Counting geodesics of given commutator length

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.