The Right Angled Artin Group Functor as a Categorical Embedding

TL;DR

This paper establishes an algebraic characterization of right-angled Artin groups (RAAGs) and their graph homomorphisms, providing a new method to recover the defining graph from the group structure.

Contribution

It introduces a purely algebraic criterion to identify RAAGs within groups and relates graph homomorphisms to group homomorphisms, along with an algorithm for reconstructing the graph.

Findings

Algebraic characterization of RAAGs within groups

Correspondence between graph homomorphisms and group homomorphisms

Algorithm for recovering the defining graph from a RAAG

Abstract

It has long been known that the combinatorial properties of a graph $\Gamma$ are closely related to the group theoretic properties of its right angled artin group (raag). It's natural to ask if the graph homomorphisms are similarly related to the group homomorphisms between two raags. The main result of this paper shows that there is a purely algebraic way to characterize the raags amongst groups, and the graph homomorphisms amongst the group homomorphisms. As a corollary we present a new algorithm for recovering $\Gamma$ from its raag.