Combinatorial Group Theory

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Springer, Mar 12, 2015 - Mathematics - 339 pages
From the reviews:
"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 (items)). ...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

Free Groups and Their Subgroups 1 Introduction
1
Nielsens Method
4
Subgroups of Free Groups
13
Automorphisms of Free Groups
21
Stabilizers in AutF
43
Equations over Groups
49
Quadratic Sets of Word
58
Equations in Free Groups
64
Aspherical Groups
161
Coset Diagrams and Permutation Representations
163
Behr Graphs
170
Free Products and HNN Extensions
174
HigmanNeumannNeumann Extensions and Free Products with Amalgamation
178
Some Embedding Theorems
188
Some Decision Problems
192
OneRelator Groups
198

Abstract Length Functions
65
Representations of Free Groups the Fox Calculus
67
Free Products with Amalgamation
71
Generators and Relations
87
Finite Presentations
89
Fox Calculus Relation Matrices Connections with Cohomology
99
The ReidemeisterSchreier Method
102
Groups with a Single Defining Relator
104
Magnus Treatment of OneRelator Groups
111
Geometric Methods
114
Complexes
115
Covering Maps 18
118
Cayley Complexes
122
Planar Caley Complexes
124
FGroups Continued
130
Fuchsian Complexes
133
Planar Groups with Reflections
146
Singular Subcomplexes
149
Spherical Diagrams
156
Bipolar Structures
206
The Higman Embedding Theorem
214
Algebraically Closed Groups
227
Chapter W Small Cancellation Theory
235
The Small Cancellation Hypotheses
240
The Basic Formulas
242
Dehns Algorithm and Greendlingers Lemma
246
The Conjugacy Problem
252
The Word Problem
259
The Conjugacy Problem
262
Applications to Knot Groups
267
The Theory over Free Products
274
Small Cancellation Products
280
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
Copyright

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About the author (2015)

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 Théorique 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.

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