Infinite presentations for fundamental groups of surfaces

Abstract

For any finite type connected surface $S$, we give an infinite presentation of the fundamental group $\pi_1(S,\ast)$ of $S$ based at an interior point $\ast\in{S}$ whose generators are represented by simple loops. When $S$ is non-orientable, we also give an infinite presentation of the subgroup of $\pi_1(S,\ast)$ generated by elements which are represented by simple loops whose regular neighborhoods are annuli.