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 [in: Geometry, Rigidity, and Group Actions (2011), 609–643] fails for F₂ and provides counterexamples to two conjectures in the theory of noncongruence subgroups of SL₂.Z/ by Chen [Math. Ann. 371 (2018), 41–126]. Our main result explicitly produces, for every prime power q ≥ 7, the groups SL₃.F_q/ and SU₃.F_q/ 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 [Arch. Math. (Basel) 38 (1982), 404–409]. Weisfeiler’s strong approximation theorem guarantees that our specializations are surjective for infinitely many primes, but they are not effective. To make our result effective, we give another proof of surjectivity via a careful analysis of the maximal subgroup structures of SL₃.F_q/ and SU₃.F_q/. These examples are minimal in the sense that no finite simple group of the form PSL₂.F_q/ appears as a characteristic quotient of F₂,.