Expansive actions of automorphisms of locally compact groups ${\varvec{G}}$ on $\hbox {Sub}_{{\varvec{G}}}$

Abstract

For a locally compact metrizable group G, we consider the action of $\mathrm{Aut}(G)$ on $\mathrm{Sub}_G$, 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 $\mathrm{Sub}_G$. 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 $T\in \mathrm{Aut}(G)$ acts expansively on $\mathrm{Sub}_G$ if and only if T is expansive. However, any higher dimensional p-adic vector space ${\mathbb {Q}}_p^n$, ($n\ge 2$), does not admit any automorphism which acts expansively on $\mathrm{Sub}_G$.