On the automorphisms of a graph product of abelian groups

TL;DR

This paper investigates the structure of automorphism groups of graph products of finitely-generated abelian groups, revealing a semi-direct product decomposition and exploring geometric applications.

Contribution

It introduces a natural subgroup of automorphisms, Aut* W, and establishes its structure, including a semi-direct product decomposition involving inner automorphisms.

Findings

Aut* W equals Aut W for finite vertex groups

Aut* W admits a semi-direct product decomposition with Inn W

Applications include geometric insights into automorphism groups

Abstract

We study the automorphisms of a graph product of finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = 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 of Aut* W in which one of the factors is Inn W. We also give a number of applications, some of which are geometric in nature.