Groups generated by Dehn Twists along fillings of surfaces

TL;DR

This paper constructs specific sets of simple closed curves on surfaces such that the Dehn twists along these curves generate free groups of arbitrary rank, revealing new algebraic structures in surface mapping class groups.

Contribution

It demonstrates the existence of fillings on surfaces whose Dehn twist groups are free of any prescribed rank, advancing understanding of the algebraic properties of surface mapping class groups.

Findings

Existence of fillings with Dehn twist groups isomorphic to free groups of any rank d

Construction of explicit examples of such fillings for all d ≥ 2

Shows the algebraic richness of Dehn twist groups in surface topology

Abstract

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 $d$.