Commutator Subgroups of Twin Groups and Grothendieck's Cartographical Groups

Abstract

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

Few corollaries added, acknowledgements added, comments are welcome