Complex hyperbolic reflection groups and lattices

本研究是对PU（n，1）中（真实的和复的）反射群的系统研究。主要目标是在PU（2，1）中获得新的离散子群和格，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领域中的一类重要群，更广泛地说是几何学和拓扑学中的一类。这样的群体是一个直接的几何描述和理解并不总是清楚的代数或算术定义的群体。虽然这些反射群在常曲率设置中相对较好地理解（它们在欧几里德、球形或真实的双曲n空间中是“考克斯特群”），但对它们的复双曲对应物知之甚少。“镶嵌”或晶体结构是一种以无限频繁重复的模式用不重叠的瓦片填充空间的方式。“晶格”是镶嵌的对称群。了解这些在欧几里得三维空间中的晶体结构在化学中至关重要。PI研究“双曲空间”中的类似结构。真实的和复双曲空间是以真实的或复数为模型的负曲率空间。负曲率大致上意味着不相交的“直”线倾向于在两个方向上彼此发散。2维和3维的真实的双曲空间出现在狭义相对论、洛伦兹和闵可夫斯基时空中;理解这些空间和它们的对称群在理论物理学中很重要。最后，PI将继续开展涉及本科生和K-12学生的各种外展活动。

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.