Tessellations of Homogeneous Spaces of Classical Groups of Real Rank Two*

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a tessellation if there is a discrete subgroup of G, such that F acts properly discontinuously on G/H, and the double-coset space r\G/H is compact. Note that if either H or G / H is compact, then G / H has a tessellation; these are the obvious examples. It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, Kulkarni and Kobayashi constructed examples that are not obvious when G = S0(2,2n)O or SU(2, In). Oh and Witte constructed additional examples in both of these cases, and obtained a complete classification when G = S0(2 ,2ny . We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G = SU(2, In). This includes the construction of another family of examples. The main results are obtained from methods of Benoist and Kobayashi: we fix a Cartan decomposition G = KA+K, and study the intersection (KHK) n A+. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level. Mathematics Subject Classifications (2000). 22E40, 53C30.