Rigidity of graph products of abelian groups

TL;DR

This paper proves that groups with a graph-product decomposition of finitely-generated abelian groups have two unique canonical decompositions, one with cyclic vertex groups and another with abelian vertex groups satisfying the $T_0$ property.

Contribution

It establishes the uniqueness of two canonical graph-product decompositions for groups with finitely-generated abelian vertex groups, extending prior results.

Findings

Unique decomposition with directly-indecomposable cyclic groups

Unique $T_0$ graph decomposition with finitely-generated abelian groups

Builds on and extends previous work by Droms, Laurence, and Radcliffe

Abstract

We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.