Random Cayley graphs of finite groups with optimal spectral gaps

Determine whether there exist sequences of finite groups G_N such that, when r generators are drawn independently and uniformly at random from each G_N, the Cayley graphs Cay(G_N; σ_1,…,σ_r) achieve an optimal spectral gap in the Alon–Boppana sense; that is, ascertain that the nontrivial spectrum is asymptotically contained in [−2√(2r−1), 2√(2r−1)] (equivalently, that the nontrivial spectral radius converges to 2√(2r−1)).

Background

The survey discusses strong convergence and spectral gaps for random regular graphs and then considers random Cayley graphs generated by a fixed number of uniformly random elements. In this broader setting, it is a longstanding problem to establish optimal spectral gaps analogous to the Friedman result for random regular graphs.

This question is formulated for arbitrary sequences of finite groups, not just the symmetric group, and remains open despite progress on expansion in some families and explicit constructions in special cases.