Mathematical Sciences: Discrete Subgroups of Lie Groups

该项目延续了PI之前在三个相关主题上的工作：超刚性、均匀空间的镶嵌和对圆的作用。1)马古利斯超刚性定理只适用于半单群中的格，但PI已将这一结果推广到许多其他李群中的格。2) PI打算继续研究形式为D\G/H的紧流形的结构，其中G是单连通李群，H和D是闭子群，使得D恰当地作用于G/H。一个目标是增加我们对哪些齐次空间G/H承认这样的镶嵌D\G/H的理解。3) PI证明了SL(3，Z)（以及其他高q秩的算术群）在圆或实直线上没有有趣的连续作用。尝试将这个结果推广到q秩为0或1的等差群，会导致关于r秩更高的等差群的有趣代数问题：它们是正确可序的吗？正规子群是唯一在乘法下闭合的共轭不变子集吗？许多重要的材料都是晶体。这种材料的原子结构是非常对称的，理解它的第一步是研究由所有对称结构形成的基团。这个项目研究了数学空间中晶体产生的对称群，而不是我们生活的三维宇宙。最基本的问题是了解哪些空间包含晶体，以及在什么情况下，两个不同空间中的晶体具有相同的对称性。

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 D\G/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 D\G/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.