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Tessellations of Homogeneous Spaces of Classical Groups of Real Rank Two

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Abstract

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 Γ of G, such that Γ acts properly discontinuously on G/H, and the double-coset space Γ\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=SO(2, 2n)° or SU(2, 2n). Oh and Witte constructed additional examples in both of these cases, and obtained a complete classification when G=SO(2, 2n)°. We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G=SU(2, 2n). 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=K A + K, and study the intersection (K H K)∩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.

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Author notes

  1. Alessandra Iozzi

    Present address: FIM, ETH Zentrum, CH-8092, Zürich, Switzerland

  2. Dave Witte Morris

    Present address: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20910, U.S.A

    Alessandra Iozzi

  2. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, U.S.A

    Dave Witte Morris

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  1. Alessandra Iozzi
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  2. Dave Witte Morris
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Iozzi, A., Witte Morris, D. Tessellations of Homogeneous Spaces of Classical Groups of Real Rank Two. Geometriae Dedicata 103, 115–191 (2004). https://doi.org/10.1023/B:GEOM.0000013840.18788.81

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