Finer Coarse Geometry

在几何群论和几何拓扑中，人们通常通过它们的粗糙几何来研究群和空间，只关注由qi（准等距，或具有有界加性和乘性距离畸变的映射）保留的性质。如果我们想在不指定生成集的情况下研究无限有限生成群，我们就不得不采用这种观点，因为记录每个群几何形状的图只通过qi相互关联。在这个项目中，PI提议研究在准等距下大规模但不不变的群体统计量，因此可能非平凡地依赖于生成集的选择。这种观点有利于研究群体的随机性和渐近密度。PI以这种方式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种途径。另一个特别有趣的对象是Teichmueller空间，曲面上许多几何结构的参数空间。提案包含了对正在进行和计划中的工作的描述，这些工作发展了测量Teichmueller空间、映射类群和使用准等距离破坏的数据的曲线复合体的“典型”几何特性的想法。例如，我们可以用这种方式来证明，即使Teichmueller空间不是一个双曲空间，但双曲的某些特征是平均的，或者是可以测量的。该空间具有许多负曲率的特征，但不满足任何通常的曲率条件，因此研究该空间的技术有望应用于其他设置，例如相对双曲群。该提议的第三个主要组成部分回到了准等距不变量的世界，发展了新的填充函数族，这些函数族似乎比文献中的一些前任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，在广泛的应用程序（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，这些被称为展开器；第一类例子来自群的Cayley图，后来人们意识到，在正则图上的随机化也会产生高概率的良好展开器。膨胀常数，像许多其他几何统计数据一样，是“好的”，而不是QI不变的。一般来说，通过随机抽样来理解一个对象的本质的问题出现在应用数学的许多领域，本提案中的主题都是围绕这个主题聚集的。

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.