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

Abstract

Abstract Given a free product 𝐺, we investigate the existence of faithful free representations of the outer automorphism group Out ⁡ ( G ) \operatorname{Out}(G) , or in other words of embeddings of Out ⁡ ( G ) \operatorname{Out}(G) into Out ⁡ ( F m ) \operatorname{Out}(F_{m}) for some 𝑚. This is based on work of Bridson and Vogtmann in which they construct embeddings of Out ⁡ ( F n ) \operatorname{Out}(F_{n}) into Out ⁡ ( F m ) \operatorname{Out}(F_{m}) for some values of 𝑛 and 𝑚 by interpreting Out ⁡ ( F n ) \operatorname{Out}(F_{n}) 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 Out ⁡ ( G ) \operatorname{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 Out ⁡ ( G ) \operatorname{Out}(G) when G = F d ∗ G d + 1 ∗ ⋯ ∗ G n G=F_{d}\ast G_{d+1}\ast\cdots\ast G_{n} , with F d F_{d} free of rank 𝑑 and G i G_{i} finite abelian of order coprime to n - 1 n-1 .