Commutator maps, measure preservation, and $T$-systems

Let $G$ be a finite simple group. We show that the commutator map $\alpha:G \times G \rightarrow G$ is almost equidistributed as $\vert G\vert \rightarrow\infty$. This somewhat surprising result has many applications. It shows that a for a subset $X \subseteq G$ we have $\alpha^{-1}(X)/\vert G\vert^2 = \vert X\vert/\vert G\vert + o(1)$, namely $\alpha$ is almost measure preserving. From this we deduce that almost all elements $g \in G$ can be expressed as commutators $g = [x,y]$ where $x,y$ generate $G$.

This enables us to solve some open problems regarding $T$-systems and the Product Replacement Algorithm (PRA) graph. We show that the number of $T$-systems in $G$ with two generators tends to infinity as $\vert G\vert \rightarrow \infty$. This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of $G$ with two generators.

Some of our results apply for more general finite groups and more general word maps.

Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function $\zeta^G(s) = \sum_{\chi \in \operatorname{Irr}(G)}\chi(1)^{-s}$ plays a key role in the proofs.