Winding and Unwinding and Essential Intersections in H

Let G = 〈A,B〉 be a non-elementary two generator subgroup of Isom(H). If G is discrete and free and geometrically finite, its quotient is a pair of pants and in [5] we produced a formula for the number of essential self intersections (ESIs) of any primitive geodesic on the quotient. An ESI is a point where the geodesic has a self-intersection on a seam. Self-intersections of geodesics on arbitrary hyperbolic surfaces have recently been studied by Basmajian [1] and Chas [2]. Here we extend our results to groups two generator groups G ⊂ Isom(H) which are discrete, free and geometrically finite. We generalize our definition of ESIs and give a geometric interpretation of them in the quotient manifold. We show that they satisfy the same formulas.