Complex hyperbolic reflection groups and lattices

提出的研究是对PU(n，1)中(实和复)反射群的系统研究。其主要目的是得到PU(2，1)中新的离散子群和格，更具体地说，是许多新的非算术格。双曲反射群是李群中离散子群和格的领域中的一类重要的群，更广泛地说，在几何和拓扑学中也是如此。这样的群可以通过直接的几何描述和理解来获得，这对于用代数或算术定义的群来说并不总是清楚的。虽然这些反射群在常曲率设置下相对容易理解(它们在欧几里得、球面或实双曲n维空间中是“Coxeter群”)，但对它们的复杂双曲对应关系知之甚少。“镶嵌”或晶体结构是一种用不重叠的瓷砖填充空间的方式，这种图案经常无限重复。“晶格”是细分的对称群。了解欧几里德三维空间中的这些晶体结构在化学中是至关重要的。PI研究“双曲空间”中的相似结构。实双曲空间和复双曲空间是以实数或复数为模型的负曲率空间。负曲率大致是指不相交的“直线”在两个方向上倾向于彼此发散。在洛伦兹和明可夫斯基时空中，2维和3维的实双曲空间出现在狭义相对论中，理解这些空间及其对称群在理论物理中具有重要意义。最后，国际和平协会将继续他的各种外展活动，包括本科生和K-12学生。

The proposed research is a systematic investigation of (real and complex) reflection groups in PU(n,1). The main goal is to obtain new discrete subgroups and lattices in PU(2,1), more specifically many new non-arithmetic lattices. Hyperbolic reflection groups are an important class of groups in the realm of discrete subgroups and lattices in Lie groups, and more generally in geometry and topology. Such groups are accessible to a direct geometric description and understanding which are not always clear for groups defined algebraically or arithmetically. While these reflection groups are relatively well understood in the constant curvature setting (they are then "Coxeter groups" in Euclidean, spherical or real hyperbolic n-space), very little is known about their complex hyperbolic counterparts.A "tessellation" or crystal structure is a way of filling space with non-overlapping tiles in a pattern that repeats infinitely often. A "lattice" is the symmetry group of a tessellation. Understanding these crystallographic structures in Euclidean 3-space is crucial in Chemistry. The PI studies the analogous structures in "hyperbolic spaces". Real and complex hyperbolic spaces are spaces of negative curvature modelled on the real or complex numbers. Negative curvature means loosely that non-intersecting "straight" lines tend to diverge from each other in both directions. Real hyperbolic spaces of dimensions 2 and 3 appear in special relativity, in Lorentz and Minkowski space-time; understanding these spaces and their symmetry groups is important in theoretical Physics. Finally, the PI will continue his various outreach activities involving undergraduates and K-12 students.