DISCONTINUOUS GROUPS AND CLIFFORD-KLEIN FORMS OF PSEUDO-RIEMANNIAN HOMOGENEOUS MANIFOLDS

Toshiyuki KOBAYASHI University of Tokyo and Institut Mittag-Leffler

Abstract

Let G be a Lie group and H its subgroup. A Clifford-Klein form of a homogeneous manifold G/H is the double coset space Γ\G/H if Γ is a subgroup of G acting properly discontinuously and freely on G/H. For example, any closed Riemann surface M with genus ≥ 2 is biholomorphic to a compact Clifford-Klein form of the Poincaré plane G/H = P SL(2, ℝ)/SO(2). On the other hand, there is no compact Clifford-Klein form of the hyperboloid of one sheet G/H = P SL(2, ℝ)/SO(1, 1). Even more, there is no infinite discrete subgroup of G which acts properly discontinuously on G/H (the Calabi-Markus phenomenon). As is observed in the second example, not all discrete subgroup of G can act properly discontinuously on a homogeneous manifold G/H if H is noncompact. We discuss recent developments in the theory of discontinuous groups acting on a homogeneous manifold G/H where G is a real reductive Lie group and H is a noncompact reductive subgroup. Geometric ideas of various methods together with a number of examples are presented regarding fundamental problems: “which homogeneous manifolds G/H admit properly discontinuous actions of infinite discrete subgroups of G ?”, and “which homogeneous manifolds G/H admit compact Clifford-Klein forms ?”

Contents

0. Introduction to Clifford-Klein forms.

1. Discontinuous actions (discrete and continuous version).

2. Calabi-Markus phenomenon.

3. Generalized Hirzebruch’s proportionality principle and its application.

4. Compact Clifford-Klein forms of non-Riemannian homogeneous spaces.

5. Open problems.

Typeset by AMS-TEX