Research in Geometry and Topology

代数与几何的相互作用是数学中的经典主题之一。传统上，人们通过几何物体的对称性来研究它们。在几何群论中，情况是相反的：一个人从一个代数对象的一组对称开始（例如，另一组），并构造一个具有相同对称性的几何对象。这里的重点是研究表面的对称性。令人惊讶的是，它的几何在很大程度上是由高斯、洛巴切夫斯基、庞加莱等人的双曲几何所支配的。我们的目标是更好地理解这种现象。申请人的博士生和博士后研究人员也参与了部分调查。关于经典映射类动力学的最重要的结果是尼尔森-瑟斯顿分类，它解释了简单封闭曲线的迭代行为。在无限型曲面的情况下，同样的目标看起来要困难得多，但肯定值得系统研究。局部有限图的属性同伦等价群族是大映射类群族的姐妹族，它们有许多相似之处，也有许多有趣的不同之处，这是研究者想要探索的。更经典的是，研究者想要建立经典映射类群复合体的连续逼近的“塔”，以及与自由群的自同构群相关的复合体，就像泰勒级数一样。建造这些塔的过程被称为“解体”，初步的工作表明，它成功地完成了弧形和曲线的复合体，并为它们的大规模几何结构提供了新的视角。这是建立与自由群的自同构群相关的络合物具有有限渐近维数的另一种尝试，也许是关于这个群的大尺度几何的主要突出的开放问题。关于遍历测量的问题在许多年前就已经提出了，但是需要新的方法来回答这些问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The interplay between algebra and geometry is one of the classicalthemes in mathematics. Traditionally, one studies geometric objectsvia their symmetries. In geometric group theory the situation isreversed: one starts with a group of symmetries of an algebraic object(for example, another group) and constructs a geometric object withthe same symmetries. Here the focus is on the study of symmetries ofsurfaces. Surprisingly, its geometry is to a large degree governed byhyperbolic geometry that goes back to Gauss, Lobacevski, Poincare andothers. The goal is to better understand this phenomenon. Theproposer's PhD students and postdoctoral researchers are alsoinvolved in parts of this investigation.The most important result about the dynamics of classical mappingclasses is the Nielsen-Thurston classification, which explains thebehavior of iterates of simple closed curves. The same goal in thecase of surfaces of infinite type looks to be much harder, butcertainly worthy of a systematic study. The family of groups of properhomotopy equivalences of a locally finite graph is a sister family tothe family of big mapping class groups, with many similarities andintriguing differences that the investigator wants to explore. Moreclassically, the investigator wants to build ``towers'' of successiveapproximations to the classical mapping class group complexes, as wellas complexes associated to automorphism groups of free groups, muchlike the Taylor series. The process of building these towers is called``disintegration'' and preliminary work shows that it succeeds for thearc and curve complexes and sheds new light on their large-scalegeometry. This is yet another attempt to establish that the complexesassociated with automorphism groups of free groups have finiteasymptotic dimension, perhaps the main outstanding open question aboutthe large-scale geometry of this group. The questions about ergodicmeasures could have been asked many years ago, but new methods areneeded to answer them.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.