Viveka Erlandsson: Counting geodesics on surfaces

Abstract

Counting closed geodesics on hyperbolic surfaces is both a classical and very active area of research. The geometric prime number theorem (first proved by Huber and then generalized by Margulis and others) states that the number of closed geodesics of length bounded by L grows exponentially in L, in fact, is asymptotic to exp(L)/L. There has been many generalizations of this result, among them there is an active theme of counting specific subsets of closed geodesics. In this talk I will mention some results in this direction, in particular counting geodesics of a fixed type (à la Mirzakhani) and counting geodesics with fixed commutator length.