Outer automorphism groups of free groups: linear and free representations
Abstract
We study the existence of homomorphisms between Out(Fn) and Out(Fm) 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(Fn) 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(Fn) to GL(n,Z) coming from the action of Out(Fn) on the abelianisation of Fn. We obtain similar results about linear representation theory of Out(F4) and Out(F5).
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