@inbook{d5498446f0294558a2e7d2ab0c9356d2,

title = "Winding and unwinding and essential intersections in H3",

abstract = "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{\textquoteright}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{\textquoteright}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{\textquoteright}s and generalize from the algebraic definition.",

keywords = "Discrete groups, Fuchsian groups, Geodesics, Hyperbolic geometry, Intersections, Schottky groups",

author = "Jane Gilman and Linda Keen",

year = "2017",

month = jan,

day = "1",

doi = "10.1090/conm/696/14019",

language = "English (US)",

series = "Contemporary Mathematics",

publisher = "American Mathematical Society",

pages = "125--138",

booktitle = "Contemporary Mathematics",

address = "United States",

}