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arXiv · 1103.1624

Outer automorphism groups of free groups: linear and free representations

Dawid Kielak
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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).

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Dawid Kielak. 2011-03-08. Outer automorphism groups of free groups: linear and free representations. https://doi.org/10.1112/jlms%2Fjds077

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