Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products,9780824703196

Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products

by ;
ISBN13: 9780824703196
ISBN10: 0824703197
Format: Hardcover
Pub. Date: 1999-07-27
Publisher(s): CRC Press
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Summary

A survey of one-relator products of cyclics or groups with a single defining relation, extending the algebraic study of Fuchsian groups to the more general context of one-relator products and related group theoretical considerations. It provides a self-contained account of certain natural generalizations of discrete groups.

Table of Contents

Preface v
Introduction: One-Relator Groups and Discrete Groups
1(4)
Preliminaries from Combinatorial Group Theory
5(34)
Preliminaries from Combinatorial Group Theory
5(1)
Free Groups and Free Products
5(6)
Group Amalgams
11(4)
Subgroup Theorems for Amalgams
15(1)
Bass-Serre Theory
16(2)
Nielsen Transformations
18(7)
Geometric Group Theory: Hyperbolic Groups
25(6)
Geometric Group Theory: Arboreal Group Theory
31(3)
Geometric Group Theory: Automatic Groups
34(5)
One-Relator Groups
39(38)
One-Relator Groups and the Freiheitssatz
39(1)
The Freiheitssatz and Magnus' Method
40(6)
Geometric Versions of the Freiheitssatz
46(3)
Subgroup Theory of One-Relator Groups
49(4)
Cyclically Pinched One-Relator Groups
53(6)
Conjugacy Pinched One-Relator Groups
59(5)
Fully Residually Free Groups and the Tarski Problem
64(6)
Small Cancellation Theory
70(4)
Some Further Results
74(3)
Discrete Groups
77(34)
Linear Fractional Transformations
77(2)
Discrete Groups
79(10)
Algebraic Analysis of Fuchsian Groups
89(14)
NEC Groups
103(3)
Classification of Two-Generator Discrete Groups in PSL2(R)
106(5)
One-Relator Products
111(26)
One-Relator Products
111(2)
The Freiheitssatz and Locally Indicable Groups
113(8)
The Freiheitssatz for High Powered Relators
121(9)
The Kervaire Conjecture and Klyachko's Solution
130(7)
One-Relator Products of Cyclics
137(22)
One-Relator Products of Cyclics
137(2)
Essential and Essentially Faithful Representations
139(5)
Essential Representations of One-Relator Products
144(5)
Freiheitssatz for One-Relator Amalgamated Products
149(3)
Faithful Representations of One-Relator Products
152(7)
Linear Properties of One-Relator Products of Cyclics
159(62)
Linear Properties of One-Relator Products of Cyclics
159(1)
The Tits Alternative and SQ-universality
160(6)
The Generalized Triangle Groups
166(42)
The Ordinary Triangle Groups
168(3)
The Tits Alternative - Generators of Finite Order
171(23)
The Tits Alternative - Generators of Infinite Order
194(7)
The Finite Generalized Triangle Groups
201(7)
The Virtual Torsion-Free Property
208(3)
Free Product with Amalgamation Decompositions
211(5)
Euler Characteristics
216(5)
Groups of F-Type
221(32)
Groups of F-type
221(2)
Freiheitssatz and Subgroup Theorems
223(8)
Linear Properties of Groups of F-Type
231(7)
Additional Results
238(10)
Decision Problems in One-Relator Products of Cyclics
248(5)
Related Generalizations of Discrete Groups
253(38)
Related Generalizations of Discrete Groups
253(1)
Groups of Special NEC Type
254(12)
The FHS and Essential Representations for Groups of SN-Type
255(3)
Linearity Results for Special NEC Groups
258(6)
Rank Conditions for Certain NEC Groups
264(2)
The Generalized Tetrahedron Groups
266(25)
Essential Representations
269(5)
Linearity Properties for Generalized Tetrahedron Groups
274(10)
A Freiheitssatz and the Virtual Torsion-Free Property
284(4)
Euler Characteristic for Generalized Tetrahedron Groups
288(1)
Finite Generalized Tetrahedron Groups
289(2)
Bibliography and References 291(18)
Subject Index 309(4)
Index of Names 313

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