The congruence subgroup problem for branch groups

TL;DR

This paper investigates the congruence subgroup problem for branch groups acting on rooted trees, analyzing the structure of the congruence kernel and providing explicit examples, including the Hanoi tower group.

Contribution

It introduces a new framework for understanding the congruence kernel in branch groups, proving properties of the branch and rigid kernels, and explores their behavior through examples.

Findings

The branch kernel is Abelian in regular branch groups.

The rigid kernel has finite exponent.

Explicit examples of non-trivial congruence kernels are provided.

Abstract

We state and study the congruence subgroup problem for groups acting on rooted tree, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we split into two parts: the branch kernel and the rigid kernel. In the case of regular branch groups, we prove that the first one is Abelian while the second has finite exponent. We also establish some rigidity results concerning these kernels. We work out explicitly known and new examples of non-trivial congruence kernels, describing in each case the group action. The Hanoi tower group receives particular attention due to its surprisingly rich behaviour.