by Antoni Brzoska, Courtney George, Samantha Jarvis, Luke G. Rogers, Alexander Teplyaev
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 $$z^2−1$$. This group is an important example in the class of self-similar amenable but not elementary amenable finite automata groups studied by Grigorchuk, Żuk, Šunić, Bartholdi, Virág, Nekrashevych, Kaimanovich, Nagnibeda et al. We prove that the spectrum of the Laplacian has infinitely many gaps and, on a generic blowup, is pure point with localized eigenfunctions.