Proper actions on strongly regular homogeneous spaces

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 ⊂ G locally isomorphic to SL(2, R). We classify all such spaces.

Non-virtually abelian discontinuous group actions vs. proper SL(2,R)-actions on homogeneous spaces

“…For many important classes of homogeneous spaces C2 and C3 are equivalent. For example, apart from irreducible symmetric spaces, this holds for some strongly regular homogeneous spaces ( [2],Theorem 2 and Corollary 1). On the other hand, there are known exactly two examples of spaces which fulfill the condition C2 but not C3 [2,27].…”

“…For example, apart from irreducible symmetric spaces, this holds for some strongly regular homogeneous spaces ( [2],Theorem 2 and Corollary 1). On the other hand, there are known exactly two examples of spaces which fulfill the condition C2 but not C3 [2,27]. Thus for a space of reductive type we only have the following:…”

“…The known two examples of homogeneous spaces G/H which are C2 but not C3 are obtained in [2] and [27] Our aim is to solve this problem. We create algorithms verifying C2 versus C3 and implement them in the computational algebra system GAP4 [15].…”

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

“…It is worth noting that it is a difficult open problem to decide if there exists Γ which is not a discrete subgroup of SL(2, R) and which acts properly on G/H. Actually there are two examples of homogeneous spaces G/H which admit a proper action of a discrete subgroup Γ ⊂ G, but do not admit a proper action of L locally isomorphic SL(2, R) [2].…”