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arXiv · 2606.07430

Spectral properties of the Schreier graphs of the basilica group

Kyle Ambrose·Noah Dunham·Michael Morris·Luke G. Rogers·Alexander Teplyaev
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Abstract

We study the spectral properties of Laplacians on the Schreier graphs $\Gamma_n$ of the basilica group, the iterated monodromy group of the polynomial $z^2 - 1$, which is an important example in the theory of self-similar, amenable but not elementarily amenable, automaton groups. Building heavily on results by Brzoska, Jarvis, George, Rogers and Teplyaev about certain subgraphs of the basilica graphs, we develop a new recursive framework for computing the characteristic polynomials of these Laplacians. Our analysis reveals a simple underlying dynamical system and proves approximation results for the Kesten-von Neumann-Serre (KNS) spectral measure.

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Kyle Ambrose, Noah Dunham, Michael Morris, Luke G. Rogers, Alexander Teplyaev. 2026-06-05. Spectral properties of the Schreier graphs of the basilica group. https://arxiv.org/abs/2606.07430

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