Deformation of proper actions on reductive homogeneous spaces

By Fanny Kassel. 2009

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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, that is, of quotients of G/H by discrete groups Γ that are uniform lattices in some 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 Γ keeps acting properly discontinuously and cocompactly on G/H. 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2fdb4195add3d575429edec6cf7dc3bd538670f6

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Anosov representations: domains of discontinuity and applications (2011, methodology): “In particular, as is noted in [51], Johnson and Millson [47] constructed explicit bending deformations with Zariski dense image in SO(2, 2n) when Γ is an arithmetic lattice.”