Combinatorial group theory: A topological approach

Daniel E. Cohen0521341337, 9780521349369, 9780521341332, 0521349362

In this book, developed from courses taught at the University of London, the author aims to show the value of using topological methods in combinatorial group theory. The topological material is given in terms of the fundamental groupoid, giving results and proofs that are both stronger and simpler than the traditional ones. Several chapters deal with covering spaces and complexes, an important method, which is then applied to yield the major Schreier and Kurosh subgroup theorems. The author presents a full account of Bass-Serre theory and discusses the word problem, in particular, its unsolvability and the Higman Embedding Theorem. Included for completeness are the relevant results of computability theory.

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Table of contents :
Table of Contents……Page 6
Introduction……Page 3
1.1 Free groups……Page 8
1.2 Generators and relators……Page 24
1.3 Free products……Page 29
1.4 Pushouts and amalgamated free products……Page 34
1.5 HNN extensions……Page 39
2.1 Some point-set topology……Page 54
2.2 Paths and homotopies……Page 59
3.1 Groupoids……Page 67
3.2 Direct limits……Page 75
4.1 The fundamental groupoid and the fundamental group……Page 79
4.2 Van Kampen’s theorem……Page 88
4.3 Covering spaces……Page 101
4.4 The circle and the complex plane……Page 106
4.5 Joins and weak joins……Page 112
5.1 Graphs……Page 118
5.2 Complexes and their fundamental groups……Page 126
5.3 Free groups and their automorphisms……Page 137
5.4 Coverings of complexes……Page 144
5.5 Subdivisions……Page 151
5.6 Geometric realisations……Page 154
6. Coverings of Spaces and Complexes……Page 156
7. Coverings and Group Theory……Page 172
8.1 Trees and free groups……Page 187
8.2 Nielsen’s method……Page 193
8.3 Graphs of groups……Page 203
8.4 The structure theorems……Page 209
8.5 Applications of the structure theorems……Page 216
8.6 Construction of trees……Page 238
9.1 Decision problems in general……Page 248
9.2 Some easy decision problems in groups……Page 256
9.3 The word problem……Page 259
9.4 Modular machines and unsolvable word problems……Page 270
9.5 Some other unsolvable problems……Page 273
9.6 Higman’s embedding theorem……Page 279
9.7 Groups with one relator……Page 286
10.1 Small cancellation theory……Page 291
10.2 Other topics……Page 294
Notes and References……Page 296
Bibliography……Page 302
Index……Page 311