Atoroidal surface bundles

TL;DR

The paper constructs new examples of atoroidal surface bundles over surfaces by establishing a homomorphism from the figure-eight knot complement's fundamental group to a mapping class group, revealing infinitely many pseudo-Anosov subgroups.

Contribution

It provides the first known examples of compact atoroidal surface bundles over surfaces and links knot complements to surface group representations.

Findings

Existence of a homomorphism from the figure-eight knot complement to a mapping class group.

Infinitely many commensurability classes of pseudo-Anosov surface subgroups.

First examples of compact atoroidal surface bundles over surfaces.

Abstract

We show that there is a type-preserving homomorphism from the fundamental group of the figure-eight knot complement to the mapping class group of the thrice-punctured torus. As a corollary, we obtain infinitely many commensurability classes of purely pseudo-Anosov surface subgroups of mapping class groups of closed surfaces. This gives the first examples of compact atoroidal surface bundles over surfaces.