Cutoff for product replacement on finite groups

We analyze a Markov chain, known as the product replacement chain , on the set of generating n -tuples of a fixed finite group G . We show that as n →∞ , the total-variation mixing time of the chain has a cutoff at time 3/2 n log n with window of order n . This generalizes a result of Ben-Hamou and Peres (who established the result for G = ℤ/2 ) and confirms a conjecture of Diaconis and Saloff-Coste that for an arbitrary but fixed finite group, the mixing time of the product replacement chain is O(n log n) .