Counting geodesics of given commutator length

Abstract

Let Σ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in Σ having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in Σ. In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.