Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a calculational 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 intersted in finding classes of homogeneous spaces G/H admitting proper actions of discrete nonvirtually 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.