Finite simple characteristic quotients of the free group of rank 2

Abstract

In this paper, we describe how to explicitly construct infinitely many finite simple groups as characteristic quotients of the rank 2 free group $F_2$. This shows that a “baby” version of the Wiegold conjecture [in: Geometry, Rigidity, and Group Actions (2011), 609–643] fails for $F_2$ and provides counterexamples to two conjectures in the theory of noncongruence subgroups of ${\mathrm{SL}}_2(\mathbb{Z})$ by Chen [Math. Ann. 371 (2018), 41–126]. Our main result explicitly produces, for every prime power $q\ge 7$, the groups ${\mathrm{SL}}_3({\mathbb{F}}_q)$ and ${\mathrm{SU}}_3({\mathbb{F}}_q)$ as characteristic quotients of $F_2$. Our strategy is to study specializations of the Burau representation for the braid group $B_4$, exploiting an exceptional relationship between $F_2$ and $B_4$ 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 ${\mathrm{SL}}_3({\mathbb{F}}_q)$ and ${\mathrm{SU}}_3({\mathbb{F}}_q)$. These examples are minimal in the sense that no finite simple group of the form ${\mathrm{PSL}}_2({\mathbb{F}}_q)$ appears as a characteristic quotient of $F_2$.