Word distributions in groups

OVERVIEW The PI proposes to study word maps of groups G -- |that is, maps Mw : Gd --> G that are obtained by substitution into a fixed element w in the free group on d generators. There are three parts of this project. In the first part, the PI will study quantitative aspects of word maps for compact groups. This will be expressed via the word measure, which is the push-forward of the Haar measure on Gd by Mw. The goal of this part is to bound the Fourier coefficients of the word measure, as well as to understand its density at the identity. In the second part, the PI will study word maps in higher-rank arithmetic groups, trying to determine whether words have finite widths in these groups. In the third part, the PI will study word maps in finite groups. The big question there is whether a word is determined by its distributions on all finite groups. A major part of this project will be about applications of word maps. Most notably are applications in logic (the model theory of arithmetic lattices), representation theory (Kirillov's orbit method for algebraic groups over local rings of positive characteristic), and connections of distributions of word values and low-dimensional topology. INTELLECTUAL MERIT At the fundamental level, this project is devoted to a new family of Diophantine equations. The preliminary results we have suggest that the quantitative study of word maps has far-reaching applications in many branches of mathematics. BROADER IMPACT The PI will publish his work in all the usual venues. Many parts of this project are good starting points for independent research by graduate students. In fact, some aspects of them are currently being pursued by graduate student. Although not related to the scientific part of this project, the PI led a math circle in the last few years and intends to continue doing this.