A word on d letters is an element of the free group of rank d, say, with basis x1,…,xd. Given a word w=w(x1,…,xd) on d letters, for every group G, there is a word map w:Gd→ G given by substituting the xi with elements of G. We say that a word w has a finite width n in the group G if any element in the subgroup generated by w(G) is a product of at most n element of w(G) or their inverses. In this talk, I will survey results about word width in several families of groups and then restrict the focus to the family of higher rank arithmetic groups. I will present a conjecture about word width in higher-rank arithmetic groups and explain some consequences, most notably, to the Congruence Subgroup Problem.
This video is part of the Institute for Advanced Study‘s Special year research seminar.
