Quasi-isometric geometry of groups

这一建议的大部分都是在格罗莫夫提出的一个庞大计划的保护伞下提出的：根据有限生成的群的准等距几何对其进行分类。研究群的几何的一种方法是通过一个称为渐近锥的空间；在一些重要的情况下，已经发现群的有趣的代数结构由其渐近锥的拓扑编码。通过对渐近锥的研究，PI将致力于了解几类特殊群的几何，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，着眼于证明与映射类群有关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。群是对空间的对称性进行编码的一种代数方式。结果表明，一类重要的群(有限生成的群)本身与某个几何对象相关联。这个对象是通过将每个对称看作一个点，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的数目来构建的。在对有限生成群的研究中，对其几何的洞察通常可以通过从“无限远处”观察群来获得--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体。PI提出了从这个渐近的观点来研究群的某些方法。

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.