Finitely generated subgroups of branch groups and subdirect products of just infinite groups

TL;DR

This paper characterizes finitely generated subgroups of certain branch groups, introduces the concept of block subgroups, and proves these groups are subgroup separable, with broader implications for subdirect products of just infinite groups.

Contribution

It provides a structural description of finitely generated subgroups in branch groups and establishes subgroup separability for these groups, extending to subdirect products of just infinite groups.

Findings

Finitely generated subgroups are described via block subgroups.

All groups in the studied family are subgroup separable (LERF).

Results apply to subdirect products of just infinite groups.

Abstract

The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups.