Commutator Subgroups of Twin Groups and Grothendieck's Cartographical groups

Soumya Dey, Krishnendu Gongopadhyay

Published 2018-04-15Version 1

Let $TW_n$ be the twin group on $n$ arcs. The group $TW_{n+2}$ is isomorphic to Grothendieck's $n$-dimensional cartographical group $\mathcal C_n$, $n \geq 1$. In this paper we give a finite presentation of the commutator subgroup $TW_{n+2}'$. We further prove that the commutator subgroup $TW_{n+2}'$ has rank $2n-1$, $n \geq 1$. As corollaries, we derive that $TW_{n+2}'$ is free if and only if $n \leq 3$. From this it follows that the automorphism group of $TW_{n+2}$ is finitely presented for $n \leq 3$.