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SIN actions on coset spaces in totally disconnected, locally compact (t.d.l.c.) groups

Banff International Research Station for Mathematical Innovation and Discovery

Let $G$ be a locally compact group, let $K$ be a closed subgroup of $G$, and let $H$ be a group of automorphisms of $G$ such that $h(K) = K$ for all $h in H$. When is the action of $H$ on $G/K$ a small invariant neighbourhoods (SIN) action, i.e. when is there a basis of neighbourhoods of the trivial coset consisting of $H$-invariant sets? In general, the SIN property is a strong restriction, but when $G$ is totally disconnected and $H$ is compactly generated, it turns out to be equivalent to the seemingly weaker condition that the action of $H$ on $G/K$ is distal on some neighbourhood of the trivial coset. (The analogous statement is false in the connected case: compact nilmanifolds give rise to counterexamples.) This has some general consequences for the structure of t.d.l.c. groups: for example, given any compact subset $X$ of a t.d.l.c. group $G$, there is an open subgroup containing $X$ that is the unique smallest such up to finite index.

Mathematics; Topological groups, lie groups; Abstract harmonic analysis

video/mp4

Author affiliation: University of Newcastle

2017-12-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63885

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/