有限生成离散群的拟等距刚性及其渐近锥的拓扑

Research the algebraic properties of discrete groups by study the geometric metric space which these groups action by isometric is central topics in the geometry group theory for nearly 30 years. It has made a series of important results (such as B.Farb on Abelian-by-cyclic group quasi isometric classification). But the research of the topology of asymptotic cone of some solvable groups (which isomorphic to the semidirect-product of its two subgroups) and the classification of hyperbolic groups are still no involved. This project aims to study the quasi isometric classification of a class of discrete solvable groups and the topological properties of asymptotic cone of these groups. Through the study of the topological and geometrical properties of special metric space (including hyperbolic group, hyperbolic geometry, CAT (0) space) and the asymptotic cone of them, try our best to find out the relationship between the algebraic properties of discrete groups and the geometry properties of this kind of space, and then obtain the quasi isometry invariants and the topological properties of asymptotic cone of these discrete groups.The topic of this project is the B.Farb’s extension on discrete solvable group classification. We will partly solve the basic Gromov group on finitely generated group asymptotic cone countability; we will extend the discrete quasi isometric range of classification group, and we will give some examples to explain that the algebraic and geometric problems can be transformed into each other.

通过离散群等距作用的度量空间的几何研究刻画群的代数性质是近30年几何群论研究的中心问题,取得了一系列重要成果(如B.Farb关于Abelian-by-cyclic群的拟等距分类等)，但对可解群(可分裂为两子群的半次积)的拟等距分类和双曲群的渐近锥拓扑的研究仍待解决。本项目拟研究一类离散可解群的拟等距分类以及它们的渐近锥的拓扑性质，通过对特殊度量空间(包括双曲群、双曲几何、CAT(0)空间)的几何及其渐近锥的拓扑性质的研究,致力于找出该类离散群的代数与这类空间的几何之间的关系,进而发现这类离散群的拟等距不变量及其渐近锥的拓扑。该项目是B.Farb关于离散可解群分类工作的延伸，拟部分解决Gromov关于有限生成群渐近锥的基本群可数性问题,扩大可解离散群的拟等距分类范围，给出代数问题与几何问题可相互转化的范例。

通过有限生成群的字度量空间或该群几何作用的度量空间的几何拓扑研究刻画群的代数性质是近30年几何群论研究的中心问题,取得了一系列重要成果(如B.Farb关于Abelian-by-cyclic群的拟等距分类等)，但对可解群(可分裂为两子群的半次积)的拟等距分类和双曲群的渐近锥拓扑的研究仍待解决。本项目研究了一类离散可解群的拟等距分类以及它们的渐近锥的拓扑性质，致力于对特殊度量空间(包括双曲群、双曲几何、CAT(0)空间)的几何及其渐近锥的拓扑性质的研究,试图找出该类离散群的代数与这类空间的几何之间的关系,进而发现这类离散群的拟等距不变量及其渐近锥的拓扑结构。通过研读、讨论几何群论经典文献，学习最新研究工具，本项目组在非正曲率流形的几何拓扑方面做了相关研究，给出了非正曲率流形的谱一些几何刻画，并试图建立起可几何作用于非正曲率流形的群的代数量与它的几何量之间的联系。项目组按照研究计划组织项目组成员定期讨论项目中需要用到的数学工具，提升了项目组成员的整体数学素养和科学研究能力，为后期的进一步的科学研究奠定了基础。本项目是B.Farb关于离散可解群分类工作的延伸，对非正曲率流形的几何量有了更深的了解。