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.
https://arxiv.org/pdf/2405.12067
How this paper works is that there is this Birman exact sequence. The fundamental group injects into the mapping class as point pushes. We apply a homomorphism that sends hyperbolic fundamental group elements of the 8-knot complement to pA mapping groups. Purely hyperbolic groups translate into purely pA groups.