Schreier graphs of the Basilica group

Abstract

With 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 $\{{\Gamma }_n{\}}_{n\ge 1}$, 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 $\partial T$ gives rise to orbital Schreier graphs ${\Gamma }_{\xi }$, $\xi \in \partial T$. 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 $\{({\Gamma }_n,{\xi }_n){\}}_{n\ge 1}$ 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 $\xi$.

32 pages

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