Non-closed subgroups of weakly branch groups

TL;DR

This paper constructs continuous families of non-closed subgroups within weakly branch groups, revealing complex subgroup topologies and properties related to residual finiteness and closure in various topologies.

Contribution

It introduces two novel constructions of non-closed subgroups in weakly branch groups, one with non-ERF subgroups and another with ERF subgroups under additional assumptions.

Findings

Constructed a family of non-closed, non-ERF subgroups.

Constructed a family of non-closed ERF subgroups under extra conditions.

Revealed intricate subgroup topologies in weakly branch groups.

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 $G$.