Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a calculational approach

TL;DR

This paper develops an algorithm to identify homogeneous spaces of real simple Lie groups that admit proper actions by discrete, non virtually abelian subgroups, providing a classification for groups of rank up to 8.

Contribution

It introduces a calculational approach and algorithm for classifying such homogeneous spaces, specifically for groups with rank up to 8 and maximal proper semisimple subgroups.

Findings

List of all non-compact homogeneous spaces G/H with proper actions by non virtually abelian subgroups for rank ≤ 8.

Algorithm successfully identifies these spaces in the specified case.

Provides a systematic method for studying proper actions of discrete subgroups on homogeneous spaces.

Abstract

Let G be a simple non-compact linear connected Lie group and H be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non virtually abelian subgroups of G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup of G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup.