Research in Geometry and Topology

代数和几何之间的相互作用是数学的经典主题之一。传统上，人们研究几何对象通过他们的对称性。在几何群论中，情况正好相反：人们从一个代数对象的一组对称性（例如，另一组）开始，用同样的对称性构造一个几何对象。这里的重点是研究曲面的对称性。令人惊讶的是，它的几何形状在很大程度上是由双曲几何，可以追溯到高斯，罗巴切夫斯基，庞加莱等。我们的目标是更好地理解这一现象。关于经典映射类的动力学最重要的结果是Nielsen-Thurston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困难得多，但肯定值得系统地研究。局部有限图的自同伦等价群族是大映射类群族的姐妹族，它们之间有许多相似之处，也有许多令人感兴趣的不同之处。更经典的是，研究者希望建立“塔”的连续近似的经典映射类群复形，以及复形相关的自同构群的自由团体，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，它成功地用于弧形和曲线复合体，并为它们的大规模几何形状提供了新的光线。这是另一个尝试，以建立与自由群的自同构群相关联的复体具有有限的渐近维数，也许是关于这个群的大规模几何的主要悬而未决的公开问题。关于遍历测量的问题在很多年前就已经提出了，但是需要新的方法来回答它们。这个奖项反映了NSF的法定使命，并且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，被认为是值得支持的。

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.