On the characteristic polynomial and spectrum of Basilica Schreier graphs

The Basilica group is one of the most studied automaton groups, and many papers have been devoted to the investigation of the characteristic polynomial and spectrum of the associated Schreier graphs Γn , even if an explicit description of them has not been given yet. Our approach to this issue is original, and it is based on the use of the Coefficient Theorem for signed graphs. We introduce a signed version Γn of the Basilica Schreier graph Γn , and we prove that there exist two fundamental relations between the characteristic polynomials of the signed and unsigned versions. The first relation comes from the cover theory of signed graphs. The second relation is obtained by providing a suitable decomposition of Γn into three parts, using the self-similarity of Γn, via a detailed investigation of its basic figures. By gluing together these relations, we find out a new recursive equation which expresses the characteristic polynomial of as a function of the characteristic polynomials Γn of the three previous levels. We are also able to give an explicit description of the eigenspace associated with the eigenvalue 2 , and to determine how the eigenvalues are distributed with respect to such eigenvalue.