A Class of Non-Contracting Branch Groups with Non-Torsion Rigid Kernels
Sagar Saha, K. V. Krishna
TL;DR
The paper constructs an explicit infinite family of branch groups in the non-contracting self-similar setting, $G_d$ with odd $d\ge 3$, and analyzes their CSP-related kernels. It proves $G_d$ is branch and very strongly fractal with exponential growth, while providing a precise description of its rigid kernel (non-torsion) and the nontrivial branch kernel. Through a detailed examination of level and rigid stabilizers, it derives the quotient structures and establishes the Hausdorff dimension of the closure as $\dim_H(\overline{G_d}) = 1 - \frac{\log 2}{d\log d!}$. The results yield the first explicit family of non-contracting branch groups with infinite rigid kernels, expanding the landscape of CSP phenomena in branch groups and offering exact growth and dimensional data for these new examples.
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.