On the automorphisms of a graph product of abelian groups

Abstract

We study the automorphisms of a graph product of finitely generated abelian groups $W$. More precisely, we study a natural subgroup ${\mathrm{Aut}}^{\ast }W$ of $\mathrm{Aut}W$, with ${\mathrm{Aut}}^{\ast }W=\mathrm{Aut}W$ whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition ${\mathrm{Aut}}^{\ast }W=(\mathrm{Inn}W⋊{\mathrm{Out}}^{0}W)⋊{\mathrm{Aut}}^{1}W$. We also give a number of applications, some of which are geometric in nature.