Counting geodesics of given commutator length

Source abstract

Abstract Let $\Sigma$ be a closed hyperbolic surface. We study, for fixed g , the asymptotics of the number of those periodic geodesics in $\Sigma$ 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 $\Sigma$ . In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.