Clifford–Klein Forms and a-Hyperbolic Rank. (28th July 2014)

• Record Type: Journal Article
• Authors: Bocheński, Maciej; Tralle, Aleksy
• Is Part Of: International mathematics research notices. Volume 2015:Number 15(2015)
• Page Start: 6267
• Page End: 6285
• Publication Date: 2014-07-28
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnu123
• Languages: English
• ISSNs: 1073-7928

Abstract

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 2 k +1, or E 6 . One of the advantages of this approach is the fact, that we do not 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).