Joint IAS/PU Groups and Dynamics Seminar

The geodesic flow (for the hyperbolic metric) of an infinite Riemann surface is ergodic if and only if Brownian motion is recurrent, which is also equivalent to the divergence of the Poincaré series. Surfaces with ergodic geodesic flows are most similar to compact surfaces. We give various sufficient conditions on the Fenchel-Nielsen parameters that guarantee a surface falls into this class. Notably, we prove a version of the conjecture of J. Kahn and V. Markovic, which states that any surface with arbitrarily large cuff lengths and one topological end will have an ergodic geodesic flow when the twists are appropriately chosen. These findings are the result of several collaborative projects with Ara Basmajian, Hrant Hakobyan, and Michael Pandazis.