A pseudo-Riemannian homogeneous space \(M\) of finite volume can be presented as \(M=G/H\), where \(G\) is a Lie group acting transitively and isometrically on \(M\), and \(H\) is a closed subgroup of \(G\).
The condition that \(G\) acts isometrically and thus preserves a finite measure on \(M\) leads to strong algebraic restrictions on \(G\). In the special case where \(G\) has no compact semisimple normal subgroups, it turns out that the isotropy subgroup \(H\) is a lattice, and that the metric on \(M\) comes from a bi-invariant metric on \(G\).
This result allows us to recover Zeghib’s classification of Lorentzian compact homogeneous spaces, and to move towards a classification for metric index \(2\).
As an application we can investigate which pseudo-Riemannian homogeneous spaces of finite volume are Einstein spaces. Through the existence questions for lattice subgroups, this leads to an interesting connection with the theory of transcendental numbers, which allows us to characterize the Einstein cases in low dimensions.
This talk is based on joint works with Oliver Baues, Yuri Nikolayevsky and Abdelghani Zeghib.