On subgroups of finite index in branch groups

TL;DR

This paper provides a detailed structural analysis of normal subgroups within finite index subgroups of branch groups, enhancing understanding of their lattice structures and conditions for commensurability.

Contribution

It offers a new structural description of normal subgroups in finite index subgroups of branch groups using rigid stabilizers, advancing the theory of their structure lattices.

Findings

Normal subgroups characterized via rigid stabilizers

Insight into the structure lattices of branch groups

Condition for abstract commensurability of branch groups

Abstract

We give a structural description of the normal subgroups of subgroups of finite index in branch groups in terms of rigid stabilizers. This gives further insight into the structure lattices of branch groups introduced by the second author. We derive a condition concerning abstract commensurability of branch groups acting on the p-ary tree for any prime p.