Expansive Automorphisms on Locally Compact Groups

Abstract

We show that any connected locally compact group which admits an expansive automorphism is nilpotent. We also show that for any locally compact group $G$, $\alpha \in \mathrm{Aut}(G)$ is expansive if and only if for any $\alpha$-invariant closed subgroup $H$ which is either compact or normal, the restriction of $\alpha$ to $H$ is expansive and the quotient map on $G/H$ corresponding to $\alpha$ is expansive. We get a structure theorem for locally compact groups admitting expansive automorphisms. We prove that an automorphism on a non-discrete locally compact group can not be both distal and expansive.

19 pages, Remark 2.4 is added and minor changes are made, including one in the statement of Theorem 2.9 (which was Theorem 2.8 in the earlier version)