A note on semicompleteness of graph products of abelian groups

TL;DR

This paper characterizes when a graph product of finitely generated abelian groups is semicomplete, showing it depends on the absence of a separating star in the defining graph.

Contribution

It provides a precise criterion linking the graph's structure to the automorphism kernel of the graph product of abelian groups.

Findings

Semicompleteness characterized by absence of separating star

Kernel of automorphism homomorphism equals inner automorphisms under certain conditions

Graph structure determines automorphism properties

Abstract

In this short note we prove that a graph product $G_\Gamma$ of finitely generated abelian groups is semicomplete -- that is the kernel of the natural homomorphism ${\rm Aut}(G_\Gamma)\to{\rm Aut}(G_\Gamma^{ab})$ induced by the abelianization of $G_\Gamma$ is equal to the inner automorphisms -- if and only if $\Gamma$ does not have a separating star.