We explore the mixing behavior of random walks on Cayley
graphs of non-Abelian groups, focusing on nilpotent and dihedral
groups, both of which can be viewed as “almost Abelian.” For nilpotent
groups, we study both walks driven by k randomly chosen generators and
conjugacy-invariant walks, showing in different settings how their
mixing is governed by the Abelianization. In particular, for k random
generators, cutoff occurs with high probability throughout the regime
1 ≪ log k ≪ log |G|, at the entropic time of the projected walk. For
dihedral groups, the cutoff time behaves differently from the entropic
time on the cyclic subgroup. A common perspective behind these results
comes from analyzing the entropy evolution of suitable auxiliary
processes, reflecting the underlying commutativity structure of the
group. This talk is based on several works, including joint work with
Jonathan Hermon and Renyu Rao.