Some Properties of Distal Actions on Locally Compact 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.

27 pages, main results are revised and improved, some preliminary results are removed and some new results are added, some proofs are revised and some are made shorter

References in corpus (3)

• The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth
• Distal actions and shifted convolution property
• Orbits of Distal Actions on Locally Compact Groups

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• Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups
• On the distality and expansivity of certain maps on spheres
• Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub${}_G$ and Applications to Lattices in Lie Groups