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

Abstract

Let G be a simple non-compact linear connected Lie group and H⊂G 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 Γ⊂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 Γ⊂G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup, provided that the pair (g,h) is contained in a database created by De Graaf and Marrani.