Deformation of proper actions on reductive homogeneous spaces

Abstract

Let G be a real reductive Lie group and H a closed reductive subgroup of G. We investigate the deformation of "standard" compact quotients of G/H, i.e., of quotients of G/H by discrete subgroups Gamma of G that are uniform lattices in a closed reductive subgroup L of G acting properly and cocompactly on G/H. For L of real rank 1, we prove that after a small deformation in G, such a group Gamma remains discrete in G and its action on G/H remains properly discontinuous and cocompact. More generally, we prove that the properness of the action of any convex cocompact subgroup of L on G/H is preserved under small deformations, and we extend this result to reductive homogeneous spaces G/H over any local field. As an application, we obtain compact quotients of SO(2n,2)/U(n,1) by Zariski-dense discrete subgroups of SO(2n,2) acting properly discontinuously.