Clifford-Klein forms and a-hyperbolic rank

Abstract

The porpose 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 almost compact Clifford-Klein forms. 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 types A, D or E6. 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 G/H of reductive type of the real form of E6 (of the IV type) admits compact Clifford-Klein forms if and only if H is compact. Inspired by the work of Okuda on symmetric spaces G/H we classify all 3-symmetric spaces admitting almost compact Clifford-Klein forms.

Slight change of the title and numerous corrections suggested by the referee of IMRN, appears in International Mathematics Research Notices, 2014

References in corpus (1)

• Classification of semisimple symmetric spaces with proper SL(2,R)-actions

Cited by in corpus (3)

• Proper SL(2,R)-actions on homogeneous spaces
• Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a calculational approach
• A restriction on proper actions on homogeneous spaces of reductive type