Normal automorphisms of relatively hyperbolic groups

Minasyan, A.; Osin, D.

doi:10.1090/S0002-9947-2010-05067-6

Normal automorphisms of relatively hyperbolic groups
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by A. Minasyan and D. Osin

Trans. Amer. Math. Soc. 362 (2010), 6079-6103

DOI: https://doi.org/10.1090/S0002-9947-2010-05067-6

Published electronically: June 16, 2010

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Abstract:

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 show that for any such 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 finite non-trivial 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 infinitely many ends.

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