[Topology and Geometry Seminar] “Cusp shape and tunnel number (part 1)" by Prof Jessica Purcell (Monash University)

The seminar aims to introduce research topics in topology, geometry and its interactions with other sciences. Anyone interested in mathematics is welcome to attend.

This week, Professor Jessica Purcell (Monash University) will discuss "Cusp shape and tunnel number".

Abstract: Associated to a cusped hyperbolic 3-manifold is a cusp shape, which is a point in the Teichmuller space of the torus. It is natural to ask which points in Teichmuller space arise. In the 1990s, Nimmersheim showed that the cusp shapes of finite volume hyperbolic 3-manifolds, which form a countable set, are dense in Teichmuller space. However, the 3-manifolds constructed in that theorem are very complicated topologically. Restrict instead to simpler manifolds. For example, every 3-manifold has a Heegaard splitting. If we restrict to simple Heegaard splittings, of bounded genus g, which cusp shapes arise? We show that for fixed genus g, cusp shapes of finite volume 3-manifolds of genus g are still dense in Teichmuller space. This is joint with Vinh Dang.

Part 1: I will introduce cusp shapes and tunnel number, defining the terms in the abstract. I will also give background behind the questions asked and several examples.

Part 2: I will give more details on the proof. Time permitting, I will show a contrasting result: that a well-known family of Berge knots, which have tunnel number one, all have cusp shapes limiting to a single point in Teichmuller space.