CAT(0) cubical complexes for graph products of finitely generated abelian groups

TL;DR

This paper demonstrates that graph products of finitely generated abelian groups can act properly and cocompactly on CAT(0) cubical complexes, generalizing known complexes for specific groups and connecting to existing embeddings.

Contribution

It introduces a construction of CAT(0) cubical complexes for graph products of finitely generated abelian groups, extending previous complexes for right-angled Artin and Coxeter groups.

Findings

Every such graph product acts properly and cocompactly on a CAT(0) cubical complex.

The complex generalizes the Salvetti and Coxeter complexes, up to subdivision.

The approach relates to embeddings into right-angled Coxeter groups.

Abstract

We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it is related to the embedding into a right-angled Coxeter group described by Davis and Januszkiewicz. We compare the approaches and also adapt the argument that the action extends to finite index supergroup that is a graph product of finite groups.