Discrete Groups in Complex Hyperbolic Geometry

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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.