On finitely generated normal subgroups of right-angled Artin groups and graph products of groups

TL;DR

This paper extends Schreier's classical result from free groups to right-angled Artin groups and graph products, showing that finitely generated normal subgroups have specific algebraic structures and decidability properties.

Contribution

It proves that finitely generated normal subgroups of RAAGs lead to abelian-by-finite quotients and extends this to graph products, also establishing their algorithmic properties.

Findings

Finitely generated normal subgroups of RAAGs have abelian-by-finite quotients.

The result extends to graph products of groups.

Normal subgroups in RAAGs have decidable word, conjugacy, and membership problems.

Abstract

A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we deduce, among others, that finitely generated normal subgroups of RAAGs have decidable word, conjugacy and membership problems and that they are hereditarily conjugacy separable.