Cleanliness of geodesics in hyperbolic 3-manifolds Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman
Imagine you are inside a cusped hyperbolic 3-manifold, and you have situated yourself at infinity, and now you are looking down the cusp. You see a pattern of horoballs within the fundamental parallelogram for the cusp subgroup fixing $\infty$. Imagining that horoballs are opaque, how many are visible? Do finitely many horoballs suffice to cover the fundamental parallelogram? This question was first asked by Darren Long, and this paper answers it.
Specifically, we derive conditions guaranteeing the existence of geodesics avoiding the cusps and use these geodesics to show that in “almost all” finite volume hyperbolic 3-manifolds, infinitely many horoballs in the universal cover corresponding to a cusp are visible in a fundamental domain of the cusp when viewed from infinity.