Smallest non-cyclic quotients of the automorphism group of free groups

TL;DR

This paper provides an elementary proof that the smallest non-cyclic quotients of the automorphism group of free groups are linear groups over the field of two elements, confirming a conjecture and detailing the structure of minimal quotients.

Contribution

It offers a new, elementary proof of a known result about the minimal non-cyclic quotients of automorphism groups of free groups, and clarifies their structure.

Findings

The smallest non-cyclic quotients are linear groups over GF(2).

All minimal quotients are obtained via standard projections and automorphisms.

The proof simplifies previous approaches and confirms a conjecture.

Abstract

We give an new, elementary proof of the result that the smallest non-cyclic quotients of automorphism group of free group is the linear group over the field of two elements, and moreover all minimal quotients are obtained by the standard projection composed with an automorphism of the image. This result, originally due to Baumeister-Kielak-Pierro, proves a conjecture of Mecchia-Zimmermann.