Branch groups with infinite rigid kernel

TL;DR

This paper develops a framework for calculating rigid kernels in self-similar branch groups and introduces new examples with infinite rigid kernels, expanding understanding of their structure and properties.

Contribution

It provides a new theoretical method for computing rigid kernels and presents the first examples of self-similar branch groups with infinite rigid kernels.

Findings

Established a framework for calculating rigid kernels.

Constructed new examples of branch groups with infinite rigid kernels.

Identified the rigid kernel as an infinite Cartesian power of cyclic groups.

Abstract

A theoretical framework is established for explicitly calculating rigid kernels of self-similar regular branch groups. This is applied to a new infinite family of branch groups in order to provide the first examples of self-similar, branch groups with infinite rigid kernel. The groups are analogs of the Hanoi Towers group on 3 pegs, based on the standard actions of finite dihedral groups on regular polygons with odd numbers of vertices, and the rigid kernel is an infinite Cartesian power of the cyclic group of order 2, except for the original Hanoi group. The proofs rely on a symbolic-dynamical approach, related to finitely constrained groups.