Topological and Geometric Aspects of Discrete Groups

所提出的研究涉及无限离散群的几何和拓扑方面。 该方案分为两个主要部分：群的大尺度几何和群在紧致流形上的作用。 这两个结果都可以看作是Mostow和Margulis关于李群格的著名刚性结果的广泛推广。 几何群论的大量研究致力于理解李群中格的大尺度几何。 在这种情况下，Eskin，Fisher和PI最近的结果建立了某些可解李群的刚性。 所涉及的技术才刚刚开始被理解，PI建议进一步开发它们。 特别是PI希望应用这些技术来显示更一般的李群中的可解根的不变性和幂零李群的准等距分类。 该建议的第二个焦点是研究可以作为紧流形的拓扑对称出现的群。 关于这个问题已经有很多的工作，但结果几乎都适用于特定的低维流形。 甚至对于圆的同胚也有许多未解决的问题。 PI建议研究某些类的高维流形，这些流形在同伦的水平上具有大的对称群，并确定这些对称是否可以实现为同胚群。粗略地说，大尺度（或粗糙）几何是研究对象的几何性质"从远处看"。从这个角度来看，任何有界对象都无法与点区分开来，而一条点线与实线也无法区分开来。这种几何最近在数学的许多领域都很有影响力，特别是群论、拓扑学和几何分析。这项研究将扩展PI以前的工作，探索几类数学对象的大规模几何，无论是经典的几何空间和对象，现在才被视为在几何方式。 该提案的一个特别重点是研究由熟悉的空间（如球体或其他表面）构建的相对简单物体的拓扑对称性组的可能粗糙几何。

The research proposed is concerned with the geometric and topological aspects of infinite discrete groups. The proposal divides into two main parts: the large scale geometry of groups and the actions of groups on compact manifolds. Both of these can be viewed as broad generalizations of the celebrated rigidity results of Mostow and Margulis for lattices in Lie groups. A significant amount of research in geometric group theory is devoted to understanding the large scale geometry of lattices in Lie groups. In this vein, recent results of Eskin, Fisher, and the PI establish rigidity for certain solvable Lie groups. The techniques involved are only beginning to be understood, and the PI proposes to develop them further. In particular the PI hopes to apply these techniques to show the invariance of the solvable radical in more general Lie groups and to the quasi-isometric classification of nilpotent Lie groups. The second focal point of the proposal is the study of groups can appear as topological symmetries of compact manifolds. There has been much work on this problem, but the results almost all apply to specific low dimensional manifolds. Even for the homeomorphisms of the circle there are many unresolved questions. The PI proposes studying some classes of higher dimensional manifolds which do have large groups of symmetries on the level of homotopy and to determine whether these symmetries can be realized as groups of homeomorphism.Roughly speaking, large scale (or coarse) geometry is the study of geometric properties of objects "seen from far away". From this perspective, any bounded object is indistinguishable from a point, and a line of dots is indistinguishable from a solid line. This sort of geometry has been influential recently in many areas of mathematics, notably group theory, topology, and geometric analysis. This research will extend previous work of the PI exploring the large scale geometry of several classes of mathematical objects, both classical geometric spaces and objects only now being viewed in a geometric manner. A particular focus of the proposal is the study of the possible coarse geometries of groups of topological symmetries of relative simple objects built out of familiar spaces, like spheres or other surfaces.