A note on semicompleteness of graph products of abelian groups

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 $\mathrm{Aut}(G_{\Gamma })\to \mathrm{Aut}(G_{\Gamma }^{ab})$ induced by the abelianization of $G_{\Gamma }$ is equal to the inner automorphisms -- if and only if $\Gamma$ does not have a separating star.

arXiv commentTypos corrected, to appear in Communications in Algebra