Infinite presentations for fundamental groups of surfaces

Abstract

For any finite type connected surface $S$, we give an infinite presentation of the fundamental group ${\pi }_1(S,*)$ of $S$ based at an interior point $*\in S$ whose generators are represented by simple loops. When $S$ is non-orientable, we also give an infinite presentation of the subgroup of ${\pi }_1(S,*)$ generated by elements which are represented by simple loops whose regular neighborhoods are annuli.