Special Year Research Seminar

A word on d letters is an element of the free group of rank d, say, with basis x_1,…,x_d. Given a word w=w(x_1,…,x_d) on d letters, for every group G, there is a word map w:G^d—> G given by substituting the x_i’s 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.