FREE GROUP ROOTS OF akbl AND [ ak,b]

Abstract

Let F be a free group with basis a, b, …, and let u, w ∈ F . We say that u is a root of w if w belongs to the normal closure of u in F . We show that if w is of the form akbl then w has only a finite number of conjugacy classes of roots, and that given k and l the set of roots of w can be algorithmically determined. We also classify the roots of elements of the form akba-kb-1.