  Let $TW_n$ be the twin group on $n$ arcs, $n \geq 2$. The group $TW_{m+2}$ is
isomorphic to Grothendieck's $m$-dimensional cartographical group $\mathcal
C_m$, $m \geq 1$. In this paper we give a finite presentation for the
commutator subgroup $TW_{m+2}'$, and prove that $TW_{m+2}'$ has rank $2m-1$. We
derive that $TW_{m+2}'$ is free if and only if $m \leq 3$. From this it follows
that $TW_{m+2}$ is word-hyperbolic and does not contain a surface group if and
only if $m \leq 3$. It also follows that the automorphism group of $TW_{m+2}$
is finitely presented for $m \leq 3$.
