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.
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. The proposed research was a systematic investigation of (real and complex) reflection groups in PU(n,1). The main goal was to obtain new discrete subgroups and lattices in PU(2,1), more specifically many new non-arithmetic lattices. This goal was attained beyond expectations. 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. The geometric objects in play live in complex hyperbolic 2-space, which is a curved 4-dimensional space and as such are diificult to visualize, even for people working with them on a daily basis. The accompanying pictures illustrate some of the (2- or 3-dimensional) images that we form of the 4-dimensional objects in order to visualize them.
