Free representations of outer automorphism groups of free products via characteristic abelian coverings

TL;DR

This paper constructs faithful free representations of outer automorphism groups of free products by algebraically lifting automorphisms through characteristic abelian covers, inspired by topological methods.

Contribution

It provides an algebraic construction for embedding $ ext{Out}(G)$ into $ ext{Out}(F_m)$ for free products, extending topological techniques to algebraic settings.

Findings

Faithful free representations exist for certain free products.

Embedding $ ext{Out}(G)$ into $ ext{Out}(F_m)$ is achieved via characteristic abelian covers.

Construction applies to free products with finite abelian factors coprime to $n-1$.

Abstract

Given a free product $G$, we investigate the existence of faithful free representations of the outer automorphism group $\text{Out}(G)$, or in other words of embeddings of $\text{Out}(G)$ into $\text{Out}\left(F_m\right)$ for some $m$. This is based on a work of Bridson and Vogtmann in which they construct embeddings of $\text{Out}\left(F_n\right)$ into $\text{Out}\left(F_m\right)$ for some values of $n$ and $m$ by interpreting $\text{Out}\left(F_n\right)$ as the group of homotopy equivalences of a graph $X$ of genus $n$, and by lifting homotopy equivalences of $X$ to a characteristic abelian cover of genus $m$. Our construction for a free product $G$, using a presentation of $\text{Out}(G)$ due to Fuchs-Rabinovich, is written as an algebraic proof, but it is directly inspired by Bridson and Vogtmann's topological method and can be interpreted as lifting homotopy equivalences of a graph of groups. For instance, we obtain a faithful free representation of $\text{Out}(G)$ when $G=F_d\ast G_{d+1}\ast\cdots\ast G_n$, with $F_d$ free of rank $d$ and $G_i$ finite abelian of order coprime to $n-1$.