Compact Homogeneous Universes

TL;DR

This paper classifies compact homogeneous universe topologies, detailing their degrees of freedom and geometric structures, except for hyperbolic spaces, by analyzing isometry groups and Teichmüller parameters.

Contribution

It provides a comprehensive classification of compact homogeneous universes, including their degrees of freedom and explicit constructions, extending previous work to a broader class of geometries.

Findings

Complete classification of compact homogeneous universes (except hyperbolic)

Explicit construction of isometry subgroups and Teichmüller parameters

Identification of degrees of freedom from universal cover and Teichmüller space

Abstract

A thorough classification of the topologies of compact homogeneous universes is given except for the hyperbolic spaces, and their global degrees of freedom are completely worked out. To obtain compact universes, spatial points are identified by discrete subgroups of the isometry group of the generalized Thurston geometries, which are related to the Bianchi and the Kantowski-Sachs-Nariai universes. Corresponding to this procedure their total degrees of freedom are shown to be categorised into those of the universal covering space and the Teichm\"uller parameters. The former are given by constructing homogeneous metrics on simply connected manifold. The Teichm\"uller spaces are also given by explicitly constructing expressions for the discrete subgroups of the isometry group.