Abelian covers of graphs and maps between outer automorphism groups of free groups
Group Theory
2011-08-01 v3
Abstract
We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.
Keywords
Cite
@article{arxiv.1007.2598,
title = {Abelian covers of graphs and maps between outer automorphism groups of free groups},
author = {Martin R. Bridson and Karen Vogtmann},
journal= {arXiv preprint arXiv:1007.2598},
year = {2011}
}
Comments
Final version, to appear in Mathematische Annalen. Minor errors and typos corrected, including range of n in Theorem B