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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
DOI: https://doi.org/10.1090/S0002-9939-2014-12578-0
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
Email: kate.juschenko@gmail.com

Tatiana Nagnibeda
Affiliation: Section de mathématiques, Université de Genève, 2-4, rue du Lièvre c.p. 64, 1211 Genève, Switzerland
Email: tatiana.smirnova-nagnibeda@unige.ch

DOI: https://doi.org/10.1090/S0002-9939-2014-12578-0
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