Topologically unique maximal elementary Abelian group actions on compact oriented surfaces

TL;DR

This paper classifies all finite maximal elementary abelian group actions on compact oriented surfaces of genus at least 2, identifying unique actions and their extensions, and constructs explicit equations for certain Riemann surfaces.

Contribution

It provides a complete classification of topologically unique elementary abelian group actions on surfaces and details the construction of defining equations for specific cases.

Findings

All such group actions are classified up to topological equivalence.

Group extensions for special classes are explicitly determined.

Explicit defining equations are constructed for certain Riemann surfaces.

Abstract

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus $\sigma\geq 2$ which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.