Clifford–Klein Forms and a-Hyperbolic Rank

Abstract

The purpose of this article is to introduce and investigate properties of a tool (the a-hyperbolic rank) which enables us to obtain new examples of homogeneous spaces G/H which admit and do not admit a discontinuous action of a non virtually-abelian discrete subgroup. We achieve this goal by exploring in greater detail the technique of adjoint orbits developed by Okuda combined with the well-known conditions of Benoist. We find easy-to-check conditions on G and H expressed directly in terms of the Satake diagrams of the corresponding Lie algebras, in cases when G is a real form of a complex Lie group of type A n , D 2k+1 or E 6 . One of the advantages of this approach is the fact, that we don't need to know the embedding of H into G. Using the a-hyperbolic rank we also show, that the homogeneous space E 6 IV /H of reductive type admits a discontinuous action of a non virtually abelian discrete subgroup if and only if H is compact. Also, inspired by the work of Okuda on symmetric spaces G/H we find a list of simply connected 3-symmetric spaces admitting a discontinuous action of a non virtually-abelian discrete subgroup. This list yields almost complete classification of such spaces (there is one exception).

Nonexistence of standard compact Clifford–Klein forms of homogeneous spaces of exceptional Lie groups

“…We will need the notion of the a-hyperbolic rank [5], [6] which will be used in the sieving procedure eliminating some possibilities for compact Clifford-Klein forms. Let t be a split Cartan subalgebra in g containing a, and Σ ⊂ a * be a restricted root system of g. Choose a subset of positive roots Σ + ⊂ Σ.…”

“…Note that calculations of rank a-hyp g are done in [6]. If G is a connected linear real Lie group with the semisimple real Lie algebra g, the symbol G c denotes the connected complex Lie group corresponding to g c and such that G ⊂ G c .…”

Proper actions on strongly regular homogeneous spaces

“…A-hyperbolic dimensions of simple real Lie algebras can be calculated using data in Table 1 (for a detailed description how to calculate the a-hyperbolic rank of a simple Lie algebra please refer to [2]).…”

“…The problem of proper actions was studied for example in [12], [10], [9], [1], [13], [7] and [2]. We also refer the reader to the excellent survey [8].…”

A restriction on proper group actions on homogeneous spaces of reductive type

“…It is natural to ask when does a closed subgroup of G act properly on a space of reductive type G/H. This problem was treated, inter alia, in [1], [2], [4], [6], [9], [10], [12] and [14]. In [6] one can find a very important criterion for a proper action of a subgroup L reductive in G. To state this criterion we need to introduce some additional notation.…”