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 fails for $F_2$, and provides counterexamples to two conjectures in the theory of noncongruence subgroups of ${\text{SL}}_2(\mathbb{Z})$. Our main result explicitly produces, for every prime power $q\ge 7$, the groups ${\text{SL}}_3({\mathbb{F}}_q)$ and ${\text{SU}}_3({\mathbb{F}}_q)$ 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. 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 ${\text{SL}}_3({\mathbb{F}}_q)$ and ${\text{SU}}_3({\mathbb{F}}_q)$. We also show that our examples of ${\text{PSL}}_3({\mathbb{F}}_q)$ and ${\text{PSU}}_3({\mathbb{F}}_q)$ are minimal in the sense that no group of the form ${\text{PSL}}_2({\mathbb{F}}_q)$ is a characteristic quotient of $F_2$.

This is the accepted version, to appear in Commentarii Mathematici Helvetici. Added a section proving that PSL(2,q) is never a characteristic quotient of a rank two free group. 23 pages