Research in Geometry and Topology

代数和几何之间的相互作用是数学中的经典主题之一。传统上，人们通过几何对象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(例如，另一个群)的一组对称开始，然后构造一个具有相同对称性的几何对象。这里的重点是研究曲面的对称性。令人惊讶的是，它的几何在很大程度上是由双曲几何决定的，这可以追溯到高斯、洛巴耶夫斯基、庞加莱和其他人。目的是为了更好地理解这一现象。作者的博士生和博士后研究人员也参与了部分研究。关于经典映射类动力学的最重要的结果是Nielsen-瑟斯顿分类，它解释了简单闭曲线的迭代行为。在无限型曲面的情况下，同样的目标看起来要难得多，但肯定值得系统地研究。局部有限图的适当同伦等价群族是大型映射类群族的姊妹族，具有许多相似之处和有趣的不同之处，这是研究者想要探索的。更经典的是，研究者想要建立与经典映射类群复形以及与自由群的自同构群相关的复形的“塔”，很像泰勒级数。建造这些塔的过程被称为“解体”，初步工作表明，它成功地建造了圆弧和曲线复合体，并为它们的大尺度几何测量提供了新的曙光。这是建立与自由群的自同构群相关的复数有有限个符号维的又一次尝试，这也许是关于这个群的大规模几何的主要悬而未决的问题。关于遍历措施的问题本可以在很多年前提出，但需要新的方法来回答这些问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.