CAREER: Large Scale Geometry and Dynamics in Group Theory

DMS-0349290凯文M.为什么通过凯莱图的拟等距几何来研究离散群的计划是格罗莫夫在1982年国际数学家大会上的演讲中提出的。 这个程序有两个互补的部分：找到不变量，以表明不同的群体是不是准等距，并构建准等距，以表明类似的群体是准等距。 PI在过去对此计划做出了一些贡献，包括最近证明了积分仿射变换群和曲面群自同构群的拟等距刚性（后者与Lee Mosher共同工作）。 PI提出了显着的扩展这些结果的背景下，发展的主题大尺度动力学。除了刚性现象，拟议的研究包括一个详细的研究刚性较低的群体，这应该有助于澄清问题的许多代数和组合性质的群体的准等距不变性。 粗略地说，大尺度（或粗糙）几何是研究“从远处看”的物体的几何特性。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。本研究将探讨几类数学对象的大尺度几何，包括经典的几何空间和现在才以几何方式观察的对象。 最近，人们对这一领域的研究产生了极大的兴趣，该领域的许多专家都在芝加哥地区的一所大学工作。 该提案包括各种计划，以发展数学家之间的互动，特别是他们的研究生，在这个社区。

DMS-0349290Kevin M. WhyteThe program of studying discrete groups via the quasi-isometric geometry of their Cayley graphs was suggested by Gromov in his address at the 1982 International Congress of Mathematicians. This program has two complementary parts: finding invariants to show that different groups are not quasi-isometric, and constructing quasi-isometries to show that similar groups are quasi-isometric. The PI has made several contributions to this program in the past, including a recent proof of quasi-isometric rigidity for the group of integral affine transformations and the group of automorphisms of a surface group (the latter in joint work with Lee Mosher). The PI proposes significant extensions of these results in the context of the development of the subject of large scale dynamics. In addition to rigidity phenomena, the proposed research includes a detailed study of less rigid groups, which should help to clarify questions about quasi-isometric invariance of many algebraic and combinatorial properties of groups. 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 explore the large scale geometry of several classes of mathematical objects, both classical geometric spaces and objects only now being viewed in a geometric manner. There has been a tremendous amount of interest in this area of study recently, with many of the experts in the field located at one of the universities in the Chicago area. The proposal includes a variety of programs to develop interaction between mathematicians, and especially their graduate students, within this community.