Separability properties of automorphisms of graph products of groups

We study properties of automorphisms of graph products of groups. We show that graph product Gamma G has nontrivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has nontrivial pointwise inner automorphisms. We use this result to study residual finiteness of Out(Gamma 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 Out(Gamma G) is residually finite. Finally, we generalize this result to graph products of residually p-finite groups to show that if Gamma 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 Out(Gamma 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.