TY - JOUR

T1 - Small spectral radius and percolation constants on non-amenable Cayley graphs

AU - Juschenko, Kate

AU - Nagnibeda, Tatiana

N1 - Publisher Copyright:
© 2014 American Mathematical Society.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated nonamenable group Γ, does there exist a generating set S such that the Cayley graph (Γ, S), without loops and multiple edges, has non-unique percolation, i.e., pc(Γ, S) < pu(Γ, S)? We show that this is true if Γ contains an infinite normal subgroup N such that Γ/N is non-amenable. Moreover for any finitely generated group G containing Γ there exists a generating set S' of G such that pc(G, S') < pu(G, S'). In particular this applies to free Burnside groups B(n, p) with n ≥ 2, p ≥ 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.

AB - Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated nonamenable group Γ, does there exist a generating set S such that the Cayley graph (Γ, S), without loops and multiple edges, has non-unique percolation, i.e., pc(Γ, S) < pu(Γ, S)? We show that this is true if Γ contains an infinite normal subgroup N such that Γ/N is non-amenable. Moreover for any finitely generated group G containing Γ there exists a generating set S' of G such that pc(G, S') < pu(G, S'). In particular this applies to free Burnside groups B(n, p) with n ≥ 2, p ≥ 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.

KW - Bernoulli percolation

KW - Cayley graph

KW - Isoperimetric constant

KW - Non-amenable group

KW - Spectral radius

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U2 - 10.1090/S0002-9939-2014-12578-0

DO - 10.1090/S0002-9939-2014-12578-0

M3 - Article

AN - SCOPUS:84923253554

VL - 143

SP - 1449

EP - 1458

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 4

ER -