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 $\{Γ_n\}_{n\geq 1}$, where $Γ_n$ 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 $Γ_ξ$, $ξ\in \partial T$. Denoting by $ξ_n$ the prefix of length $n$ of the infinite ray $ξ$, the rooted graph $(Γ_ξ,ξ)$ is then the limit of the sequence of finite rooted graphs $\{(Γ_n,ξ_n)\}_{n\geq 1}$ in the sense of pointed Gromov-Hausdorff convergence. In this paper, we give a complete classification (up to isomorphism) of the limit graphs $(Γ_ξ,ξ)$ associated with the Basilica group acting on the binary tree, in terms of the infinite binary sequence $ξ$.