Geometry of Defining Relations in GroupsThe main feature of this book is a systematic application of elementary geometric and topological techniques for solving problems that arise naturally in algebra. After an account of preliminary material, there is a discussion of a geometrically intuitive interpretation of the derivation of consequences of defining relations of groups. A study is made of planar and certain other two-dimensional maps connected with well-known problems in general group theory, such as the problems of Burnside and O. Yu. Schmidt. The method of cancellation diagrams developed here is applied to these and to a series of other problems. This monograph is addressed to research workers and students in universities, and may be used as a basis for a series of specialized lectures or seminars. |
Contents
General concepts of group theory | 1 |
Main types of groups and subgroups | 37 |
Elements of twodimensional topology | 73 |
Surfaces and their cell decomposition | 85 |
10 Topological invariants of surfaces | 96 |
Consequences for graphs | 97 |
Orientable surfaces | 101 |
The fundamental group of a cell decomposition | 104 |
Further remarks | 134 |
13 Graded diagrams | 137 |
Grading maps and diagrams | 138 |
Compatible sections | 142 |
Asphericity of presentations | 146 |
Atoricity | 148 |
Amaps | 152 |
95 | 194 |
Computation of the fundamental groups of surfaces | 106 |
Diagrams over groups | 112 |
The concept of a diagram | 115 |
von Kampens lemma | 117 |
Annular diagrams subdiagrams | 119 |
0refinements of diagrams | 122 |
Cancellable pairs of cells | 124 |
12 Small cancellation theory | 126 |
Diagrams over small cancellation groups | 127 |
Dehns algorithm | 131 |
Golbergs example | 133 |
Maps with partitioned boundaries of cells | 218 |
Partitions of relators | 270 |
Construction of groups with prescribed properties | 296 |
Extensions of aspherical groups | 331 |
Presentations in free products | 364 |
Applications to other problems 4 10 | 410 |
Conjugacy relations | 447 |
Bibliography | 473 |
| 497 | |
Common terms and phrases
A-edges A-map a₁ abelian group arbitrary assume B-map boundary C-map called cell decomposition cell of rank circular diagram combinatorial commutator condition conjugacy conjugate in rank contains contiguity arcs Corollary coset cyclic group defining relations definition Dehn's algorithm denote distinguished contiguity submap E₁ edges elements equal example finite group free abelian group free group fundamental group G₁ graded group G Hence homomorphism homotopic induction inequality infinite involutions isomorphic j-pair Lemma length long section non-trivial normal subgroup obtain p-groups p₁ P₂ path period of rank periodic groups proof of Lemma q₁ quotient group R-cells rank i-1 reduced diagram references to Lemmas replaced satisfies section of rank section q sides simple in rank small cancellation smooth section subgroup of G subpath subset subword Theorem tion topological topological space V₁ verbal subgroup vertex vertices word y₁ П₁ П½