Word Distributions in Groups

Northwestern University, Evanston IL

Investigators

Abstract

Consider the collection of all rotations, of all possible angles, around a point in the plane. Pick one---let's call it x---at random, such that every angle has the same probability to appear. Now, apply the rotation x twice. In mathematical notation, this is x*x. It is not hard to show that the rotation x*x is again random, and every angle has the same probability to appear as the angle of rotation of x*x. Surprisingly, this is false in three dimensions: if x is a random rotation in space (each such looks like a rotation around some axis) chosen in such a way that each rotation has the same probability to appear, then some rotations have a higher probability of being x*x and some have lower probability, although this probability is never zero. Similarly, one can take two random rotations x and y and consider other expressions such as x*y*x*x*y and understand their distributions. More generally, one can ask similar questions replacing the collection of rotations by the collection of symmetries of other mathematical objects. This award provides support for graduate students working in areas related to the project. In technical terms, the PI will study word maps of groups---that is, maps from G^d to 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 G^d by the word map. 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. 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.