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JMD
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
$\{\Gamma_n\}_{n\geq 1}$, 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 $\{(\Gamma_n,\xi_n)\}_{n\geq 1}$ in the sense
of pointed Gromov-Hausdorff convergence. In this paper, we give a
complete classification (up to isomorphism) of the limit graphs
$(\Gamma_{\xi},\xi)$ associated with the Basilica group acting on
the binary tree, in terms of the infinite binary sequence
$\xi$.
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