## Combinatorial Group Theory: A Topological ApproachIn 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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### Contents

COMBINATORIAL GROUP THEORY | 1 |

SPACES AND THEIR PATHS | 49 |

GROUPOIDS | 62 |

THE FUNDAMENTAL GROUPOID AND THE FUNDAMENTAL | 74 |

COMPLEXES | 113 |

COVERINGS OF SPACES AND COMPLEXES | 151 |

BASSSERRE THEORY | 182 |

DECISION PROBLEMS | 243 |

FURTHER TOPICS | 286 |

297 | |

306 | |

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amalgamated free product automorphism bijection Britton's Lemma cancellation complex component conjugate construction contains Corollary corresponding coset countable cyclically reduced definition denote easy to check edge-pair element of G elementary reduction equivalence classes equivalence relation Exercise finitely presented group follows free group free product fundamental group G-tree given gives Godel numbering graph of groups group G group theory groupoid Hence HNN extension homomorphism identity inclusion indexed cell induces infinite integer inverse irreducible path isomorphism length Let F Let G Let H locally path-connected maximal tree monomorphism non-trivial obtained one-one partial algorithm particular path-connected permutation Plainly Proof Let Proposition prove pushout recursive reduced as written reduced form reduced word represents an element result sending sequence simply connected solvable word problem space stabilisers stable letters starting subgroup of G subset subspace Suppose Tietze transformations topology trivial unique van Kampen's Theorem vertex vertex group vertices