Automorphisms of Hyperbolic Groups and Graphs of Groups |
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Authors: | Email author" target="_blank">Gilbert?LevittEmail author |
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Institution: | (1) LMNO, UMR CNRS 6139, Université de Caen, 5186, Caen Cedex, BP, 14032, France |
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Abstract: | Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups Ge are rigid (i.e. Out(Ge) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups. |
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Keywords: | automorphism groups graphs of groups hyberbolic groups mapping class groups JSJ decomposition tree automorphisms |
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