A random tunnel number one 3-manifold does not fiber over the circle

There are two natural notions of “random” for tunnel number
one 3-manifolds (that is, a manifold obtained by attaching a disk to a
genus 2 handlebody). With respect to both notions of random,
experiments show that a random manifold does not fiber over $S^1 $ when
the manifold is large enough. We prove it with respect to one notion.
The question is motivated by the virtual fibration conjecture. We use
techniques of Brown to turn the question into a group theory question
and techniques of Agol, Hass, and Thurston to study the question for
such large manifolds.

This is joint work with Nathan Dunfield.