2019/08/28 by Antoni Brzoska, Courtney George, Brzoska, Antoni +7 · 1 citation
Mathematics · Physics and Astronomy · #(05C25 #05C50 #20E08 #28A80 #31C25 #37A30 #37B15 #37F10 #60J10 #81Q35) #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1908.10505
openalex publication_date 2019/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide the foundation of the spectral analysis of the Laplacian on the orbital Schreier graphs of the Basilica group, the iterated monodromy group of the quadratic polynomial z2-1. This group is an important example in the class of self-similar amenable but not elementary amenable finite automata groups studied by Grigorchuk, Żuk, \v Sunić, Bartholdi, Virág, Nekrashevych, Kaimanovich, Nagnibeda et al. We prove that the spectrum of the Laplacian has infinitely many gaps and that the support of the KNS Spectral Measure is a Cantor set. Moreover, on a generic blowup, the spectrum coincides with this Cantor set, and is pure point with localized eigenfunctions and eigenvalues located at the endpoints of the gaps.