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January  2010, 4(1): 167-205. doi: 10.3934/jmd.2010.4.167

Schreier graphs of the Basilica group

Daniele D'angeli 1, , Alfredo Donno 2, , Michel Matter 3, and  Tatiana Nagnibeda 3,

1. 

Department of Mathematics, Technion Institute of Technology, Technion City, Haifa 32 000, Israel
2. 

"Sapienza" Università di Roma, Dipartimento di Matematica "Guido Castelnuovo", P.le AldoMoro, 2, 00185 Roma, Italy
3. 

Université deGenève, Section demathématiques, 2-4, rue du Lièvre, c.p. 64, 1211 Genève 4, Switzerland, Switzerland

Received  November 2009 Revised  March 2010 Published  May 2010

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: random weak limit of graphs, graph isomorphism, action by automorphisms of a rooted tree, Self-similar group, Schreier graph, infinite binary sequence.
Mathematics Subject Classification: Primary: 20E08; Secondary: 20F69, 05C63, 37E2.
Citation: Daniele D'angeli, Alfredo Donno, Michel Matter, Tatiana Nagnibeda. Schreier graphs of the Basilica group. Journal of Modern Dynamics, 2010, 4 (1) : 167-205. doi: 10.3934/jmd.2010.4.167
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