四元数双曲几何与拟共形映射

The quaternionic hyperbolic space is an important space among the rank-one symmetric space of non-compact type. Quaternionic hyperbolic geometry is significantly different from real or complex hyperbolic geometry. Due to the difficulty caused by the noncommutativity of quaternions,quaternionic hyperbolic geometry is less well undersood. However, because real or complex hyperbolic space is a subspace of quaternionic hyperbolic space, any breakthrough in quaternionic hyperbolic geometry will affect the theory of real or complex hyperbolic geometry. Using tools and methods of related fields such as Lie group, Riemanian geometry and geometric function theory, we will investigate the following three problems:.(1) the algebraic and geometric characteristic of the Hurwitz-Picard modular group;.(2) Heisenberg group and the quasiconformal mappings on it;.(3) the deformation theory of discrete subgroups acting on quaternionic hyperbolic spaces.. The above closely interrelated researches are three core problems in quaternionic hyperbolic geometry. These works will not only make contribution in quaternionic hyperbolic geometry but also supply new theory or effective tools for real and complex hyperbolic geometry. We think that this project has important theoretical significance and potential application value.

四元数双曲空间是秩为一的非紧致对称空间中一个重要的空间。四元数双曲几何与实或复双曲几何有着显著的不同。因四元数的非交换性带来的困难，人们对四元数双曲几何知之甚少。但由于实或复双曲空间是四元数双曲空间的子空间，四元数双曲几何研究上的任何突破都将影响实或复双曲几何的相关理论。本项目拟综合利用李群,黎曼几何及几何函数理论等领域的相关方法来研究如下三个方面的问题：.(1) Hurwitz-Picard模群的代数几何特征；.(2) Heisenberg群及其上拟共形映射；.(3) 复和四元数双曲空间离散群的形变理论。.以上相互紧密联系的研究内容是四元数双曲几何中的三个核心问题。这些问题深入系统的研究与解决，不仅将促进四元数双曲几何基础理论的发展，也将对实和复双曲几何的研究提供新的理论或有效工具，有重要的理论意义和潜在的应用价值。