On manifolds locally modelled on non-Riemannian homogeneous spaces
The investigation of compact manifolds locally modeled on a homogeneous space of a finite dimensional Lie group is continued. The aim of this paper is threefold. First, the proof of the main result of a previous paper [\textit{R. J. Zimmer}, J. Am. Math. Soc. 7, 159-168 (1994; Zbl 0801.22009)] is drastically simplified. The approach considered in the present paper enables to use Moore's ergodicity theorem in place of Ratner's theorem which is required in the proof given in the above mentioned paper. The superrigidity property for cocycles is used in proving the main result. Second, using the new approach, the results of the previous paper are extended to a more general situation. In particular, it is shown that \(SL (m) \backslash SL(2m)\) does not admit a cocompact lattice, nor does \(J \backslash SL(2m)\) where \(J \subset SL(m)\) is either \(SO(p,q)\), \(p + q = m\), or \(Sp(2r,R)\) with \(2r = m\). Third, all these results are generalized to the case of compact forms. Nonconnected algebraic hulls are also considered.
scientific article; zbMATH DE number 881242