Proof of Kobayashi’s rank conjecture on Clifford-Klein forms. (English) Zbl 1479.57079

Suppose \(G/H\) is a homogeneous space of reductive type and let \(K\subseteq G\) and \(K_H\subseteq H\) be maximal compact subgroups. The author proves that if \(\operatorname{rank}(G) - \operatorname{rank}(K) < \operatorname{rank}(H) - \operatorname{rank}(K_H)\), then \(G/H\) does not admit a compact Clifford-Klein form. That is, there is no discrete subgroup \(\Gamma\subseteq G\) which acts property and freely for which the quotient space \(\Gamma \backslash G/H\) is compact. This affirmatively answers a conjecture of Kobayashi.
The author had previously established [J. Differ. Geom. 100, No. 3, 533–545 (2015; Zbl 1323.53056); Publ. Res. Inst. Math. Sci. 53, No. 2, 287–298 (2017; Zbl 1371.53051)] that non-existence of a compact Clifford-Klein form would follow if a certain map on the level of Lie algebra cohomology was not injective.
The main result of the article under review is that, under the above rank inequalities, this Lie algebra cohomology map is not injective. The author proves this by constructing a spectral sequence and using it to show that the Lie algebra cohomomology map is injective if and only if another map is surjective. However, the rank condition easily implies this latter map is not surjective.


57S30 Discontinuous groups of transformations
17B56 Cohomology of Lie (super)algebras
55P62 Rational homotopy theory
57T15 Homology and cohomology of homogeneous spaces of Lie groups
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