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

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.