# Schreier graphs of the Basilica group

## TL;DR

This paper classifies the limit Schreier graphs of the Basilica group acting on a binary tree, based on the infinite binary sequences, providing a comprehensive understanding of their structure.

## Contribution

It offers a complete classification of the limit graphs associated with the Basilica group, linking their structure to infinite binary sequences.

## Key findings

- Classification of limit graphs up to isomorphism
- Connection between binary sequences and graph structures
- Complete description of Schreier graph limits for 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 $\{\Gamma_n\}_{n\geq 1}$, 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 $\xi$.