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arXiv · 1311.6254

Clifford-Klein forms and a-hyperbolic rank

Maciej Bochenski·Aleksy Tralle
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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.

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Maciej Bochenski, Aleksy Tralle. 2014-11-14. Clifford-Klein forms and a-hyperbolic rank. https://doi.org/10.1093/imrn%2Frnu123

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