Cutoff for random Cayley graphs of nilpotent groups

We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes $G=G(n)$, whose ranks and nilpotency classes are uniformly bounded. For some $k=k(n)$ such that 1log k log |G|, we pick a random set of generators $S=S(n)$ by sampling k elements Z₁,…,Zₖ from G uniformly at random with replacement, and set S:=ⱼ± 1:1 ≤ j≤ k \. We show that the simple random walk on Cay$(G,S)$ exhibits cutoff with high probability. Some of our results apply to a general set of generators. Namely, we show that there is a constant $c>0$, depending only on the rank and the nilpotency class of G, such that for all symmetric sets of generators S of size at most clog |G|/log log |G|, the spectral gap and the ε-mixing time of the simple random walk X=(Xₜ)t≥ 0 on Cay$(G,S)$ are asymptotically the same as those of the projection of X to the abelianization of G, given by [G,G]Xₜ. In particular, X exhibits cutoff if and only if its projection does.