A Cohomological Approach to the Existence Problem of Compact Clifford-Klein Forms for Indecomposable Pseudo-Riemannian Symmetric Spaces

We give examples of indecomposable but reducible pseudo-Riemannian symmetric spaces of arbitrary signature which admit no compact Clifford-Klein forms by a cohomological approach.We also show some series of even dimensional indecomposable but reducible pseudo-Riemannian symmetric spaces of arbitrary signature admit compact Clifford-Klein forms.We give another proof for the classification of indecomposable but reducible symmetric spaces with signature (2, 2) which admit compact Clifford-Klein forms which is proved in a forthcoming paper [Four dimensional compact Clifford-Klein forms of pseudo-Riemannian symmetric spaces with signature (2, 2)] by a different approach.