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Automorphisms of hyperbolic groups and graphs of groups. (English) Zbl 1107.20030
The 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.

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
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