An algebraic characterization of simple closed curves on surfaces with boundary

We prove that a conjugacy class in the fundamental group of a surface with boundary is represented by a power of a simple curve if and only if the Goldman bracket of two different powers of this class, one of them larger than two, is zero. The main theorem actually counts self-intersection number of a primitive class by counting the number of terms of the Goldman bracket of two distinct powers, one of them larger than two.