On extension of coverings

TL;DR

This paper investigates conditions under which boundary coverings of surfaces can be extended to the entire surface, focusing on infinite covers and providing group-theoretic criteria for such extensions, including regularity and connectivity.

Contribution

It establishes necessary and sufficient conditions for extending boundary coverings to surfaces, especially for infinite covers, and introduces new group-theoretic results relevant to these extensions.

Findings

Characterization of extendability of boundary coverings

Conditions for connected and regular extensions

Group-theoretic results on symmetric groups and commutators

Abstract

We address the question of when a covering of the boundary of a surface can be extended to a covering of the surface (equivalently: when is there a branched cover with a prescribed monodromy). If such an extension is possible, when can the total space be taken to be connected? When can the extension be taken to be regular? We give necessary and sufficient conditions for both finite and infinite covers (infinite covers are our main focus). In order to prove our results, we show group-theoretic results of independent interests, such as the following extension (and simplification) of the theorem of Ore}: every element of the infinite symmetric group is the commutator of two elements which, together, act transitively