Commutator Subgroups of Twin Groups and Grothendieck's Cartographical Groups
| Autor: | Dey, Soumya, Gongopadhyay, Krishnendu |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $m \leq 3$. Comment: Few corollaries added, acknowledgements added, comments are welcome |
| Databáze: | arXiv |
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