Probabilistic Waring problems for finite simple groups

TL;DR

This paper proves that certain word maps on finite simple groups are almost uniformly distributed, providing new insights into probabilistic Waring problems, geometric characterizations, and applications to algebraic structures.

Contribution

It offers the first positive solution to the probabilistic Waring problem for finite simple groups and characterizes words inducing uniform distribution on groups of Lie type.

Findings

Almost uniform distribution for products of two non-trivial words

Geometric characterization for words on Lie type groups

Existence of N such that products of N words are almost uniform

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 distribution 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 distribution 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 $N = N(l)$, 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.