Proper actions on strongly regular homogeneous spaces

TL;DR

This paper characterizes when strongly regular homogeneous spaces with certain properties admit proper actions by specific subgroups, providing a classification of all such spaces based on these actions.

Contribution

It establishes a precise criterion linking proper actions of non-virtually abelian groups to actions of SL(2,R)-like subgroups on these spaces and classifies all such spaces.

Findings

Proper actions by non-virtually abelian groups occur iff SL(2,R)-like subgroup actions exist.

Complete classification of strongly regular homogeneous spaces with these properties.

Characterization of proper actions in terms of subgroup structures.

Abstract

Let G/H be a strongly regular homogeneous space such that H is a Lie group of inner type. We show that G/H admits a proper action of a discrete non-virtually abelian subgroup of G if and only if G/H admits a proper action of a subgroup L of G locally isomorphic to SL(2,R). We classify all such spaces.