A topological version of Huber's theorem

Let $X$ be a closed hyperbolic surface. We prove that the number of topological types of primitive closed geodesics of length at most $L$ is asymptotic to $$\frac{1}{\mathrm{\mid }\text{Isom}(X)\mathrm{\mid }}\frac{e^L}{2L}\mathrm{.}$$ as $L$ grows. Thus Huber’s asymptotic remains unchanged after quotienting by topological type, up to the finite symmetry factor coming from the isometry group of $X$.