# Codistal subgroups of locally compact groups

Let $G$ be a topological group and let $H$ be a closed subgroup of $G$. Say $H$ is codistal in $G$ if the translation action of $G$ on the coset space $G/H$ is distal (meaning that no non-diagonal orbit of $G$ on $G/H \times G/H$ has an accumulation point on the diagonal).

This concept appears (although not with this name) in the paper

H. B. Keynes, 'A study of the proximal relation in coset transformation groups', Trans. Amer. Math. Soc. 128 (1967), 389–402.

Keynes shows that $H$ is codistal in $G$ if and only if $\bigcap_{U \in \mathcal{U}}HUH = H$, where $\mathcal{U}$ is a basis of identity neighbourhoods in $G$. In particular, it is enough to show that the action of $H$ on $G/H$ is distal, and if $K$ is codistal in $H$ and $H$ is codistal in $G$, then $K$ is codistal in $G$.

Beyond that point though I could not find many references. Has this property been studied further under another name? It is a generalization of some other natural situations, for instance any intersection of open subgroups is codistal. I am particularly interested in the following question:

Suppose $G$ is a locally compact group and $H$ is a codistal subgroup of $G$. When does it happen that the action of $H$ on $G/H$ has small invariant neighbourhoods, in other words, $\{HUH/H \mid U \in \mathcal{U}\}$ is a base of neighbourhoods of the trivial coset in $G/H$?

If $G/H$ is compact, it looks like one obtains a small invariant neighbourhoods action by considering the Ellis semigroup of $H$ on $G/H$, so the interesting case is when $G/H$ is not compact. All the examples I know of (at least in totally disconnected, locally compact groups) still satisfy the stronger condition that $H$ has small invariant neighbourhoods on $G/H$, but I suspect this is not true in general.

• Maybe it has not been further studied because it's not of fundamental importance? Why look specifically at the actions on pairs and not on $n$-tuples? For instance, the action of $SL_n(\mathbf{R})$ on $\mathbf{R}^n-\{0\}$ is "codistal" for $n=2$ but not $n\ge 3$ (same for $p$-adics if you rather like totally disconnectedness). Anyway this is just an impression and there are certainly good motivations. – YCor Oct 19 '16 at 7:44
• Indeed, perhaps it is a good idea to consider n-tuples as well in this context, to give progressively weaker properties of 'n-distal'/'n-codistal', although the class of codistal subgroups already seems quite large to me in the t.d.l.c. context, and I don't know a good way generalize Keynes' criterion to 3-tuples or higher. Probably 'distal' is defined in this way since distal actions have mostly been studied on compact spaces, where the distinction between pairs and n-tuples is irrelevant. – Colin Reid Oct 19 '16 at 8:33