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

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 $n-1$.