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

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 $S^{\prime }$ of $G$ such that $p_c(G,S^{\prime })<p_u(G,S^{\prime })$. In particular this applies to free Burnside groups $B(n,p)$ with $n\ge 2,p\ge 665$. 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.

References in corpus (2)

• A remark about the spectral radius
• Fixed Price of Groups and Percolation

Cited by in corpus (1)

• A remark about the spectral radius