CAREER: Metric Geometry of Solvable Groups

群的概念最初是用来描述几何对象的对称性的。格罗莫夫将群体作为几何对象进行研究的计划颠覆了这一观点。抽象地说，群是由元素的集合给出的，带有元素乘法的概念。当一个组可以被实现为几何对象的对称性时，它继承了对象本身的几何。我们可以通过分析抽象群作用于的对象的几何形状来获得有关该抽象群的信息(及其乘法的性质)。这就是几何群论的起源。这个项目将允许PI继续她对一类具有特别好的乘性的群的几何的研究：所谓的可解群。此外，PI将为中西部的研究生和博士后举办一年一度的为期一天的几何群论区域研讨会，以及两次为期五天的研讨会，以帮助初级研究人员学习如何找到问题和合作。最后，国际数学联合会将继续开展高中女生课外数学项目的工作，该项目的目的是鼓励女生攻读STEM相关领域。这个项目允许PI继续她对Gromov将有限生成的群作为几何对象进行研究的计划。特别是，PI将使用和扩展Ekin-Fisher-Whyte的“粗微分”技术来回答几何群论中的基本问题。格罗莫夫把有限生成群作为几何对象来研究的计划彻底改变了群论。它引入了几何学和分析的技术，以帮助更好地理解无限有限生成的群。在Ekin-Fisher-Whyte的突破和新的“粗微分”技术之前，可解Lie群中格的刚性一直是这一领域的主要公开问题之一。PI已经并将继续使用Ekin-Fisher-Whyte的工作来为Gromov程序中提出的基本问题提供答案，并通过证明可解群的大规模几何的刚性结果为该程序做出了贡献。

The notion of a group was originally introduced to describe the symmetries of a geometrical object. Gromov's program of studying groups as geometric objects reverses this idea. Abstractly a group is given by a collection of elements with some notion of multiplication of elements. When a group can be realized as the symmetries of a geometric object it inherits the geometry of the object itself. We can gain information about an abstract group (and the properties of its multiplication) by analyzing the geometry of the object it is acting upon. This is the genesis of geometric group theory. This project will allow the PI to continue her study of the geometry of a class of groups with particularly nice multipication: the so called solvable groups. Additionally the PI will run an annual one day regional workshop on geometric group theory for graduate students and postdocs in the midwest and two five day workshops to help beginning researchers learn how to find problems and collaborations to work on. Finally the PI will continue her work on an after school mathematics program for high schools girls whose aim is to encourage girls to pursue STEM related fields. This project allows the PI to continue her research on Gromov's program of studying finitely generated groups as geometric objects. In particular the PI will use and extend "coarse differentiation" techniques of Eskin-Fisher-Whyte to answer fundamental questions in geometric group theory. Gromov's program of studying finitely generated groups as geometric objects revolutionized group theory. It brought in techniques from geometry and analysis to help better understand infinite finitely generated groups. The rigidity of lattices in solvable Lie groups was one of the major open problems in this area until Eskin-Fisher-Whyte's breakthrough and new "coarse differentiation" techniques. The PI has used and will continue to use the work of Eskin-Fisher-Whyte to provide answers to fundamental questions raised in the Gromov program and has contributed to the program by proving rigidity results on the large scale geometry of solvable groups.