CAREER: Metric Geometry of Solvable Groups

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

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.