Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - GRANT (Groups, Representations and Analysis in Number Theory)

Teaser

The project lies at the crossroads of analytic number theory and asymptotic group theory. The basic issue on the side of group theory is the following. Consider a finite subset A of a group, that is, a set whose elements can be multiplied. Now consider the set B of all things...

Summary

The project lies at the crossroads of analytic number theory and asymptotic group theory. The basic issue on the side of group theory is the following. Consider a finite subset A of a group, that is, a set whose elements can be multiplied. Now consider the set B of all things that can be obtained as the product of, say, elements of A. Generally, B will have more elements than A, but how many more?
This question - growth - is closely related to that of the diameter of groups. In the case of permutation groups, the concept of diameter can be explained very simply. Rubik\'s cube, and other puzzles like it, can be solved by means of some combination of moves. What is the smallest number d such that every solvable position can be solved in at
most d moves? This number d is called the diameter. What makes this a question in asymptotic group theory is that we do not care so much about the exact number d; I am concerned with how d grows as, say, the side of the cube tends to infinity.
(All but finitely many groups fall into two kinds, linear algebraic groups - that is, groups of matrices, with the usual matrix multiplication - and permutation groups. The latter may be familiar to the general public as puzzles, but both kinds appear naturally in all sorts of less playful contexts within mathematics and the sciences. I originally worked on linear algebraic groups. My work was continued by other researchers, and I shifted my attention to permutation groups some years ago. The best result on this is that by myself and Seress; it has stood unmatched for six years. I believe with good reason that an eventual improvement on permutation groups may hold the key to further progress in linear algebraic groups.)
Analytic number theory often reduces its problems to the following form, that is, it finds that their crux lies in the following issue: finding that certain sums have cancellation, i.e., are much smaller than their number of terms would let us think. This can often be restated as an issue of equidistribution: given, say, a large set of points on a circle, are they distributed on it without any significant bias?
Questions of growth in groups turn out to be closely connected to questions of equidistribution, in part through the study of random walks. I propose to research several instances of cancellation and equidistribution, using at times techniques and ideas coming from my study of growth in groups.

Work performed

This is the first stage of my project; it has involved, among other things, gradually getting my research team together. I am preparing a preprint revisiting my own work on permutation groups from a unified perspective, showing that many of the ideas that gave fruit in algebraic groups can be abstracted enough to work on permutation groups aswell. On this issue, I am coordinating efforts with another ERC grant recipient whose research plan turned out to intersect mine.

On the number theoretical side, I am working in a new direction involving random walks. I am also glad to see that an elegant solution to a problem on cancellation in exponential sums has recently been found by a very young researcher inspired by the suggestions given in one of my winter-school courses; her method is based on results by myself and other people on growth in linear algebraic groups.

Final results

See above.