On the length sets of closed hyperbolic surfaces

Abstract

For every closed surface $\Sigma_g$ of genus $g\geq6$, we prove that there exists a countable union of positive-codimension algebraic subsets of Teichm\"uller 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 $\tau(d)e^{\delta(g)L}$, where the explicit constant $\delta(g)$ satisfies $\delta(g)>\frac12$.