Four-dimensional compact Clifford-Klein forms of pseudo-Riemannian symmetric spaces with signature $(2, 2)$

TL;DR

This paper classifies four-dimensional pseudo-Riemannian symmetric spaces with signature (2,2) that admit compact Clifford-Klein forms, introducing a method for solvable spaces and examining a solvable analogue of Kobayashi's conjecture.

Contribution

It provides a classification of certain symmetric spaces admitting compact forms and develops a new method applicable to solvable symmetric spaces, also exploring a related conjecture.

Findings

Classification of 4D symmetric spaces with compact Clifford-Klein forms

Development of a method for solvable symmetric spaces

Evidence on the importance of reductive assumption in Kobayashi's conjecture

Abstract

We give a classification of irreducible four-dimensional symmetric spaces $G/H$ which admit compact Clifford-Klein forms, where $G$ is the transvection group of $G/H$. For this, we develop a method that applies to particular 1-connected solvable symmetric spaces. We also examine a `solvable analogue' of Kobayashi's conjecture for reductive groups and find an evidence that the reductive assumption in Kobayashi's conjecture is crucial.