Spectral properties of graphs associated to the Basilica group

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, \.Zuk, \v Suni\'c, Bartholdi, Vir\'ag, 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.