Expansive Actions of Automorphisms of Locally Compact Groups $G$ on ${\rm Sub}_G$

Abstract

For a locally compact metrizable group $G$, we consider the action of ${\rm Aut}(G)$ on ${\rm Sub}_G$, the space of all closed subgroups of $G$ endowed with the Chabauty topology. We study the structure of groups $G$ admitting automorphisms $T$ which act expansively on ${\rm Sub}_G$. We show that such a group $G$ is necessarily totally disconnected, $T$ is expansive and that the contraction groups of $T$ and $T^{-1}$ are closed and their product is open in $G$; moreover, if $G$ is compact, then $G$ is finite. We also obtain the structure of the contraction group of such $T$. For the class of groups $G$ which are finite direct products of $\mathbb{Q}_p$ for distinct primes $p$, we show that $T\in{\rm Aut}(G)$ acts expansively on ${\rm Sub}_G$ if and only if $T$ is expansive. However, any higher dimensional $p$-adic vector space $\mathbb{Q}_{p^n}$, ($n\geq 2$), does not admit any automorphism which acts expansively on ${\rm Sub}_G$.