FREE GROUP ROOTS OF a^kb^l AND [ a^k,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.

The winding invariant

“…In fact, if w ∈ F ′ 2 then any primitive element of F 2 is a normal root: if {u, v} is a basis of F 2 , w is a product of powers of u and v, and since w ∈ F ′ 2 the total exponent of v in this expression is 0, so w ∈ N (u). Magnus studied some cases in [110], McCool [114] extended results of Steinberg [148] about normal roots of x n y m , and also characterized the normal roots of [x n , y].…”