Discrete Groups in Complex Hyperbolic Geometry

双曲几何是一种非欧几何，在科学和工程的许多领域都有应用。双曲平面的几何是世纪发现的经典平行公设的反例之一，它启发了在复数而不是真实的数上进行类似的构造。 与熟悉的真实的双曲环境相比，在复双曲环境中，刚性运动的离散群更难理解和表现，这些项目旨在大大扩展已知的这类对称群的集合。李群的离散子群的研究有一些主要的一般结果或例子家族，这些结果或例子家族在复双曲n-空间的等距群中仍然没有对应的格，特别是（非）算术子群的结果。 这些项目将研究在PU（n，1）中反射生成的格，n至少为2，算术格的几何方面，以及双曲3流形群到PU（3，1）的变形。 目标包括在PU（2，1）中产生无限多个不同的非算术格的可积性类，并获得由反射生成的格的其他无限族，例如Picard模群。

Hyperbolic geometry is a non-Euclidean geometry that has applications in many areas in sciences and engineering. The geometry of the hyperbolic plane that is one of the counterexamples found in the nineteenth century to the classic Parallel Postulate inspires a similar construction performed over the complex numbers rather than the real numbers. Discrete groups of rigid motions are harder to understand and to exhibit in the complex hyperbolic setting than in the more familiar real hyperbolic environment, and these projects aim to greatly expand the collection of known symmetry groups of this type.The study of discrete subgroups of Lie groups has some major general results or families of examples that remain without counterparts for lattices in the isometry group of complex hyperbolic n-space, especially results on (non)arithmetic subgroups. These projects will study lattices generated by reflections in PU(n, 1) for n at least 2, geometric aspects of arithmetic lattices, and deformations of hyperbolic 3-manifold groups into PU(3, 1). Goals include the production of infinitely many distinct commensurability classes of nonarithmetic lattices in PU(2, 1) and to obtain other infinite families of lattices generated by reflections, such as the Picard modular groups.