Normal automorphisms of relatively hyperbolic groups

Details Normal automorphisms of relatively hyperbolic groups Normal automorphisms of relatively hyperbolic groups

2008-09-14

arxiv.org/abs/0809.2408v4

Trans. Amer. Math. Soc. 362 (2010), no. 11, 6079-6103

Mathematics

Group Theory

Version 4: final (27 pages, 2 figures)

Scientific paper

An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no non-trivial finite normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with more than one end.

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