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

.