Compact Clifford-Klein forms of symmetric spaces - revisited

Abstract

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space G/H if H is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for G/H, followed by the most update and complete list of symmetric spaces with/without compact quotients. In the second half, as applications of general theory, we prove: (i) there exists a 15 dimensional compact pseudo-Riemannian manifold of signature (7, 8) with constant curvature, (ii) there exists a compact quotient of the complex sphere of dimension 1, 3 and 7, and (iii) there exists a compact quotient of the tangential space form of signature (p, q) if and only if p is smaller than the Hurwitz-Radon number of q.

Deformation of proper actions on reductive homogeneous spaces

“…Note that L always admits torsion-free uniform lattices by [3]. Kobayashi and Yoshino conjectured that any reductive homogeneous space G/H admitting compact quotients admits standard ones [21,Conj. 3.3.10]; this conjecture remains open.…”

Nonexistence of standard compact Clifford–Klein forms of homogeneous spaces of exceptional Lie groups

“…The aim of this paper is to prove the following. Recently, Tojo [34] proved that irreducible symmetric spaces G/H that admit standard Clifford-Klein forms are exactly those in the list [18]. Thus, there is an overlap with our results in this case.…”

Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

“…One interesting problem in differential geometry is to decide whether a given homogeneous space G/I possesses or not a compact quotient. A more general related question is to decide whether there exist compact manifolds locally modelled on (G, G/I ) (see, for instance [1,2,22,24]). …”

“…It turns out that compact quotients of S n are known to exist only for n = 1, 3 or 7. We discussed the case n = 3 above, and the existence of a compact quotient of S 7 was proved in [22]. Here, we dare ask with [22]: Conjecture 2.3 S n has no compact quotients, for n = 1, 3, 7.…”