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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.

##### 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
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