Distribution of Word Maps in Groups

DISTRIBUTION OF WORDS MAPS IN GROUPS - PROJECT SUMMARY Goals. The PI proposes to study word maps of groups G¡ªthat is, maps Mw : Gd ¡ú G that are obtained by substitution into a fixed element w in the free group on d generators. There are two parts to this research plan. The first is a study of distribution of values of general word maps in groups. The broad goal of this part is to give a quantitative answer to the question ¡°How random are word maps?¡± More specifically, the goal is to prove a bound on the Fourier coefficients of such distributions in finite and p-adic groups. These Fourier coefficients are non-commutative analogs of Gauss sums. The goal of the second part in this proposal is to explore a promising connection that the PI and Rami Aizenbud found between the distribution of values of the word [x1, y1] ¡¤ ¡¤ ¡¤ [xk, yk], the moduli space of G-local systems on a curve, the representation theories of lattices in higher rank algebraic groups, and new kind of Topological Field Theory. Intellectual Merit. At the fundamental level, this project is a study of equations in finite and p-adic groups, which is of central importance to Group Theory and all of its applications. It is a quantitative version of the recent results on the sets of word values that, in the last few years, led to the solutions to several long standing conjectures in Group Theory, such as the Ore Conjecture and Serre¡¯s conjecture on openness of finite index subgroups of pro-finite groups. The quantitative point of view taken here, however, is new, and suggests many conjectures, some partially proved, in diverse areas of algebra, such as the representation theory of lattices, random hyperbolic groups, quantizations of co-adjoint orbits in positive characteristic, and limits of Dijkgraaf¨CWitten topological field theories. Broader Impact. The PI will make his work public in scientific journals, the Arxiv, in seminars, and in conferences. In addition, as a new tenure-track professor at Northwestern University, he expects to supervise graduate students on projects stemming out of this project. He plans to use some of the money to bring experts in Group Theory to a research seminar he is organizing at Northwestern. Over the last three years, the PI taught several graduate topic courses on topics related to his research, and distributed lecture notes on Motivic Integration, Representations of Finite Linear Groups, and Lattices in Semisimple Lie Groups, which can be found on his webpage. He plans to continue doing this in the future.