Journal of Modern Dynamics (JMD)

Schreier graphs of the Basilica group

Pages: 167 - 205, Issue 1, January 2010      doi:10.3934/jmd.2010.4.167

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Daniele D'angeli - Department of Mathematics, Technion Institute of Technology, Technion City, Haifa 32 000, Israel (email)
Alfredo Donno - "Sapienza" Università di Roma, Dipartimento di Matematica "Guido Castelnuovo", P.le AldoMoro, 2, 00185 Roma, Italy (email)
Michel Matter - Université deGenève, Section demathématiques, 2-4, rue du Lièvre, c.p. 64, 1211 Genève 4, Switzerland (email)
Tatiana Nagnibeda - Université de Genève, Section de mathématiques, 2-4, rue du Lièvre, c.p. 64, 1211 Genève 4, Switzerland (email)

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.

Received: November 2009;      Revised: March 2010;      Available Online: May 2010.