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

Kate Juschenko, Tatiana Nagnibeda

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1449-1458
Number of pages10
JournalProceedings of the American Mathematical Society
Issue number4
StatePublished - 2015


  • Bernoulli percolation
  • Cayley graph
  • Isoperimetric constant
  • Non-amenable group
  • Spectral radius

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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