Memoirs of the American Mathematical Society 1995; 204 pp; softcover Volume: 117 ISBN10: 0821803611 ISBN13: 9780821803615 List Price: US$50 Individual Members: US$30 Institutional Members: US$40 Order Code: MEMO/117/561
 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. Readership 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 \(t=9\) & 22 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 MatelskiBeardon count
 A summary of notation
