Separability properties of automorphisms of graph products of groups

Abstract

We study properties of automorphisms of graph products of groups. We show that graph product $\Gamma\mathcal{G}$ has non-trivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has non-trivial pointwise inner automorphisms. We use this result to study residual finiteness of $\mathop{Out}(\Gamma \mathcal{G})$. We show that if all vertex groups are finitely generated residually finite and the vertex groups corresponding to central vertices satisfy certain technical (yet natural) condition, then $\mathop{Out}(\Gamma\mathcal{G})$ is residually finite. Finally, we generalise this result to graph products of residually $p$-finite groups to show that if $\Gamma\mathcal{G}$ is a graph product of finitely generated residually $p$-finite groups such that the vertex groups corresponding to central vertices satisfy the $p$-version of the technical condition then $\mathop{Out}(\Gamma\mathcal{G})$ is virtually residually $p$-finite. We use this result to prove bi-orderability of Torreli groups of some graph products of finitely generated residually torsion-free nilpotent groups.