Automorphisms of hyperbolic groups and graphs of groups.

*(English)*Zbl 1107.20030The author describes the outer automorphism group \(\text{Out}(G)\) of a one-ended word hyperbolic group \(G\). He provides a product structure for which \(\text{Out}(G)\) is virtually a direct product of mapping class groups and a free Abelian group. He also shows a necessary and sufficient condition for the outer automorphism group \(\text{Out}(G)\) to be infinite, when \(G\) is a one-ended hyperbolic group, and that there are only finitely many conjugacy classes of torsion elements in \(\text{Out}(G)\) and \(\operatorname{Aut}(G)\), when \(G\) is a torsion-free hyperbolic group. Moreover, for a finite graph \(\Gamma\) of group decomposition of an arbitrary group \(G\) the author describes the group of automorphisms of \(G\) preserving \(\Gamma\), by comparing it to direct products of suitably defined mapping class groups of vertex groups.

Reviewer: Ann Chi Kim (Pusan)

##### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

20F28 | Automorphism groups of groups |

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

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

20F65 | Geometric group theory |

20E08 | Groups acting on trees |

##### Keywords:

outer automorphism groups; graphs of groups; hyberbolic groups; mapping class groups; JSJ decompositions; tree automorphisms##### References:

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