Notes on word hyperbolic groups.

*(English)*Zbl 0849.20023
Ghys, É. (ed.) et al., Group theory from a geometrical viewpoint. Proceedings of a workshop, held at the International Centre for Theoretical Physics in Trieste, Italy, 26 March to 6 April 1990. Singapore: World Scientific. 3-63 (1991).

The aim of these notes is to give an accessible introduction to the ideas of hyperbolic groups, accentuating the group theoretic approach.

The paper is divided up as follows: The first chapter consists of a collection of alternative definitions, both of hyperbolic metric spaces and of hyperbolic groups including Gromov’s inner product, slim and thin triangles, Cooper’s diverging geodesics, the linear isoperimetric inequality, and Dehn’s algorithm. The next chapter consists of proofs of the equivalence of these definitions of a hyperbolic metric space, and of a hyperbolic group. We also show here that the Dehn’s algorithm definition gives immediately that there is only a finite number of conjugacy classes of torsion elements, and also provides a time efficient algorithm for solving the word problem (a result originally due to Domanski and Anshel).

Some properties of quasigeodesics are developed in Chapter 3. These are used to establish the fact that the centralizer of an element of a hyperbolic group is cyclic-by-finite, and that thus there are no \(\mathbb{Z} \times \mathbb{Z}\) subgroups in a hyperbolic group. We define the boundary of a hyperbolic metric space in Chapter 4, though we do not make much use of the construction to establish properties of hyperbolic groups, as is done say in [E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’après M. Gromov (Prog. Math. 83, 1990; Zbl 0731.20025)]. We finally build the Rips complex to show that a hyperbolic group if \(FP_\infty\). This gives another proof that there is only a finite number of conjugacy classes of torsion elements.

For the entire collection see [Zbl 0809.00017].

The paper is divided up as follows: The first chapter consists of a collection of alternative definitions, both of hyperbolic metric spaces and of hyperbolic groups including Gromov’s inner product, slim and thin triangles, Cooper’s diverging geodesics, the linear isoperimetric inequality, and Dehn’s algorithm. The next chapter consists of proofs of the equivalence of these definitions of a hyperbolic metric space, and of a hyperbolic group. We also show here that the Dehn’s algorithm definition gives immediately that there is only a finite number of conjugacy classes of torsion elements, and also provides a time efficient algorithm for solving the word problem (a result originally due to Domanski and Anshel).

Some properties of quasigeodesics are developed in Chapter 3. These are used to establish the fact that the centralizer of an element of a hyperbolic group is cyclic-by-finite, and that thus there are no \(\mathbb{Z} \times \mathbb{Z}\) subgroups in a hyperbolic group. We define the boundary of a hyperbolic metric space in Chapter 4, though we do not make much use of the construction to establish properties of hyperbolic groups, as is done say in [E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’après M. Gromov (Prog. Math. 83, 1990; Zbl 0731.20025)]. We finally build the Rips complex to show that a hyperbolic group if \(FP_\infty\). This gives another proof that there is only a finite number of conjugacy classes of torsion elements.

For the entire collection see [Zbl 0809.00017].

##### MSC:

20F65 | Geometric group theory |

20F05 | Generators, relations, and presentations of groups |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

57M07 | Topological methods in group theory |

20F38 | Other groups related to topology or analysis |