# Two-generator discrete subgroups of ${\rm PSL}(2,{\bf R})$

### About this Title

**Jane Gilman**

Publication: Memoirs of the American Mathematical Society

Publication Year
1995: Volume 117, Number 561

ISBNs: 978-0-8218-0361-5 (print); 978-1-4704-0140-5 (online)

DOI: http://dx.doi.org/10.1090/memo/0561

MathSciNet review: 1290281

MSC: Primary 20H10; Secondary 22E40, 30F35

### 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 Matelski-Beardon count
- Appendix B. A summary of notation