The congruence subgroup problem for a family of branch groups

TL;DR

This paper introduces a new family of branch groups that generalize the Hanoi towers group, analyzing their congruence subgroup problem and revealing novel properties such as trivial rigid kernel and Hausdorff dimension close to 1.

Contribution

It constructs a new family of branch groups with specific properties, expanding understanding of the congruence subgroup problem and providing new examples with non-trivial kernels.

Findings

Groups are just infinite with trivial rigid kernel

Strict bounds on the branch kernel are established

Existence of subgroups with non-trivial rigid kernel

Abstract

We construct a family of groups which generalize the Hanoi towers group and study the congruence subgroup problem for the groups in this family. We show that unlike the Hanoi towers group, the groups in this generalization are just infinite and have trivial rigid kernel. We also put strict bounds on the branch kernel. Additionally, we show that these groups have subgroups of finite index with non-trivial rigid kernel. The only previously known group where this kernel is non-trivial is the Hanoi towers group and so this adds infinitely many new examples. Finally, we show that the topological closures of these groups have Hausdorff dimension arbitrarily close to 1.