Groups generated by Dehn Twists along fillings of surfaces

Rakesh Kumar

Published 2023-07-26Version 1

Let $S_g$ denote a closed oriented surface of genus $g \geq 2$. A set $\Omega = \{ c_1, \dots, c_d\}$ of pairwise non-homotopic simple closed curves on $S_g$ is called a filling system or simply a filling of $S_g$, if $S_g\setminus \Omega$ is a union of $\ell$ topological discs for some $\ell\geq 1$. For $1\leq i\leq d$, let $T_{c_i}$ denotes the Dehn twist along $c_i$. In this article, we show that for each $d\geq 2$, there exists a filling $\Omega=\{c_1,c_2,\dots, c_d\}$ of $S_g$ such that the group $\langle T_{c_1}, T_{c_2},\dots,T_{c_d}\rangle$ is isomorphic to the free group of rank $d$.