Prime geodesic theorem and closed geodesics for large genus

Abstract

Let ${\mathcal{M}}_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil–Petersson metric. We show that for any $ϵ>0$, as $g\to \infty$, for a generic surface in ${\mathcal{M}}_g$, the error term in the Prime Geodesic Theorem is bounded from above by $g\cdot t^{3/4+ϵ}$, up to a uniform multiplicative constant. The expected value of the error term in the Prime Geodesic Theorem over ${\mathcal{M}}_g$ is also studied. As an application, we show that as $g\to \infty$, on a generic hyperbolic surface in ${\mathcal{M}}_g$ most closed geodesics of length significantly less than $\sqrt{g}$ are simple and nonseparating, and most closed geodesics of length significantly greater than $\sqrt{g}$ are not simple, which confirms a conjecture of Lipnowski–Wright. A novel effective upper bound for intersection numbers on ${\mathcal{M}}_{g,n}$ is also established, when certain indices are large compared to $\sqrt{g+n}$.