Infinite presentations for fundamental groups of surfaces

TL;DR

This paper provides infinite presentations of the fundamental groups of surfaces, using simple loops as generators, and extends to non-orientable surfaces with specific subgroup presentations.

Contribution

It introduces new infinite presentations for surface fundamental groups and their subgroups, emphasizing simple loop generators, including for non-orientable surfaces.

Findings

Infinite presentation for orientable surface groups using simple loops

Infinite presentation for certain subgroups in non-orientable surfaces

Extensions to non-orientable surfaces with annulus neighborhoods

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.