KKL, Kruskal-Katona, and Monotone Nets

We generalize the Kahn-Kalai-Linial (KKL) Theorem to random walks on Cayley and Schreier graphs, making progress on an open problem of Hoory, Linial, and Wigderson. In our generalization, the underlying group need not be abelian so long as the generating set is a union of conjugacy classes.