Combinatorial Group Theory

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Springer Science & Business Media, Jan 12, 2001 - Mathematics - 339 pages
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"This book (...) defines the boundaries of the subject now called combinatorial group theory. (...)it is a considerable achievement to have concentrated a survey of the subject into 339 pages. This includes a substantial and useful bibliography; (over 1100 ÄitemsÜ). ...the book is a valuable and welcome addition to the literature, containing many results not previously available in a book. It will undoubtedly become a standard reference." Mathematical Reviews, AMS, 1979
 

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Contents

Chapter I Free Groups and Their Subgroups
1
2 Nielsens Method
4
3 Subgroups of Free Groups
13
4 Automorphisms of Free Groups
21
5 Stabilizersin AutF
43
6 Equations over Groups
49
7 Quadratic Sets of Word
58
8 Equations in Free Groups
64
11 Aspherical Groups
161
12 Coset Diagrams and Permutation Representations
163
13 Behr Graphs
170
Chapter IV Free Products and HNN Extensions
174
2 HigmanNeumannNeumann Extensions and Free Products with Amalgmation
178
3 Some Embedding Theorems
188
4 Some Decision Problems
192
5 OneRelator Groups
198

9 Abstract Length Functions
65
10 Representations of Free Groups the Fox Calculus
67
11 Free Products with Amalgamation
71
Chapter II Generators and Relations
87
2 Finite Presentations
89
3 Fox Calculus Relation Matrices Connections with Cohomology
99
4 The ReidemeisterSchreier Method
102
5 Groups with a Single Defining Relator
104
6 Magnus Treatment of OneRelator Groups
111
Chapter III Geometric Methods
114
2 Complexes
115
3 Covering Maps
118
4 Cayley Complexes
122
5 Planar Cayley Complexes
124
6 FGroups Continued
130
7 Fuchsian Complexes
133
8 Planar Groups with Reflections
146
9 Singular Subcomplexes
149
10 Spherical Diagrams
156
6 Bipolar Structures
206
7 The Higman Embedding Theorem
214
8 Algebraically Closed Groups
227
Chapter V Small Cancellation Theory
235
2 The Small Cancellation Hypotheses
240
3 The Basic Formulas
242
4 Dehas Algorithm and Greendlingefs Lemma
246
5 The Conjugacy Problem
252
6 The Word Problem
259
7 The Conjugacy Problem
262
8 Applications to Knot Groups
267
9 The Theory over Free Products
274
10 Small Cancellation Products
280
11 Small Cancellation Theory Over Free Products with Amalgamation and HNN Extensions
285
Bibliography
295
Russian Names in Cyrillic
332
Index of Names
333
Subject Index
336
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About the author (2001)

Biography of Roger C. Lyndon

Roger Lyndon, born on Dec. 18, 1917 in Calais (Maine, USA), entered Harvard University in 1935 with the aim of studying literature and becoming a writer. However, when he discovered that, for him, mathematics required less effort than literature, he switched and graduated from Harvard in 1939.

After completing his Master's Degree in 1941, he taught at Georgia Tech, then returned to Harvard in 1942 and there taught navigation to pilots while, supervised by S. MacLane, he studied for his Ph.D., awarded in 1946 for a thesis entitled The Cohomology Theory of Group Extensions.

Influenced by Tarski, Lyndon was later to work on model theory. Accepting a position at Princeton, Ralph Fox and Reidemeister's visit in 1948 were major influencea on him to work in combinatorial group theory. In 1953 Lyndon left Princeton for a chair at the University of Michigan where he then remained except for visiting professorships at Berkeley, London, Montpellier and Amiens.

Lyndon made numerous major contributions to combinatorial group theory. These included the development of "small cancellation theory," his introduction of "aspherical" presentations of groups and his work on length functions. He died on June 8, 1988.

Biography of Paul E. Schupp

Paul Schupp, born on March 12, 1937 in Cleveland, Ohio was a student of Roger Lyndon's at the Univ. of Michigan. Where he wrote a thesis on "Dehn's Algorithm and the Conjugacy Problem." After a year at the University of Wisconsin he moved to the University of Illinois where he remained. For several years he was also concurrently Visiting Professor at the University Paris VII and a member of the Laboratoire d'Informatique Theorique et Programmation (founded by M. P. Schutzenberger).

Schupp further developed the use of cancellation diagrams in combinatorial group theory, introducing conjugacy diagrams, diagrams on compact surfaces, diagrams over free products with amalgamation and HNN extensions and applications to Artin groups. He then worked with David Muller on connections between group theory and formal language theory and on the theory of finite automata on infinite inputs. His current interest is using geometric methods to investigate the computational complexity of algorithms in combinatorial group theory.