Non-closed subgroups of weakly branch groups
Jorge Fariña-Asategui, Paul-Henry Leemann, Tatiana Nagnibeda
TL;DR
The paper investigates how subgroups of weakly branch groups interact with profinite and congruence topologies, focusing on the ERF property. It develops two complementary constructions: (i) a continuum of subgroups that are not closed in the profinite topology and are non-ERF, extending Cornulier’s ideas to groups with micro-supported actions; (ii) under suitable hypotheses, a continuum of ERF subgroups that are not closed in the congruence topology (and hence not closed in the ambient profinite topology). The results show that weakly branch groups are not ERF, while also exhibiting rich families of ERF subgroups whose non-closure occurs precisely in the congruence topology. The constructions apply to well-known groups such as the Grigorchuk group and GGS/multi-GGS groups, revealing intricate subgroup structures and clarifying the distinctions between profinite and congruence closures in self-similar group actions.
Abstract
For a weakly branch group $G$ acting on a regular enough rooted tree, we provide two constructions of continuous families of distinct subgroups that are not closed in the profinite topology on $G$. On the one hand, we construct a continuous family of distinct non-closed subgroups such that each $H$ in the family is not ERF, that is, contains subgroups not closed in the profinite topology on $H$. On the other hand, under an additional assumption on $G$, we construct a continuous family of ERF subgroups which are not closed in the congruence (and in the profinite) topology on $G$.