Homogeneous Transformation Groups of the Sphere

In this paper, we study the structure of homogeneous subgroups of the homeomorphism group of the sphere, which are defined as closed groups of homeomorphisms of the sphere that contain the rotation group. We prove two structure theorems about the behaviour and properties of such groups and present a diagram of the structure of these groups partly on the basis of these results. In addition, we prove a number of explicit relations between the groups in the diagram.