On minimal finite quotients of outer automorphism groups of free groups

TL;DR

This paper identifies the smallest nonabelian finite quotient groups of outer automorphism groups of free groups for ranks 3 and 4, suggesting a pattern that may extend to higher ranks, supported by computational subgroup analysis.

Contribution

It proves that for n=3 and 4, the minimal nonabelian finite quotient of Out F_n is PSL_n(Z_2), proposing this might hold for all n > 2.

Findings

The minimal finite quotient for n=3 and 4 is PSL_n(Z_2).

Computational analysis of low index subgroups supports the theoretical results.

The conjecture extends the pattern to all ranks n > 2.

Abstract

We prove that, for n=3 and 4, the minimal nonabelian finite factor group of the outer automorphism group Out F_n of a free group of rank n is the linear group PSL_n(Z_2) (conjecturally, this may remain true for arbitrary rank n > 2). We also discuss some computational results on low index subgroups of Aut F_n and Out F_n, for n = 3 and 4, using presentations of these groups.