Probabilistic Waring problems for finite simple groups

Abstract

The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ 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 $L^{\infty }$ Waring problem for finite simple groups. We show that for every $l \ge 1$, there exists (an explicit) $N = N(l)=O(l^4)$, such that if $w_1, \ldots , w_N$ are non-trivial words of length at most $l$ in pairwise disjoint sets of variables, then their product $w_1 \cdots w_N$ is almost uniform on finite simple groups with respect to the $L^{\infty }$ norm. The dependence of $N$ on $l$ is genuine. This result implies that, for every word $w = w_1 \cdots w_N$ 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 $\Gamma $, a random homomorphism from $\Gamma $ to a finite simple group $G$ is surjective with probability tending to $1$ as $|G| \to \infty $.