On the length sets of closed hyperbolic surfaces

Abstract

For every closed surface $\hat{I}{\pounds }_g$ of genus $g\ge 6$, we prove that there exists a countable union of positive-codimension algebraic subsets of Teichmüller space such that, for every hyperbolic metric $d$ outside this exceptional set, the number of distinct primitive closed-geodesic lengths at most $L$ is bounded below by $\ddot{I}„(d)e^{\hat{I}´(g)L}$, where the explicit constant $\hat{I}´(g)$ satisfies $\hat{I}´(g)>\frac{1}{2}$.