Geometry and Topology Seminar

A pseudo-Riemannian manifold is a manifold where each tangent space is endowed with a quadratic form that is non-degenerate, but not necessarily positive definite. A typical example is the hyperbolic space H(p,q), which is a pseudo-Riemannian manifold of signature (p,q) and constant negative sectional curvature. It is homogeneous, as it admits a transitive isometric action of the Lie group SO(p,q+1). A long standing question is to determine for which values of (p,q) one can find a discrete subgroup of SO(p,q+1) acting properly discontinuously and cocompactly on H(p,q). In this talk I will show that there is no such action when p is odd and q >0. The proof relies on a computation of the volume of the corresponding quotient manifold. The proof also implies that, when p is even, this volume is essentially rational. I will discuss in more details the case of H(2,1) (the 3-dimensional anti-de Sitter space), for which compact quotients exist and have been described by work of Kulkarni-Raymond and Kassel.