Collaborative Research: Atoroidal Surface Bundles

Systems of equations describe most phenomena in mathematics, physics, and science more generally. Solutions to these systems are typically not simply numbers, but families of numbers arranged into spaces called manifolds. A fundamental problem in mathematics is developing tools to study all manifolds, and hence the solutions to any systems of equations. The general problem of studying manifolds is exceedingly difficult, but some manifolds can naturally be reconstructed from 2-dimensional slices, called surfaces. This is similar to how a CT scan describes 3-dimensional objects in terms of 2-dimensional images. The investigators have recently discovered infinitely many manifolds constructed from surface slices, and this project aims to show that these manifolds admit a certain structure called Gromov hyperbolicity, which provides numerous useful algorithmic tools to understand them. The Broader Impacts include an active mentoring program for graduate students and young researchers in the field, writing expository articles for the general public and organizing conferences. The PIS will continue their active involvement in various outreach activities that include giving public lectures, participation in several high school outreach events, aiming at attracting talented students to STEM disciplines and preparing the future STEM workforce. In 2024, the investigators gave the first examples of purely pseudo-Anosov surface subgroups of mapping class groups, thus proving the existence of closed atoroidal surface bundles over surfaces. This was done by constructing a type-preserving representation of the fundamental group of the figure-eight knot complement into the mapping class group of the thrice-punctured torus. The investigators will study the geometry, topology, and algebra of these bundles from a variety of viewpoints. One of the key goals, involves joint work with J. Russell, and is to prove that the fundamental groups of these bundles are Gromov hyperbolic, by proving the monodromy images are convex cocompact. The key tool is a theorem of the investigators in their joint work with M. Bestvina and K. Bromberg and reduces the problem to showing that their homomorphism is undistorted. The investigators, together with Russell, will develop the structure of associated Teichmüller spaces and curve complexes to achieve this goal. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.