Closed and transitive transformation groups of a surface

TL;DR

This paper surveys the structure of closed, transitive transformation groups on surfaces, detailing their relations on the sphere, and describing groups acting on tori and higher genus surfaces.

Contribution

It provides a comprehensive classification and relations of transformation groups acting on various surfaces, including the sphere, torus, and higher genus surfaces.

Findings

Relations between groups on the sphere containing rotations

Diagram of group connections on the sphere

Descriptions of transformation groups on tori and higher genus surfaces

Abstract

The purpose of this paper is to survey the structure of closed and transitive transformation groups acting on a closed surface. In particular, we prove a number of relations between groups acting on the sphere that contain the rotation group, together with a diagram of how these groups are connected. In addition, we describe transformation groups acting on the torus and higher genus surfaces.