Schreier graphs of the Basilica group

Daniele D'angeli; Alfredo Donno; Michel Matter; Tatiana Nagnibeda

doi:10.3934/jmd.2010.4.167

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2010, Volume 4, Issue 1: 167-205. Doi: 10.3934/jmd.2010.4.167

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Schreier graphs of the Basilica group

1.
Department of Mathematics, Technion Institute of Technology, Technion City, Haifa 32 000
2.
"Sapienza" Università di Roma, Dipartimento di Matematica "Guido Castelnuovo", P.le AldoMoro, 2, 00185 Roma
3.
Université deGenève, Section demathématiques, 2-4, rue du Lièvre, c.p. 64, 1211 Genève 4

Received: October 31, 2009

Revised: February 28, 2010

Published: April 2010

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Abstract

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 $\xi$.

Keywords:

Self-similar group,
action by automorphisms of a rooted tree,
Schreier graph,
infinite binary sequence,
random weak limit of graphs,
graph isomorphism.

Mathematics Subject Classification: Primary: 20E08; Secondary: 20F69, 05C63, 37E25.

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