  Let G be a finite simple group. We show that the commutator map $a : G \times
G \to G$ is almost equidistributed as the order of G goes to infinity. This
somewhat surprising result has many applications. It shows that for a subset X
of G we have $a^{-1}(X)/|G|^2 = |X|/|G| + o(1)$, namely $a$ 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 the order of G goes to infinity. 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 plays a key role in
the proofs.
