## Topics in Geometric Group TheoryIn this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A recognized expert in the field, de la Harpe adopts a hands-on approach, illustrating key concepts with numerous concrete examples. The first five chapters present basic combinatorial and geometric group theory in a unique and refreshing way, with an emphasis on finitely generated versus finitely presented groups. In the final three chapters, de la Harpe discusses new material on the growth of groups, including a detailed treatment of the "Grigorchuk group." Most sections are followed by exercises and a list of problems and complements, enhancing the book's value for students; problems range from slightly more difficult exercises to open research problems in the field. An extensive list of references directs readers to more advanced results as well as connections with other fields. |

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### Contents

Introduction | 1 |

Free products and free groups | 17 |

HI Finitelygenerated groups | 43 |

Finitelygenerated groups viewed as metric spaces | 75 |

Finitelypresented groups | 117 |

Growth of finitelygenerated groups | 151 |

Groups of exponential or polynomial growth | 187 |

The first Grigorchuk group | 211 |

265 | |

295 | |

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### Common terms and phrases

abelian group algebra automorphism Baumslag-Solitar groups Cayley graph Chapter Check commensurable compact Complement compute Consider contains Corollary countable curvature defined Definition denote element example Exercise exists exponential growth F is finitely F x F finite group finite index finite set finitely presented finitely-generated group finitely-presented group follows free group free product free subgroups fundamental group genus g geometric Grigorchuk Gromov group F group of order group of rank growth function growth series homomorphism hyperbolic groups infinite integer isometries isomorphic Item lattice Lemma Let F Lie group linear mapping Math metric space modular group nilpotent non-abelian free normal subgroup Observe particular polynomial growth proof properties Proposition quasi-convex quasi-isometric quotient random walk reduced word residually finite result Riemannian manifold Section simple groups solvable spherical growth subgroup of F subgroup of finite surface of genus Theorem topology torsion-free uncountably uniformly exponential growth vertex vertices word length

### Popular passages

Page 278 - G. HIGMAN, Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27(1952), 73-81. 12. ST Hu, "Homotopy Theory," Academic Press, New York/London, 1959. 13. AG KUROSH, "The Theory of Groups,

Page 280 - On everywhere dense imbedding of free groups in Lie groups, Nagoya Math.

Page 278 - Subgroups of finitely presented groups. Proc. Royal Soc. London Ser. A 262, 455-475 (1961).

Page 277 - P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. 4 (1954), 419-436.

Page 272 - Geodesic automation and growth functions for Artin groups of finite type, Math. Ann. 301 (1995), 307-324.