Distal actions on coset spaces in totally disconnected locally compact groups

Abstract

Let [Formula: see text] be a totally disconnected locally compact (t.d.l.c.) group and let [Formula: see text] be an equicontinuously (for example, compactly) generated group of automorphisms of [Formula: see text]. We show that every distal action of [Formula: see text] on a coset space of [Formula: see text] is a SIN action, with the small invariant neighborhoods arising from open [Formula: see text]-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups of a t.d.l.c. group. For example, it follows that for every compactly generated subgroup [Formula: see text] of [Formula: see text], there is a compactly generated open subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and such that every open subgroup of [Formula: see text] containing a finite index subgroup of [Formula: see text] contains a finite index subgroup of [Formula: see text]. We also show that for a large class of closed subgroups [Formula: see text] of [Formula: see text] (including for instance all closed subgroups [Formula: see text] such that [Formula: see text] is an intersection of subnormal subgroups of open subgroups), every compactly generated open subgroup of [Formula: see text] can be realized as [Formula: see text] for an open subgroup of [Formula: see text].