Finer Coarse Geometry

在几何群论和几何拓扑学中，人们通常通过群和空间的粗略几何来研究它们，只关注由QI(拟等距，或具有有界可加和乘性距离扭曲的映射)保持的性质。如果我们想要研究无限有限生成的群而不指定生成集，这个观点是强加给我们的，因为记录每个群的几何的图只通过QI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变的群统计，因此可能不是平凡的依赖于生成集的选择。这一观点有助于研究群体中的随机性和渐近密度。PI用这种方法研究了自由阿贝尔群和离散海森堡群，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，它是曲面上各种几何结构的参数空间。该提案描述了正在进行的和计划的工作发展思路，利用被准等距破坏的数据来测量Teichmueller空间、映射类组和曲线的复杂的“典型”几何性质。例如，人们可以用这种方式表明，即使Teichmueller空间不是双曲空间，双曲性的某些特征也是平均的，或达到一定程度。这个空间有许多负曲率的特征，但不满足任何通常的曲率条件，为研究这个空间而开发的技术有望应用于其他环境，如相对双曲群。该提案的第三个主要组成部分回到了准等距不变量的世界，开发了新的填充函数族，这些函数似乎比文献中的一些前身具有更精细的几何区别。这个项目属于一个具有越来越多实际应用的思想世界。例如，具有稀疏但连接良好的图形在广泛的应用程序(如通信网络)中很有用。在数学上，这些被称为扩张器；第一族的例子来自群的Cayley图，后来人们意识到，在正则图上的随机化也产生了高概率的好的扩张器。像许多其他几何统计一样，膨胀常数是“精细的”，并不是QI不变的。一般来说，通过随机抽样对象的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一主题。

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.