Discrete Groups and Non-riemannian Homogeneous Spaces

A basic question in geometry is to understand compact locally homogeneous manifolds, i.e., those compact manifolds that can be locally modelled on a homogeneous space J\H of a finite-dimensional Lie group H. This means that there is an atlas on a manifold M consisting of local diffeomorphisms with open sets in J\H where the transition functions between these open sets are given by translations by elements of H. A basic example is the case of M = J\H jr where r cHis a discrete subgroup acting freely and properly discontinuously on J j H, with a compact quotient. (These examples can be abstractly characterized as the "complete" case. See [G].) We then call r a co compact lattice on J\H. When J is compact, J\H has an H-invariant Riemannian metric, so that M itself becomes a Riemannian manifold in a natural way. Thus, the class of locally homogeneous Riemannian manifolds includes (but is not exhausted by) the locally symmetric spaces. For H = Aff(lRn) and J = GL(n, lR) , the spaces J\Hjr are the complete affinely flat manifolds, a non-Riemannian example which has been extensively studied. When H is simple and J is not compact, the situation is far from being well understood. Some special series of such homogeneous spaces admitting a cocompact lattice have been constructed by Kulkarni [Ku] and by T. Kobayashi [K]. For example, Kulkarni shows the existence of such a discrete group r for the homogeneous spaces SO(l, 2n)\SO(2, 2n) and Kobayshi does the same for U(l, n)\SO(2, 2n). On the other hand, the constructions they give for these (and other similar) series are quite special, and there has been the general suspicion that most homogeneous spaces J\H with H simple and J not compact do not admit a cocompact lattice. (We recall that, when J is compact, i.e. the Riemannian case, such r always exist by a result of Borel [B].) One general situation in which it is known that no such r exists is the case in which J\H is noncompact and of reductive type and lR-rank(H) = lR-rank(J). This situation was studied by Calabi-Markus [CM], Wolf [W], and Kobayashi [K], and the nonexistence of r in this situation is a special case of a more general result known as the Calabi-Markus phenomenon. Kobayashi and Ono [KO] use the Hirzebruch proportionality principle to show that some other special series do