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

TL;DR

This paper investigates automorphisms acting expansively on the space of closed subgroups of locally compact groups, revealing structural properties and characterizations, especially for groups like finite products of p-adic numbers.

Contribution

It characterizes groups with automorphisms acting expansively on subgroup spaces and describes their structure, including contraction groups and conditions for expansiveness.

Findings

Groups with expansive automorphisms are totally disconnected and have closed contraction groups.

For finite products of dic numbers, automorphisms act expansively iff they are expansive.

Higher-dimensional p-adic vector spaces do not admit such expansive automorphisms.

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 ${\rm Sub}_G$.