Low dimensional free and linear representations of $\mathrm{Out}(F_3)$

TL;DR

This paper investigates low-dimensional linear representations and homomorphisms of the outer automorphism group of a free group of rank 3, showing they are highly constrained and mostly factor through known algebraic structures.

Contribution

It proves that all low-dimensional linear representations of 0ut(F_3) factor through 0ut(Z^3) and that homomorphisms to 0ut(F_5) have finite images, revealing structural rigidity.

Findings

All 0ut(F_3) linear representations of dimension 0 are trivial or factor through 0ut(Z^3)

Homomorphisms from 0ut(F_3) to 0ut(F_5) have finite image

Low-dimensional representations are highly constrained and do not produce new infinite images.

Abstract

We study homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$, and $\mathrm{GL}(m,K)$ for $m < 7$, where $K$ is a field of characteristic other than 2 or 3. We conclude that all $K$-linear representations of dimension at most 6 of $\mathrm{Out}(F_3)$ factor through $\mathrm{GL}(3,Z)$, and that all homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$ have finite image.