A constructive proof that the Hanoi towers group has non-trivial rigid kernel

TL;DR

This paper provides a simplified proof that the Hanoi towers group has a rigid kernel isomorphic to the Klein 4 group, also analyzing its structure as a self-similar, recurrent, regular branch group.

Contribution

It offers a more straightforward proof of the rigid kernel's structure and details the group's self-similar and branch properties, expanding understanding of its algebraic features.

Findings

Rigid kernel is the Klein 4 group

Group is self-similar and recurrent

Group is a regular branch group

Abstract

In 2012, Bartholdi, Siegenthaler, and Zalesskii computed the rigid kernel of the Hanoi towers group. We present a simpler proof that the rigid kernel is the Klein 4 group. In the course of the proof, we also compute the rigid stabilizers and present proofs that this group is a self-similar, recurrent, regular branch group.