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

Memoirs of the American Mathematical Society
1995; 204 pp; softcover
Volume: 117
ISBN-10: 0-8218-0361-1
ISBN-13: 978-0-8218-0361-5
List Price: US$50
Individual Members: US$30
Institutional Members: US$40
Order Code: MEMO/117/561
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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.


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

Table of Contents

  • I. Introduction
  • The acute triangle theorem
  • Discreteness theorem proof outline
  • II. Preliminaries
  • Triangle groups and their tilings
  • Pentagons
  • Hyperbolic formulae & geometry
  • Extending Knapp & Poincaré
  • III. Geometric equivalence and the discreteness theorem
  • The standard acute triangles
  • Nielsen eq: \((2,3,n)t=3; k=3\)
  • Nielsen eq: \((2,4,n)t=2; k=2\)
  • Pentagon \(t=9\) & 2-2 spectrum
  • The seven & geometric eq \(t=9\)
  • Discreteness theorem proof
  • IV. The real number algorithm and the Turing machine algorithm
  • Forms of the algorithm
  • V. Appendix
  • Verify Matelski-Beardon count
  • A summary of notation
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