# Distal actions and shifted convolution property

C. R. E. Raja and R. Shah

Arxiv ID: 0806.1820•Last updated: 6/24/2020

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.

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