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**Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems.**
*(English)*
Zbl 1093.20025

Mem. Am. Math. Soc. 843, 100 p. (2006).

Originally, the notion of a relatively hyperbolic group was proposed by M. Gromov [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] in order to generalize various examples of algebraic and geometric nature of fundamental groups of finite-volume non-compact Riemannian manifolds of pinched negative curvature. Gromov’s idea has been elaborated by B. H. Bowditch in his preprint [Relatively hyperbolic groups (Univ. Southampton) (1998)]. In 1994 B. Farb, in his PhD Univ. Princeton thesis [published as article in Geom. Funct. Anal. 8, No. 5, 810-840 (1998; Zbl 0985.20027)], proposed an alternative approach. We refer the reader to the article by I. Bumagin [in Contemp. Math. 372, 189-196 (2005; Zbl 1091.20029)] for a discussion of the differences between these two points of view.

In this hundred pages long article, the author presents a different characterization of relative hyperbolicity in terms of isoperimetric inequalities and adobts techniques based on van Kampen diagrams to the study of algebraic and algorithmic properties of relatively hyperbolic groups. We can say that the language of the article is close to topology/combinatorial group theory. In the paper are considered: relative Dehn functions, algebraic properties of relatively hyperbolic groups (elements of finite order, cyclic subgroups and translation numbers), the word problem and algorithmic problems for hyperbolic elements.

The author mentions, that the article allows him to establish the background for the subsequent paper “Small cancellations over relatively hyperbolic groups and embedding theorems” where he uses relative hyperbolicity to prove embedding theorems for countable groups and to construct groups with certain “exotic” properties. For instance, he constructs first examples of finitely generated groups other than \(\mathbb{Z}_2\) with exactly 2 conjugacy classes.

In this hundred pages long article, the author presents a different characterization of relative hyperbolicity in terms of isoperimetric inequalities and adobts techniques based on van Kampen diagrams to the study of algebraic and algorithmic properties of relatively hyperbolic groups. We can say that the language of the article is close to topology/combinatorial group theory. In the paper are considered: relative Dehn functions, algebraic properties of relatively hyperbolic groups (elements of finite order, cyclic subgroups and translation numbers), the word problem and algorithmic problems for hyperbolic elements.

The author mentions, that the article allows him to establish the background for the subsequent paper “Small cancellations over relatively hyperbolic groups and embedding theorems” where he uses relative hyperbolicity to prove embedding theorems for countable groups and to construct groups with certain “exotic” properties. For instance, he constructs first examples of finitely generated groups other than \(\mathbb{Z}_2\) with exactly 2 conjugacy classes.

Reviewer: Andrzej Szczepański (Gdańsk)

### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

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

20F06 | Cancellation theory of groups; application of van Kampen diagrams |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

20F65 | Geometric group theory |

20F69 | Asymptotic properties of groups |

57M05 | Fundamental group, presentations, free differential calculus |

57M07 | Topological methods in group theory |