Distal actions and shifted convolution property

TL;DR

This paper investigates the shifted convolution property (SCP) in locally compact groups, characterizes groups with SCP via conjugation actions, and explores the distality of factor actions and their implications.

Contribution

It provides a characterization of groups with SCP through point-wise distality of conjugation actions and examines the distality of factor actions in the context of SCP.

Findings

Groups with SCP are characterized by point-wise distality of conjugation actions.

Distality of factor actions holds under certain conditions, such as invariance under compact groups.

Results have implications for distality of actions on probability measure spaces and related measures.

Abstract

A locally compact group $G$ is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure $\mu$ on $G$, either $\sup_{x\in G} \mu ^n (Cx) \ra 0$ for all compact subsets $C$ of $G$, or there exist $x\in G$ and a compact subgroup $K$ normalised by $x$ such that $\mu^nx^{-n} \ra \omega_K$, the Haar measure on $K$. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors under compact groups invariant under the action and for factors under the connected component of identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (point-wise distality). This has some interesting corollaries to distality of certain actions and Choquet Deny measures which actually motivated SCP and point-wise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.