Studying totally geodesic surfaces has been essential in understanding the geometry and topology of hyperbolic 3-manifolds. Recently, Bader-Fisher-Miller-Stover showed that containing infinitely many such surfaces compels a manifold to be arithmetic. We are hence interested in counting totally geodesic surfaces in hyperbolic 3-manifolds in the finite (possibly zero) cases. We expand an obstruction, due to Calegari, to the existence of these surfaces using Euler class and Thurston's norm. On the flipside, we prove the uniqueness of known totally geodesic surfaces by considering their behavior in the universal cover. This talk will explore this progress for both the uniqueness and the absence of totally geodesic surfaces in knot complements with small crossing number. Joint work with Khánh Lê.
Totally geodesic surfaces in knot complements with small crossing number
Rebekah Palmer, Temple University
2022 AWM Research Symposium
Women in Groups, Geometry, and Dynamics