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

TL;DR

This paper extends the study of essential self-intersections of primitive geodesics from hyperbolic surfaces to hyperbolic three-space, providing formulas and geometric interpretations for these intersections in the 3D setting.

Contribution

It generalizes the concept of essential self-intersections to subgroups of isometries of hyperbolic 3-space and derives corresponding formulas and geometric interpretations.

Findings

Formulas for ESIs in hyperbolic 3-space

Geometric interpretation of ESIs in 3D manifolds

Extension of previous 2D results to 3D setting

Abstract

Let $G = \langle A,B \rangle$ 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.