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**Structure and rigidity in hyperbolic groups. I.**
*(English)*
Zbl 0818.20042

This paper is on hyperbolic groups in the sense of Gromov. The authors introduce the notion of a Dehn twist automorphism of a group given as an amalgamated product or HNN extension. They call an automorphism obtained by a sequence of Dehn twists and inner automorphisms an internal automorphism. They prove that for a torsion-free hyperbolic group, the group of internal automorphisms is of finite index. They give a proof of a theorem of Gromov which asserts that if \(\Gamma\) is a hyperbolic group and \(\Gamma_ 1\) a finitely presented, torsion-free, freely indecomposable non-cyclic subgroup, then \(\Gamma\) contains at most finitely many conjugacy classes of subgroups isomorphic to \(\Gamma_ 1\). They obtain also the following results: If \(\Gamma\) is a hyperbolic group and \(\Gamma_ 1\) a finitely presented torsion-free, freely indecomposable subgroup which has a finitely generated normalizer in \(\Gamma\), then \(\operatorname{Aut} (\Gamma_ 1)\) is finitely generated. The automorphism group of a torsion-free hyperbolic group is finitely generated.

The authors introduce “weakly rigid groups”, a notion which they say will be important in the second author’s approach to the study of the automorphism group of a free group. The paper has a short appendix which summarizes results (mainly due to the first author and to Bestvina and Feighn) on the theory of group actions on trees which is used in the paper.

The authors introduce “weakly rigid groups”, a notion which they say will be important in the second author’s approach to the study of the automorphism group of a free group. The paper has a short appendix which summarizes results (mainly due to the first author and to Bestvina and Feighn) on the theory of group actions on trees which is used in the paper.

Reviewer: A.Papadopoulos (Strasbourg)

### MSC:

20F28 | Automorphism groups of groups |

57M07 | Topological methods in group theory |

20F65 | Geometric group theory |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20E08 | Groups acting on trees |

57M60 | Group actions on manifolds and cell complexes in low dimensions |

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

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

20E07 | Subgroup theorems; subgroup growth |

### Keywords:

finitely presented torsion-free subgroups; weakly rigid groups; Dehn twist automorphisms; amalgamated products; HNN extensions; Dehn twists; inner automorphisms; torsion-free hyperbolic groups; internal automorphisms; freely indecomposable subgroups; automorphism groups; automorphism groups of free groups; group actions on trees
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\textit{E. Rips} and \textit{Z. Sela}, Geom. Funct. Anal. 4, No. 3, 337--371 (1994; Zbl 0818.20042)

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