Width in Groups

This project will study equations which arise from the study of mathematical structures called arithmetic groups. These play a central role in many branches of modern mathematics, from Number Theory, to Geometry, to Mathematical Physics. There are infinitely many equations in this class and the goal is to understand which of those have solutions that are integral numbers. The primary goal of this project is to prove a conjectural criterion for the solvability of an equation in this class. Proving this conjectural criterion will answer several important questions about the algebraic, logical, and geometric properties of arithmetic groups. A secondary goal of the project is to count the number of such solutions when they exist. This part is more analytic in nature and will study a new and unexplored statistical-mechanics model of random matrices. A good example of an equation in this project is the following: given two elements g,h in an arithmetic group G and a positive integer m, write h as a product of m conjugates of g. The conjugating matrices are the variables in this equation. The main conjecture is a kind of local-to-global criterion saying that, for higher-rank arithmetic groups, if a solution exists in every completion of G, then a solution exists in G itself, except that one may need to increase the number m. This conjecture has consequences for many other problems, including: the Congruence Subgroup Problem; bounded generation for isotropic arithmetic groups; model theory of higher rank lattices; and conjugation-invariant norms on higher rank arithmetic groups. 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.