Seminars & Colloquia Calendar
Distribution of holonomy on compact hyperbolic 3-manifolds
Lindsay Dever (Bryn Mawr)
Date & time: Tuesday, 01 February 2022 at 2:00PM - 3:00PM
Date: January 25 2:00 - 3:00pm
The study of hyperbolic 3-manifolds draws deep connections between number theory, geometry, topology, and quantum mechanics. Specifically, the closed geodesics on a manifold are intrinsically related to the eigenvalues of Maass forms via the Selberg trace formula and are parametrized by their length and holonomy, which describes the angle of rotation by parallel transport along the geodesic. The trace formula for spherical Maass forms can be used to prove the Prime Geodesic Theorem, which provides an asymptotic count of geodesics up to a certain length. I will present an asymptotic count of geodesics (obtained via the non-spherical trace formula) by length and holonomy in prescribed intervals which are allowed to shrink independently. This count implies effective equidistribution of holonomy and substantially sharpens the result of Sarnak and Wakayama in the context of compact hyperbolic 3-manifolds. I will then discuss new results regarding biases in the finer distribution of holonomy.