Low dimensional linear representations of SAut(Fn)

TL;DR

This paper proves that the group SAut(Fn), a subgroup of automorphisms of a free group, cannot be represented linearly in low dimensions over fields with characteristic not two.

Contribution

It establishes a non-existence result for low-dimensional linear representations of SAut(Fn), highlighting its algebraic rigidity.

Findings

No non-trivial linear representations of degree d<n over fields with characteristic not two.

SAut(Fn) exhibits algebraic rigidity with respect to low-dimensional linear representations.

The result applies to all ranks n of free groups.

Abstract

We prove that SAut(Fn), the unique subgroup of index two in the automorphism group of a free group of rank n, admits no non-trivial linear representation of degree d<n for any field of characteristic not equal to two.