eBook ISBN:  9781470401405 
Product Code:  MEMO/117/561.E 
List Price:  $50.00 
MAA Member Price:  $45.00 
AMS Member Price:  $30.00 
eBook ISBN:  9781470401405 
Product Code:  MEMO/117/561.E 
List Price:  $50.00 
MAA Member Price:  $45.00 
AMS Member Price:  $30.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 117; 1995; 204 ppMSC: Primary 30; 32; 20
The discreteness problem is the problem of determining whether or not a twogenerator 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.
ReadershipResearchers working in Kleinian groups, Teichmüller theory or hyperbolic geometry.

Table of Contents

Chapters

I. Introduction

1. Introduction

2. The triangle algorithm and the acute triangle theorem

3. The discreteness theorem

II. Preliminaries

4. Triangle groups and their tilings

5. Pentagons

6. A summary of formulas for the hyperbolic trigonometric functions and some geometric corollaries

7. The Poincaré polygon theorem and its partial converse; Knapp’s theorem and its extension

III. Geometric equivalence and the discreteness theorem

8. Constructing the standard acute triangles and standard generators

9. Generators and Nielsen equivalence for the $(2,3, n)t = 3$; $k = 3$ case

10. Generators and Nielsen equivalence for the $(2,4, n)t = 2$; $k = 2$ case

11. Constructing the standard $(2,3,7)k=2;\ t=9$ pentagon: Calculating the 2–2 spectrum

12. Finding the other seven and proving geometric equivalence

13. The proof of the discreteness theorem

IV. The real number algorithm and the Turing machine algorithm

14. Forms of the algorithm

V. Appendix

Appendix A. Verify MatelskiBeardon count

Appendix B. A summary of notation


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The discreteness problem is the problem of determining whether or not a twogenerator 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.
Researchers working in Kleinian groups, Teichmüller theory or hyperbolic geometry.

Chapters

I. Introduction

1. Introduction

2. The triangle algorithm and the acute triangle theorem

3. The discreteness theorem

II. Preliminaries

4. Triangle groups and their tilings

5. Pentagons

6. A summary of formulas for the hyperbolic trigonometric functions and some geometric corollaries

7. The Poincaré polygon theorem and its partial converse; Knapp’s theorem and its extension

III. Geometric equivalence and the discreteness theorem

8. Constructing the standard acute triangles and standard generators

9. Generators and Nielsen equivalence for the $(2,3, n)t = 3$; $k = 3$ case

10. Generators and Nielsen equivalence for the $(2,4, n)t = 2$; $k = 2$ case

11. Constructing the standard $(2,3,7)k=2;\ t=9$ pentagon: Calculating the 2–2 spectrum

12. Finding the other seven and proving geometric equivalence

13. The proof of the discreteness theorem

IV. The real number algorithm and the Turing machine algorithm

14. Forms of the algorithm

V. Appendix

Appendix A. Verify MatelskiBeardon count

Appendix B. A summary of notation