SCHREIER GRAPHS OF THE BASILICA GROUP

To any self-similar action of a finitely generated group G of automorphisms of a regular rooted tree T can be naturally associated an infinite sequence of finite graphs {Gamma(n)}(n >= 1), where Gamma(n) is the Schreier graph of the action of G on the n-th level of T. Moreover, the action of G on partial derivative T gives rise to orbital Schreier graphs Gamma(xi), xi is an element of partial derivative T. Denoting by xi(n) the prefix of length n of the infinite ray xi, the rooted graph (Gamma(xi), xi) is then the limit of the sequence of finite rooted graphs {(Gamma(n), xi(n))}(n >= 1) in the sense of pointed Gromov-Hausdorff convergence. In this paper, we give a complete classification (up to isomorphism) of the limit graphs (Gamma(xi), xi) associated with the Basilica group acting on the binary tree, in terms of the infinite binary sequence xi.