Suppose we draw NxN permutation matrices A,B,C,D uniformly at random fromthe set of all such matrices that satisfy [A,B][C,D]=1. What can we sayabout the spectral gap of noncommutative polynomials of these matrices,asymptotically as N goes to infinity? I will aim to explain the origin ofof such questions of strong convergence, their implications for the studyof spectral gaps of surfaces, and how a recent new development, thepolynomial method, now makes it possible to prove such results. This talkis based on joint work with Michael Magee and Doron Puder.