Compact Clifford-Klein forms of homogeneous spaces of SO(2,n)

TL;DR

This paper classifies certain homogeneous spaces of SO(2,n) that admit compact Clifford-Klein forms, providing new examples for even n and exploring conditions under which such forms do not exist, with additional insights into SL(3,R).

Contribution

It offers a complete classification for even n and partial results for odd n regarding homogeneous spaces of SO(2,n) with compact Clifford-Klein forms, and analyzes noncompact forms of SL(3,R).

Findings

Classified all closed, connected subgroups H of SO(2,n) for even n with compact Clifford-Klein forms.

Identified new examples of homogeneous spaces of SO(2,n) with compact Clifford-Klein forms when n is even.

Proved nonexistence of compact Clifford-Klein forms for certain subgroups of SL(3,R) when neither H nor G/H is compact.

Abstract

A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein form, but our classification is not quite complete when n is odd. The work reveals new examples of homogeneous spaces of SO(2,n) that have compact Clifford-Klein forms, if n is even. Furthermore, we show that if H is a closed, connected subgroup of G = SL(3,R), and neither H nor G/H is compact, then G/H does not have a compact Clifford-Klein form, and we also study noncompact Clifford-Klein forms of finite volume.