Groups generated by Dehn Twists along fillings of surfaces

Abstract

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