Schreier spectrum of the Hanoi Towers group on three pegs

TL;DR

This paper computes the spectra of graphs related to the Hanoi Towers game on three pegs using finite dimensional representations of the Hanoi Towers group, including the limiting case on the boundary of the tree.

Contribution

It introduces a spectral analysis of the Hanoi Towers group via Schreier graphs, providing explicit spectra for finite and infinite cases.

Findings

Spectra of finite Schreier graphs are explicitly calculated.

The spectrum of the infinite boundary graph is characterized.

Finite dimensional representations are key to spectral computation.

Abstract

Finite dimensional representations of the Hanoi Towers group are used to calculate the spectra of the finite graphs associated to the Hanoi Towers Game on three pegs (the group serves as a renorm group for the game). These graphs are Schreier graphs of the action of the Hanoi Towers group on the levels of the rooted ternary tree. The spectrum of the limiting graph (Schreier graph of the action on the boundary of the tree) is also provided.