Quasi-isometric geometry of groups

群的拟等距几何

• 批准号: 0812513
• 负责人: Jason Behrstock
• 金额: $ 4.75万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-31 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0812513&HistoricalAwards=false
• 关键词: Quasi; isometric; geometry; groups

Most of this proposal lies under the umbrella of a vast program proposed by Gromov: classify finitely generated groups by their quasi-isometric geometry. One way to study the geometry of a group is via a space called the asymptotic cone; in several important cases, interesting algebraic structure of groups has been found encoded by the topology of their asymptotic cones. Through a study of asymptotic cones, the PI will work towards understanding the geometry of several particular classes of groups, including mapping class groups, relatively hyperbolic groups, and three-manifold groups. In particular, the goal of the project is to develop new geometric invariants with an eye towards proving one of the main geometric conjectures concerning mapping class groups, namely quasi-isometric rigidity. This conjecture asserts that no other groups geometrically look like mapping class groups.Groups are an algebraic way of encoding the symmetries of a space. It turns out that an important class of groups (the finitely generated ones) are themselves associated with a certain geometric object. This object is built by taking each symmetry to be a point and declaring the distance between a pair of points to be the number of simple symmetries one has to apply to get from one to the other. In the study of finitely generated groups, insight into their geometry can often be gained by looking at the group from ``infinitely far away'' -- this notion can be made mathematically precise and yields what is called an asymptotic cone of the group. The PI proposes certain ways to study groups from this asymptotic viewpoint.

该提案的大部分内容都属于格罗莫夫提出的一个庞大计划的范畴：通过准等距几何对有限生成的群进行分类。研究群几何的一种方法是通过称为渐近锥的空间。在几个重要的例子中，我们发现了由渐近锥的拓扑编码的有趣的群代数结构。通过渐近锥的研究，PI 将致力于理解几个特定类群的几何形状，包括映射类群、相对双曲群和三流形群。特别是，该项目的目标是开发新的几何不变量，着眼于证明有关映射类组的主要几何猜想之一，即准等距刚性。这个猜想断言没有其他群在几何上看起来像映射类群。群是编码空间对称性的代数方式。事实证明，一类重要的群（有限生成的群）本身与某个几何对象相关联。该对象的构建方法是将每个对称性视为一个点，并将一对点之间的距离声明为从一个点到另一个点所必须应用的简单对称性的数量。在对有限生成群的研究中，通常可以通过从“无限远”观察群来深入了解其几何形状——这个概念可以在数学上变得精确，并产生所谓的群的渐近锥。 PI 提出了从渐近观点研究群的某些方法。