On the length sets of closed hyperbolic surfaces

Abstract

For every closed surface $Σ_g$ of genus $g\geq6$, 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 $τ(d)e^{δ(g)L}$, where the explicit constant $δ(g)$ satisfies $δ(g)>\frac12$.