Winding and unwinding and essential intersections in H³

Let G = 〈A, B〉 be a non-elementary two generator subgroup of the isometry group of H². If G is discrete, free and generated by two hyperbolic elements with disjoint axes, its quotient is a pair of pants and in [3] we produced a recursive formula for the number of essential self intersections (ESIs) of any primitive geodesic on the quotient. An alternative non-recursive proof of the formula was given in [10]. We used the formula to give a geometric description of how the primitive geodesic winds around the “cuffs” of the pants. On a pair of pants, an ESI is a point of a “seam” where a primitive geodesic has a self-intersection. Here we generalize the definition of ESI’s to non-elementary purely loxodromic two generator groups G ⊂ Isom(H³) which are discrete and free. We also associate an open geodesic to each generalized ESI we call a Connector. In the quotient manifold the connectors correspond to “opened up self-intersections”. We then define a class of groups we call Winding groups; the connectors for these groups give us a way generalize the concept of winding of primitive geodesics in the quotient manifold to this context. For winding groups, the ESI’s and connectors satisfy the same formula as they do in the two-dimensional case. Our techniques involve using the stopping generators we defined in [3] for the group G to make it a model. We then use the model to obtain an algebraic rather than geometric definition for ESI’s and generalize from the algebraic definition.