Colloquium . Matt Stover . Feb 18th 2021. A geometric characterization of arithmeticity

Abstract: A very old, fundamental problem is classifying closed n-manifolds admitting a metric of constant curvature. The most mysterious case is constant curvature -1, that is, hyperbolic manifolds, and these divide further into "arithmetic" and "nonarithmetic" manifolds. However, it is not at all evident from the definitions that this distinction has anything to do with the differential geometry of the manifold. Recently, Uri Bader, David Fisher, Nicholas Miller and I gave a geometric characterization of arithmeticity in terms of properly immersed totally geodesic submanifolds, answering a question due independently to Alan Reid and Curtis McMullen. I will give an overview, assuming only basic differential topology, of how (non)arithmeticity and totally geodesic submanifolds are connected, then describe how this allows us to import tools from ergodic theory and homogeneous dynamics originating in groundbreaking work of Margulis to prove our characterization, along the way highlighting current and former OSU mathematicians that play important roles in the story.