Probabilistic Waring problems for finite simple groups

The probabilistic Waring problem for finite simple groups asks whether every word of the form w₁w₂, where w₁ and w₂ are non-trivial words in disjoint sets of variables, induces almost uniform distributions on finite simple groups with respect to the L ¹ norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distributions on finite simple groups of Lie type of bounded rank, and study related random walks. Our second main result concerns the probabilistic L^∞ Waring problem for finite simple groups. We show that for every l≥1, there exists (an explicit) N=N(l)=O(l⁴), such that if w₁,...,w_N are non-trivial words of length at most l in pairwise disjoint sets of variables, then their product w₁...w_N is almost uniform on finite simple groups with respect to the L^∞ norm. The dependence of N on l is genuine. This result implies that, for every word w=w₁...w_N as above, the word map induced by w on a semisimple algebraic group over an arbitrary field is a flat morphism. Applications to representation varieties, subgroup growth, and random generation are also presented. In particular, we show that, for certain one-relator groups Γ, a random homomorphism from Γ to a finite simple group G is surjective with probability tending to 1 as |G|→∞.