## Combinatorial Group TheoryFrom 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 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 |

336 | |

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### Common terms and phrases

abelian algebraically closed group algorithm Amer Aut(F automorphism Baumslag boundary cycle boundary label Cayley complex combinatorial conjugacy problem conjugate contains cosets cyclic groups cyclic words cyclically reduced Dehn's algorithm diagram edge element of G elementary embedded equations F-group factor finite groups finite index finitely generated subgroup finitely presented group follows free group free product Fuchsian groups fundamental group group G hence Higman HNN extension homomorphism hypothesis implies induction infinite integer isomorphic Karrass Lemma length Let F Let G London Math.Soc loop Lyndon Magnus Math matrix minimal modulo Neumann non-trivial element normal closure normal form normal subgroup obtained one-relator groups path presentation G Proc product with amalgamation proof Proposition quotient group rank recursively enumerable reduced form reduced word region residually finite residually finite groups result satisfies sequence Solitar solvable word problem subgroup of G subset subword suppose theorem trivial vertex vertices whence Zieschang