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Small spectral radius and percolation constants on non-amenable Cayley graphs

Authors: Kate Juschenko and Tatiana Nagnibeda
Journal: Proc. Amer. Math. Soc. 143 (2015), 1449-1458
MSC (2010): Primary 20F65, 97K50; Secondary 20F05
Published electronically: December 9, 2014
MathSciNet review: 3314060
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Abstract: Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $\Gamma$, does there exist a generating set $S$ such that the Cayley graph $(\Gamma ,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(\Gamma ,S)<p_u(\Gamma ,S)$? We show that this is true if $\Gamma$ contains an infinite normal subgroup $N$ such that $\Gamma /N$ is non-amenable. Moreover for any finitely generated group $G$ containing $\Gamma$ there exists a generating set $S’$ of $G$ such that $p_c(G,S’)<p_u(G,S’)$. In particular this applies to free Burnside groups $B(n,p)$ with $n \geq 2, p \geq 665$. We also explore how various non-amenability numerics, such as the isoperimetric constant and the spectral radius, behave on various growing generating sets in the group.

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Additional Information

Kate Juschenko
Affiliation: École polytechnique fédérale de Lausanne, Route Cantonale, 1015 Lausanne, Switzerland
MR Author ID: 780620

Tatiana Nagnibeda
Affiliation: Section de mathématiques, Université de Genève, 2-4, rue du Lièvre c.p. 64, 1211 Genève, Switzerland

Keywords: Non-amenable group, Cayley graph, spectral radius, Bernoulli percolation, isoperimetric constant
Received by editor(s): August 29, 2013
Published electronically: December 9, 2014
Additional Notes: The authors acknowledge the support of the Swiss National Foundation for Scientific Research and of the Mittag-Leffler Insitute
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society