Two-generator discrete subgroups of PSL (2, R) / Jane Gilman.

Format:

Book

Author/Creator:

Gilman, Jane, 1945- author.

Series:

Memoirs of the American Mathematical Society ; Volume 117, Number 561.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 117, Number 561

Language:

English

Subjects (All):

Fuchsian groups.

Kleinian groups.

Teichmüller spaces.

Physical Description:

1 online resource (221 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 1995.

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""I: Introduction""; ""1 Introduction""; ""1.1 Overview Intersecting Axes""; ""1.2 Overview of the intertwining cases""; ""1.3 Why an algorithm is needed""; ""2 The Acute Triangle Theorem""; ""2.1 Nielsen equivalence""; ""2.2 Idea of proof: Acute triangle theorem""; ""2.3 Labeling Conventions""; ""2.4 Ascending order conventions""; ""2.5 The Triangle Algorithm""; ""2.6 Q and the last triangle along A""; ""2.7 Combining triangle algorithm steps""; ""2.8 The sides and heights converge to 0""; ""2.9 Acute triangle theorem: proof""; ""3 Discreteness Theorem Proof Outline""

""3.1 The Discreteness Theorem""""3.2 Discreteness theorem""; ""3.3 Geometric equivalence theorems""; ""II: Preliminaries""; ""4 Triangle Groups and their Tilings""; ""4.1 Basic facts about triangle groups""; ""4.2 Minimal tiling distances""; ""4.3 The wedge at a vertex""; ""4.4 Proofs of lemmas and theorems""; ""4.5 Additional Notation""; ""4.6 Distances in the extended wedge""; ""5 Pentagons""; ""5.1 Constructing the pentagon, P[sub(A,B)]""; ""5.2 Notation""; ""5.3 Applying the Poincare Polygon Theorem""; ""5.4 Pentagon Tilings""; ""5.5 Distances in the shingling""

""8.4 Pentagon distances (2,3, n) t = 3 k = 3""; ""9 Nielsen Eq: (2,3, n) t = 3; k = 3""; ""9.1 Introduction""; ""9.2 Types of triples: distances""; ""9.3 Locating t[sub(2)] and t[sub(3)]""; ""10 Nielsen Eq: (2,4, n) t = 2; k = 2""; ""10.1 Introduction""; ""10.2 Types of triples""; ""10.3 Location of t[sub(1)],t[sub(2)] and t[sub(3)]""; ""11 Pentagon t = 9 & 2�2 Spectrum""; ""11.1 Step 1: Label the wedge""; ""11.2 Step 2: Double and Extend""; ""11.3 Step 3: Drop perpendiculars""; ""11.4 The two�two spectrum""; ""11.5 More distance computations""; ""11.6 Distances to q�[sub(0)]""

""11.7 Locate three order two points""""12 The Seven & Geometric Eq t = 9""; ""12.1 Introduction""; ""12.2 The variation of h and b""; ""12.3 Rule out a seven on the β side""; ""12.4 Rule out a seven on the D side""; ""12.5 Interior sevens""; ""12.6 Notation""; ""12.7 Geometric equivalence (2,3, 7) t = 9; k = 2""; ""13 Discreteness Theorem Proof""; ""13.1 The Proof of the Discreteness Theorem""; ""13.2 The proof of sufficiency""; ""IV: The Real Number Algorithm and the Turing Machine Algorithm""; ""14 Forms of the Algorithm""; ""14.1 What is an algorithm?""

""14.2 The Elliptic Order Algorithm""

Notes:

"September 1995, volume 117, number 561 (fourth of 5 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0140-1