  In the mid eighties Goldman proved an embedded curve could be isotoped to not
intersect a closed geodesic if and only if their Lie bracket (as defined in
that work) vanished. Goldman asked for a topological proof and about extensions
of the conclusion to curves with self-intersection. Turaev, in the late
eighties, asked about characterizing simple closed curves algebraically, in
terms of the same Lie structure. We show how the Goldman bracket answers these
questions for all finite type surfaces. In fact we count self-intersection
numbers and mutual intersection numbers for all finite type orientable
orbifolds in terms of a new Lie bracket operation, extending Goldman's. The
arguments are purely topological, or based on elementary ideas from hyperbolic
geometry.
  These results are intended to be used to recognize hyperbolic and Seifert
vertices and the gluing graph in the geometrization of three manifolds. The
recognition is based on the structure of the String Topology bracket of three
manifolds.
