Some Properties of Distal Actions on Locally Compact Groups

TL;DR

This paper investigates the properties of distal actions on locally compact groups, establishing equivalences, structural decompositions, and conditions for distality and contraction groups, with implications for Lie group approximations.

Contribution

It provides new equivalences for distality, structural decomposition theorems, and criteria for automorphism distality in locally compact groups.

Findings

Equivalence of distality and pointwise distality for certain group actions

Existence of a compact normal subgroup with distal quotient and ergodic conjugacy action

Characterization of distality of automorphisms via contraction groups

Abstract

We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We also show that a compactly generated locally compact group of polynomial growth has a compact normal subgroup $K$ such that $G/K$ is distal and the conjugacy action of $G$ on $K$ is ergodic; moreover, if $G$ itself is (pointwise) distal then $G$ is Lie projective. We prove a decomposition theorem for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient condition for distality of an automorphism in terms of its contraction group. We compare classes of (pointwise) distal groups and groups whose closed subgroups are unimodular. In particular, we study relations between distality, unimodularity and contraction subgroups.