Outer automorphism groups of free groups: linear and free representations

TL;DR

This paper investigates homomorphisms and linear representations of Out(F_n), revealing that most such maps factor through finite groups or natural projections, thus classifying low-dimensional linear representations.

Contribution

It classifies low-dimensional linear representations of Out(F_n) over characteristic zero fields and shows most homomorphisms between Out(F_n) and Out(F_m) factor through finite groups.

Findings

Homomorphisms between Out(F_n) and Out(F_m) are trivial unless m=n.

Low-dimensional linear representations of Out(F_n) factor through GL(n,Z).

Results extend to Out(F_4) and Out(F_5).

Abstract

We study the existence of homomorphisms between Out(F_n) and Out(F_m) for n > 5 and m < n(n-1)/2, and conclude that if m is not equal to n then each such homomorphism factors through the finite group of order 2. In particular this provides an answer to a question of Bogopol'skii and Puga. In the course of the argument linear representations of Out(F_n) in dimension less than n(n+1)/2 over fields of characteristic zero are completely classified. It is shown that each such representation has to factor through the natural projection from Out(F_n) to GL(n,Z) coming from the action of Out(F_n) on the abelianisation of F_n. We obtain similar results about linear representation theory of Out(F_4) and Out(F_5).