A Class of Non-Contracting Branch Groups with Non-Torsion Rigid Kernels

TL;DR

This paper introduces a new family of non-contracting, non-torsion rigid kernel branch groups that are fractal, have exponential growth, and lack the congruence subgroup property, expanding understanding of self-similar groups.

Contribution

It provides the first example of such groups with non-torsion rigid kernels and analyzes their structure, growth, and Hausdorff dimension.

Findings

Groups are very strongly fractal and not regular branch

Groups exhibit exponential growth

Rigid kernels are non-torsion and lack the congruence subgroup property

Abstract

In this work, we provide the first example of an infinite family of branch groups in the class of non-contracting self-similar groups. We show that these groups are very strongly fractal, not regular branch, and of exponential growth. Further, we prove that these groups do not have the congruence subgroup property by explicitly calculating the structure of their rigid kernels. This class of groups is also the first example of branch groups with non-torsion rigid kernels. As a consequence of these results, we also determine the Hausdorff dimension of these groups.