Real hyperbolic on the outside, complex hyperbolic on the inside

Abstract

The rank one symmetric spaces of negative curvature come in three infinite families: real hyperbolic space H; complex hyperbolic space CH; and quaternionic hyperbolic space QH. (The Cayley plane is the remaining example.) Aside from the obvious embeddings H ↪→ CH ↪→ QH the three geometries seem fairly unrelated to each other. For instance, H admits non-arithmetic lattices in all dimensions [GrP] while QH only admits arithmetic lattices [GrS]. (See [C] for a related result.) The question of non-arithmetic lattices in CH is a basic unsolved problem [DM]. For a representation-theoretic comparison of discrete subgroups in the different rank one spaces, see [Sh]. In this paper we make a new connection between H and CH. We construct a closed hyperbolic 3-manifold which (as a diffeomorphic copy) is the ideal boundary of a complex hyperbolic 4-manifold. Up to index 2, the isometry group of CH is PU(2, 1), the group of complex projective automorphisms of the unit ball in C. The ideal boundary of CH is the unit 3-sphere S. A spherical CR structure on a 3-manifold is a system of coordinate charts into S whose transition functions are restrictions of elements of PU(2, 1). While plenty of closed Seifert fibered manifolds admit spherical CR structures [KT], our example gives the only known spherical CR structure on a closed hyperbolic 3-manifold.