On the congruence subgroup problem for branch groups

TL;DR

This paper demonstrates that the congruence subgroup problem for branch groups is independent of their action on a tree and shows that the congruence topology is determined by the group's structure graph.

Contribution

It establishes the independence of the congruence subgroup problem from the branch action and introduces a natural definition of the structure graph.

Findings

The congruence topology is determined by the group's structure graph.

The problem is independent of the branch action on a tree.

A more natural definition of the structure graph is provided.

Abstract

We answer a question of Bartholdi, Siegenthaler and Zalesskii, showing that the congruence subgroup problem for branch groups is independent of the branch action on a tree. We prove that the congruence topology of a branch group is determined by the group; specifically, by its structure graph, an object first introduced by Wilson. We also give a more natural definition of this graph.