Averaging Methods in Coarse Geometry

该提案包括两个主要部分。 第一节讨论粗几何和几何群论。PI（与D。Fisher和K. Whyte）最近发展了一种新的方法--“粗微分法”，它可以看作是拟等距的一种可微性替代。当然，传统的导数对这样的地图没有意义，因为它们甚至没有在小尺度上定义;相反，我们必须去更大更大的尺度。利用这一技巧，我们解决了该领域三个长期存在的问题，即证明了三维可解群Sol的拟等距刚性，证明了一个传递图与任何Cayley图都不是拟等距的，以及证明了两个状态和三个状态灯灯群都不是拟等距的.第二部分涉及有理多边形中台球的相关分析研究，阿贝尔和二次微分的模空间，以及SL（2，R）作用在这些模空间上的动力学。PI还建议研究这些空间的几何学相关问题，如它们的体积和SL（2，R）不变子流形。特别是，PI通过数值实验发现了一些多边形，这些多边形似乎具有完全意想不到的性质，并建议进一步研究它们。提案第一部分中的一些粗略几何与计算机科学有着意想不到的联系，特别是存在有效的算法，可以找到通过切割尽可能少的边来断开连接的方法。事实上，我们提出的问题之一就是取自这一领域。理性的台球，这是研究的主要课题在第二部分的建议是重要的，特别是以下原因：一些自然现象是“混乱”（即不可预测的）。这些通常是用解剖学方法研究的。另一些是“可整合的”（即可预测的和有规律的）。这两种现象之间的某个地方。多边形台球是已知的最简单的中间行为模型之一。因此，它在物理学中也得到了广泛的研究，特别是与“量子混沌”有关。

This proposal consists of two main sections. The first section dealswith coarse geometry and geometric group theory. The PI (together withD. Fisher and K. Whyte) has recently developed a new technique,"coarse differentiation", which can be viewed as a sort ofdifferentiability substitute for quasi-isometries. Of course,conventional derivatives do not make sense for such maps, since theyare not even defined on small scales; instead we must go to larger andlarger scales. Using this technique, we were able to resolve threelongstanding open problems in the field, namely proving thequasi-isometric rigidity of the three-dimensional solvable group Sol,exhibiting a transitive graph which is not quasi-isometric to anyCayley graph, and showing that the two state and the three statelamplighter groups are not quasi-isometric. We list some otherpotential applications of the method, many of which are to problemswhich seemed completely out of reach a year ago.The second section concerns the interrelated analytic study ofbilliards in rational polygons, moduli spaces of abelian and quadraticdifferentials, and the dynamics of the SL(2,R) action on these modulispaces. The PI also proposes to study related questions about thegeometry of these spaces, such as their volumes and their SL(2,R)invariant submanifolds. In particular, the PI has found by numericalexperiment some polygons which seem to have competely unexpectedproperties, and proposes to study them further.Some of the coarse geometry in the the first part of the proposal hasunexpected connections to computer science, in particular theexistence of efficient algorithms for finding ways to disconnect agraph by cutting as few edges as possible. In fact, one of ourproposed problems is taken from this field. The rational billiard, which is the main subject of study in thesecond part of the proposal is important in particular for thefollowing reason: Some natural phenomena are "chaotic"(i.e. unpredictable). These are often studied by statisticalmethods. Others are "integrable" (i.e. predictable and regular). Otherphenomena fit somewhere in between. The polygonal billiard is one ofthe simplest known models of intermediate behavior. As such it hasbeen studied extensively in physics as well, in particular inconnection to "quantum chaos".