A cohomological obstruction to the existence of compact Clifford–Klein forms
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Yosuke Morita [1]
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[1] University of Tokyo
University of Tokyo
Japón
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- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 23, Nº. 3, 2017, págs. 1931-1953
- Idioma: inglés
- Enlaces
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Resumen
In this paper, we continue the study of the existence problem of compact Clifford–Klein forms from a cohomological point of view, which was initiated by Kobayashi–Ono and extended by Benoist–Labourie and the author. We give an obstruction to the existence of compact Clifford–Klein forms by relating a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology with an upper-bound estimate for cohomological dimensions of discontinuous groups. From this obstruction, we derive some examples, e.g. SO0(p + r, q)/(SO0(p, q) × SO(r)) (p, q,r ≥ 1, q: odd) and SL(p + q, C)/SU(p, q) (p, q ≥ 1), of a homogeneous space that does not admit a compact Clifford–Klein form. To construct these examples, we apply Cartan’s theorem on relative Lie algebra cohomology of reductive pairs and the theory of ε-families of semisimple symmetric pairs.
