Joint IAS/PU Groups and Dynamics Seminar

How many closed geodesics with length less than $L$ are there on a hyperbolic surface? Classical results of Huber and Margulis say that the counting function has asymptotics $\frac{e^{L}}{L}$. When one is concerned with closed geodesics of a fixed topological type, Mirzakhani proved that the counting function has asymptotics $e^{(6g-6) L}$. In this talk, I will explain a generalization of Mirzakhani’s counting result, for non-elementary and convex-cocompact subgroups of the mapping class group. Our framework is based on a topological phenomena called the north-south dynamics. Joint work with Dongryul M. Kim.