One focus of this project is the study of tessellations of homogeneous spaces. Namely, if G/H is a non-compact, simply connected homogeneous space of a connected Lie group G, the question is whether there is a properly discontinuous subgroup D of G such that the orbit space DG/H is compact. Some special cases have been studied by L. Auslander, Y. Benoist, G. A. Margulis, R. J. Zimmer, H. Oh, and others. If there is a compact subset C of G with CHC = G, then we say that H is a Cartan-decomposition subgroup, and it is easy to see that G/H does not have a tessellation. The project will use Cartan-decomposition subgroups as a starting point in a study of tessellations of homogeneous spaces of semisimple groups of real rank two. The other main focus of this project is generalizing the PI's theorem that arithmetic groups of Q-rank at least two have no interesting continuous actions on connected one-manifolds. (In algebraic terms, the theorem asserts that no central extension of any such arithmetic group is right orderable.) The PI conjectured that the assumption on Q-rank can be replaced by the same assumption on the real rank, and this leads to interesting questions about the algebraic structure of arithmetic groups.
Many important materials are crystals. The atomic structure of such a material is very symmetric, and a first step toward understanding the physical properties of the material is to study the group formed by all the symmetries of the structure. This project investigates the symmetry groups that arise from crystals in mathematical spaces other than the 3-dimensional universe we live in. A basic problem addressed by this project is to understand which spaces do contain crystals. Also, this project will study the structure of the symmetry group of the crystals that do exist. For example, the PI and others have shown, in many cases, that crystals in two different spaces cannot have the same group of symmetries. However, it is often still not known whether the group of symmetries can be the same as the symmetries of a one-dimensional non-rigid (and, hence, non-crystalline) structure.
