The Fundamental Group of a Compact Riemann Surface via Branched Covers
Meirav Amram, Michael Chitayat, Yaacov Kopeliovich
TL;DR
The paper constructs a concrete, algorithmic bridge between describing a compact Riemann surface X as a branched cover of $\mathbb{P}^1$ and the classical fundamental-group presentation $\pi_1(X,x) \cong \langle a_1,b_1,\dots,a_g,b_g\mid \prod_{i=1}^g [a_i,b_i] = 1\rangle$. Using Schreier theory, permutation representations from monodromy, and a sequence of five steps, it produces a sequence of isomorphisms from the cover’s unramified fundamental subgroup to the classical surface-group presentation, explicitly tracking generators and the single relator. The method is illustrated on a simple branched-cover example and on hyperelliptic curves, demonstrating how to obtain the commutator relation from branch data and how to express it in terms of the original branching permutations. This provides a constructive link between Nielsen classes, Hurwitz spaces, and the algebraic description of function-field extensions of $\mathbb{C}(z)$, with potential for algorithmic implementation and broader applications in the study of branched covers.
Abstract
Let $X$ be a compact Riemann surface of genus $g$ and let $x \in X$. We derive the classical presentation of $π_1(X,x)$ (i.e the one given by $2g$ generators $a_1,b_1, \dots, a_g,b_g$ and the relation $\prod_{i=1}^g[a_i,b_i] = 1$) from the description of $X$ as a branched cover $f : X \to \mathbb{C}\mathbb{P}^1$.