Presentations for the punctured mapping class groups in terms of Artin groups

Abstract

Consider an oriented compact surface $F$ of positive genus, possibly with boundary, and a finite set $\mathcal{P}$ of punctures in the interior of $F$, and define the punctured mapping class group of $F$ relatively to $\mathcal{P}$ to be the group of isotopy classes of orientation-preserving homeomorphisms $h:F\to F$ which pointwise fix the boundary of $F$ and such that $h\left(\mathcal{P}\right)=\mathcal{P}$. In this paper, we calculate presentations for all punctured mapping class groups. More precisely, we show that these groups are isomorphic with quotients of Artin groups by some relations involving fundamental elements of parabolic subgroups.