Separability properties of automorphisms of graph products of groups

TL;DR

This paper investigates automorphism properties of graph products of groups, establishing conditions for residual finiteness and bi-orderability, thereby advancing understanding of their algebraic structure and automorphism groups.

Contribution

It characterizes when automorphisms are pointwise inner and extends residual finiteness results to graph products of residually p-finite groups.

Findings

Graph product automorphisms are pointwise inner iff vertex groups at central vertices have non-trivial pointwise inner automorphisms.

Under certain conditions, the outer automorphism group of a graph product is residually finite.

The results imply bi-orderability of Torreli groups for specific graph products.

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.