A classification of Thurston geometries without compact quotients

TL;DR

This paper provides a comprehensive classification of 3-dimensional simply connected manifolds with Lie group actions that are transitive and have compact isotropy groups, focusing on geometries without compact quotients.

Contribution

It offers a complete classification of Thurston geometries in the non-compact case, expanding understanding of 3D geometric structures.

Findings

Classification of all such geometries achieved

Identification of new non-compact Thurston geometries

Framework for analyzing similar geometric structures

Abstract

We classify pairs $(M,G)$ where $M$ is a $3$--dimensional simply connected smooth manifold and $G$ a Lie group acting on $M$ transitively, effectively with compact isotropy group.