Finite simple characteristic quotients of the free group of rank 2

In this paper we describe how to explicitly construct infinitely many finite simple groups as characteristic quotients of the rank 2 free group F2. This shows that a "baby" version of the Wiegold conjecture fails for F2, and provides counterexamples to two conjectures in the theory of noncongruence subgroups of SL2(Z). Our main result explicitly produces, for every prime power q≥7, the groups SL3(Fq) and SU3(Fq) as characteristic quotients of F2. Our strategy is to study specializations of the Burau representation for the braid group B4, exploiting an exceptional relationship between F2 and B4 first observed by Dyer, Formanek, and Grossman. Weisfeiler's strong approximation theorem guarantees that our specializations are surjective for infinitely many primes, but it is not effective. To make our result effective, we give another proof of surjectivity via a careful analysis of the maximal subgroup structures of SL3(Fq) and SU3(Fq). We also show that our examples of PSL3(Fq) and PSU3(Fq) are minimal in the sense that no group of the form PSL2(Fq) is a characteristic quotient of F2.