A topological construction of families of Galois covers of the line

TL;DR

This paper introduces a new topological method for constructing families of Galois covers of the line, offering an alternative to existing approaches and correcting previous inaccuracies for non-abelian groups.

Contribution

It presents a novel topological construction of Galois covers that bypasses Teichmüller theory and Fuchsian groups, improving upon prior methods especially for non-abelian groups.

Findings

Provides a new topological construction method

Corrects inaccuracies in previous non-abelian cases

Offers an alternative to Teichmüller-based approaches

Abstract

We describe a new construction of families of Galois coverings of the line using basic properties of configuration spaces, covering theory, and the Grauert-Remmert Extension Theorem. Our construction provides an alternative to a previous construction due to Gonz\'alez-D\'{i}ez and Harvey (which uses Teichm\"uller theory and Fuchsian groups) and, in the case the Galois group is non-abelian, corrects an inaccuracy therein.