Power commutator groups

TL;DR

This paper studies a class of finitely generated groups defined by power relations of commutators, showing their embeddings into Coxeter groups and establishing their linearity using elementary combinatorial group theory methods.

Contribution

It introduces a new class of groups called power commutator groups and proves their embeddings into Coxeter groups, demonstrating their linearity.

Findings

Power commutator groups embed into Coxeter groups.

These groups are linear due to their embeddings.

Includes subclasses like graph groups and Shephard groups.

Abstract

We consider the class of finitely generated groups whose relators are powers of commutators of the generators. This class contains as a small subclass graph groups (also called RAAGs), namely if all powers are one. Graph groups are the only torsionfree groups in this class. The generators are of infinite order, but we may also add torsion by assigning arbitrary orders to the generators. Then the above mentioned small subclass contains partially commutative Shephard groups. We show that these groups embed into Coxeter groups as finite index subgroups, thus establishing the linearity of these groups. The very short proof requires only elementary methods in combinatorial group theory.