Commutator subgroups of twin groups and Grothendieck's cartographical groups

Abstract

Let TWnbe the twin group on narcs, n ≥2, which is a right-angled Coxeter group. The group TWm+2is isomorphic to Grothendieck’s m-dimensional cartographical group Cm, m ≥1. In this paper we give a finite presentation for the commutator subgroup TW′m+2, and prove that TW′m+2has rank 2m −1. We derive that TW′m+2is free if and only if m ≤3.