Abstract Witte The project continues previous work of the PI on three related topics: superrigidity, tessellations of homogeneous spaces, and actions on the circle. 1) The Margulis Superrigidity Theorem applies to lattices only in semisimple groups, but the PI has extended this result to lattices in many other Lie groups. 2) The PI intends to continue investigating the structure of compact manifolds of the form DG/H, where G is a simply connected Lie group and H and D are closed subgroups, such that D acts properly on G/H. One goal is to add to our understanding of which homogeneous spaes G/H admit such a tessellation DG/H. 3) The PI proved that SL(3,Z) (and other arithmetic groups of higher Q-rank) has no interesting continuous actions on the circle or the real line. Attempts to extend this result to arithmetic groups whose Q-rank is 0 or 1 lead to interesting algebraic questions about arithmetic groups of higher R-rank: are they right orderable? are normal subgroups the only conjugation-invariant subsets that are closed under multiplication? Many important materials are crystals. The atomic structure of such a material is very symmetric, and a first step toward understanding it 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. Basic problems are to understand which spaces do contain crystals, and when it can happen that crystals in two different spaces have the same group of symmetries.
