Finer Coarse Geometry

更精细的粗略几何形状

• 批准号: 1207106
• 负责人: Moon Duchin
• 金额: $ 15.38万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207106&HistoricalAwards=false
• 关键词: Finer; Coarse; Geometry

In geometric group theory and geometric topology, one usually studies groups and spaces through their coarse geometry, focusing only on properties that are preserved by QIs (quasi-isometries, or maps with bounded additive and multiplicative distortion of distances). This viewpoint is forced on us if we want to study infinite finitely-generated groups without specifying a generating set, since the graphs that record each group's geometry are only mutually related by QIs. In this project, the PI proposes to study group statistics that are large-scale but not invariant under quasi-isometry, and so may depend nontrivially on a choice of generating set. This point of view facilitates the study of randomness and asymptotic density in groups. The PI has studied free abelian groups and the discrete Heisenberg group in this way, opening up several avenues for current and future research. Another object of particular interest is Teichmueller space, the parameter space for many kinds of geometric structures on surfaces. The proposal contains descriptions of ongoing and planned work developing ideas for measuring "typical'' geometric properties in Teichmueller space, the mapping class group, and the complex of curves using data that is destroyed by quasi-isometry. For instance, one can show in this way that even though Teichmueller space is not a hyperbolic space, certain characteristics of hyperbolicity hold on average, or up to measure. The techniques developed to study this space, which has many features of negative curvature but does not satisfy any of the usual curvature conditions, are promising for applicability in other settings, such as relatively hyperbolic groups. A third main component of the proposal returns to the world of quasi-isometry invariants, developing new families of filling functions that appear to make finer geometric distinctions than some of their predecessors in the literature.This project belongs to a world of ideas that has a growing number of practical applications. For instance, it is useful in a wide range of applications (like communications networks) to have sparse but well-connected graphs. In mathematics these are called expanders; the first families of examples came from Cayley graphs of groups, and later it was realized that randomization over regular graphs also produces good expanders with high probability. Expansion constants, like a host of other geometric statistics, are "fine" and not QI invariant. Generally, the question of understanding the nature of an object by randomly sampling its points appears across many areas of applied mathematics, and the topics in this proposal are clustered around this theme.

在几何群论和几何拓扑学中，人们通常通过粗略的几何学来研究群和空间，只关注由QIs（准等距，或具有有界加性和乘性距离失真的映射）保持的性质。 如果我们想研究无限的有限生成群而不指定生成集，这种观点是强加给我们的，因为记录每个群的几何形状的图只通过QIs相互关联。 在这个项目中，PI建议研究大规模但在准等距下不恒定的组统计量，因此可能依赖于生成集的选择。 这一观点有利于研究群的随机性和渐近密度。 PI以这种方式研究了自由交换群和离散海森堡群，为当前和未来的研究开辟了几条途径。 另一个特别感兴趣的对象是Teichmueller空间，参数空间的许多种几何结构的表面。 该提案包含描述正在进行的和计划的工作开发的想法测量“典型”的几何性质在Teichmueller空间，映射类组，和复杂的曲线使用的数据，被摧毁的准等距。 例如，我们可以用这种方式证明，即使泰希穆勒空间不是双曲空间，双曲性的某些特征平均而言还是成立的，或者说是达到了测量的标准。 开发的技术来研究这个空间，它有许多功能的负曲率，但不满足任何通常的曲率条件，是有希望的适用性在其他设置，如相对双曲群。 第三个主要组成部分的建议返回到世界的准等距不变量，开发新的家庭的填充功能，似乎使更精细的几何区别比他们的前辈在literary.This项目属于世界的想法，有越来越多的实际应用。 例如，在广泛的应用（如通信网络）中，具有稀疏但连接良好的图是有用的。 在数学中，这些被称为扩展子;第一个例子来自群的凯莱图，后来人们意识到，在正则图上的随机化也可以产生高概率的好扩展子。 膨胀常数，像其他几何统计的主机，是“罚款”，而不是QI不变。 一般来说，通过随机采样点来理解对象的性质的问题出现在应用数学的许多领域，本提案中的主题围绕着这个主题。