Power commutator 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.