# 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 $|G| \to \infty$.

## Authors

Michael Larsen

Indiana University, Bloomington, IN USA

Aner Shalev

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel

Pham Huu Tiep

Rutgers University, Piscataway, NJ USA