

MAA Member Price: $45.00

# Two-Generator Discrete Subgoups of \(PSL(2, R)\)

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*Jane Gilman*

The discreteness problem is the problem of determining whether or not a two-generator subgroup of \(PSL(2, R)\) is discrete. Historically, papers on this old and subtle problem have been known for their errors and omissions. This book presents the first complete geometric solution to the discreteness problem by building upon cases previously presented by Gilman and Maskit and by developing a theory of triangle group shinglings/tilings of the hyperbolic plane and a theory explaining why the solution must take the form of an algorithm. This work is a thoroughly readable exposition that captures the beauty of the interplay between the algebra and the geometry of the solution.

#### Readership

Researchers working in Kleinian groups, Teichmüller theory or hyperbolic geometry.

#### Table of Contents

# Table of Contents

## Two-Generator Discrete Subgoups of $PSL(2, R)$

- Contents v6 free
- I: Introduction 112 free
- 1 Introduction 314 free
- 2 The Acute Triangle Theorem 1324
- 2.1 Nielsen equivalence 1324
- 2.2 Idea of proof: Acute triangle theorem 1930
- 2.3 Labeling Conventions 2031
- 2.4 Ascending order conventions 2132
- 2.5 The Triangle Algorithm 2132
- 2.6 Q and the last triangle along A 2536
- 2.7 Combining triangle algorithm steps 2839
- 2.8 The sides and heights converge to 0 3142
- 2.9 Acute triangle theorem: proof 3142

- 3 Discreteness Theorem Proof Outline 3344

- II: Preliminaries 3950
- III: Geometric Equivalence and the Discreteness Theorem 95106
- IV: The Real Number Algorithm and the Turing Machine Algorithm 165176
- V: Appendix 193204