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

Multi-way expanders and imprimitive group actions on graphs

For n at least 2, the concept of n-way expanders was defined by various researchers. Bigger n gives a weaker notion in general, and 2-way expanders coincide with expanders in usual sense. Koji Fujiwara asked whether these concepts are equivalent to that of ordinary expanders for all n for a sequence of Cayley graphs. In this paper, we answer his question in the affirmative. Furthermore, we obtain universal inequalities on multi-way isoperimetric constants on any finite connected vertex-transitive graph, and show that gaps between these constants imply the imprimitivity of the group action on the graph.Comment: Accepted in Int. Math. Res. Notices. 18 pages, rearrange all of the arguments in the proof of Main Theorem (Theorem A) in a much accessible way (v4); 14 pages, appendix splitted into a forthcoming preprint (v3); 17 pages, appendix on noncommutative L_p spaces added (v2); 12 pages, no figure