Recent work on Beauville surfaces, structures and groups

TL;DR

This survey explores the groups used to construct Beauville surfaces, highlighting their geometric properties, open problems, and conjectures related to these complex surfaces.

Contribution

It provides a comprehensive overview of the groups involved in Beauville surface constructions and discusses open problems and conjectures in the field.

Findings

Identification of groups suitable for Beauville surfaces

Discussion of geometric properties influenced by group choices

Presentation of open problems and conjectures in the area

Abstract

Beauville surfaces are a class of complex surfaces defined by letting a finite group $G$ act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group $G$. In this survey we discuss the groups that may be used in this way. \emph{En route} we discuss several open problems, questions and conjectures.