Abelian covers of graphs and maps between outer automorphism groups of free groups

TL;DR

This paper investigates homomorphisms between outer automorphism groups of free groups, establishing conditions under which these maps are finite or embeddings, and explores lifts of actions on graphs to abelian covers.

Contribution

It characterizes when homomorphisms between Out(F_n) and Out(F_m) are finite or embeddings, based on the relationship between n and m, and analyzes lifts of group actions to abelian covers.

Findings

Homomorphisms are finite when n > 8 even and m ≤ 2n, or n odd and m ≤ 2n - 2.

Existence of embeddings for specific m related to n via r^n(n - 1) + 1 with r coprime to (n - 1).

Determined conditions for lifting Out(F_n) actions to abelian Galois covers.

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.