Distal actions on coset spaces in totally disconnected, locally compact groups

TL;DR

This paper investigates distal actions of automorphism groups on coset spaces in totally disconnected, locally compact groups, revealing they are SIN actions and deriving structural consequences for open subgroups.

Contribution

It demonstrates that all distal automorphism actions are SIN actions with open invariant neighborhoods and explores implications for the structure of open subgroups in such groups.

Findings

Distal actions are SIN actions with open invariant neighborhoods.

Existence of a compactly generated open subgroup containing a given compactly generated subgroup.

Realization of certain subgroups as intersections with open subgroups of the ambient group.

Abstract

Let $G$ be a totally disconnected, locally compact group and let $H$ be an equicontinuously (for example, compactly) generated group of automorphisms of $G$. We show that every distal action of $H$ on a coset space of $G$ is a SIN action, with the small invariant neighbourhoods arising from open $H$-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups. For example, it follows that for every compactly generated subgroup $K$ of $G$, there is a compactly generated open subgroup $E$ of $G$ such that $K \le E$ and such that every open subgroup of $G$ containing a finite index subgroup of $K$ contains a finite index subgroup of $E$. We also show that for a large class of closed subgroups $L$ of $G$ (including for instance all closed subgroups $L$ such that $L$ is an intersection of subnormal subgroups of open subgroups), every compactly generated open subgroup of $L$ can be realized as $L \cap O$ for an open subgroup of $G$.