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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)
MathSciNet review: 1290281
MSC: Primary 20H10; Secondary 22E40, 30F35

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Table of Contents


  • 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