Winding and Unwinding and Essential Intersections in ${\mathbb{H}}^3$

Abstract

Let $G=⟨A,B⟩$ be a non-elementary two generator subgroup of the isometry group of ${\mathbb{H}}^2$, the hyperbolic plane. If $G$ is discrete and free and geometrically finite, its quotient is a pair of pants and in prior work 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 and Chas. Here we extend our results to two generator subgroups $G$ of isometries of ${\mathbb{H}}^3$, hyperbolic three-space, 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.