Topics in combinatorial group theory

Gilbert Baumslag9780817629212, 9783764329211, 0817629211, 3764329211

Combinatorial group theory is a loosely defined subject, with close connections to topology and logic. With surprising frequency, problems in a wide variety of disciplines, including differential equations, automorphic functions and geometry, have been distilled into explicit questions about groups, typically of the following kind: Are the groups in a given class finite (e.g., the Burnside problem)? Finitely generated? Finitely presented? What are the conjugates of a given element in a given group? What are the subgroups of that group? Is there an algorithm for deciding for every pair of groups in a given class whether they are isomorphic or not? The objective of combinatorial group theory is the systematic development of algebraic techniques to settle such questions. In view of the scope of the subject and the extraordinary variety of groups involved, it is not surprising that no really general theory exists. These notes, bridging the very beginning of the theory to new results and developments, are devoted to a number of topics in combinatorial group theory and serve as an introduction to the subject on the graduate level.

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Table of contents :
2. The beginnings……Page 8
3. Finitely presented groups……Page 10
4. More history……Page 12
5. Higman’s marvellous theorem……Page 16
6. Varieties of groups……Page 17
7. Small Cancellation Theory……Page 23
1. Introduction……Page 27
2. The Grigorchuk-Gupta-Sidki groups……Page 29
3. An application to associative algebras……Page 40
1. Frobenius’ representation……Page 42
2. Semidirect products……Page 47
3. Subgroups of free groups are free……Page 52
4. The calculus of presentations……Page 64
5. The calculus of presentations (continued)……Page 69
6. The Reidemeister-Schreier method……Page 77
7. Generalized free products……Page 80
1. Recursively presented groups……Page 83
2. Some word problems……Page 86
3. Groups with free subgroups……Page 87
1. Background……Page 100
2. Some basic algebraic geometry……Page 101
3. More basic algebraic geometry……Page 106
4. Useful notions from topology……Page 108
5. Morphisms……Page 112
6. Dimension……Page 119
7. Representations of the free group of rank two in SL(2,C)……Page 123
8. Affine algebraic sets of characters……Page 129
1. Applications……Page 135
2. Back to basics……Page 139
3. More applications……Page 144
4. Some word, conjugacy and isomorphism problems……Page 154
1. Basic definitions……Page 158
2. Covering space theory……Page 166
3. Graphs of groups……Page 168
4. Trees……Page 172
5. The fundamental group of a graph of groups……Page 174
6. The fundamental group of a graph of groups (continued)……Page 178
7. Group actions and graphs of groups……Page 184
8. Universal covers……Page 188
9. The proof of Theorem 2……Page 192
10. Some consequences of Theorems 2 and 3……Page 193
11. The tree of SL₂……Page 197