Commutator Subgroups of Twin Groups and Grothendieck's Cartographical Groups

TL;DR

This paper investigates the algebraic structure of twin groups and their relation to Grothendieck's cartographical groups, providing finite presentations of their commutator subgroups and analyzing their properties such as freeness, hyperbolicity, and automorphisms.

Contribution

It offers a finite presentation for the commutator subgroup of twin groups and characterizes their algebraic and geometric properties, linking them to Grothendieck's cartographical groups.

Findings

The commutator subgroup $TW_{m+2}'$ has rank $2m-1$.

$TW_{m+2}'$ is free if and only if $m \u003c= 3$.

$TW_{m+2}$ is word-hyperbolic and lacks surface groups if and only if $m \u003c= 3$.

Abstract

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 $m \leq 3$.