Closed geodesics in homology classes modulo sublattices

Abstract

Let $M$ be a Weil-Petersson random hyperbolic surface of genus $g$, and let $\Gamma \subset {\mathbb{Z}}^{2g}$ be a lattice of prime index $q$. We study the distribution of primitive closed geodesics in homology classes mod $\Gamma$ in the large genus limit. Averaging over all lattices of index $q$, with $q\to \infty$, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether $\frac{X}{q\log X}$, the expected number of primitive geodesics in a given homology class mod $\Gamma$, tends to $\lambda >0$ or $\infty$). We also study the unnormalized variance $G_M(X,\Gamma )$ of the counts among homology classes, and show that as $X\to \infty$, averaged over all lattices of prime index $q$, it is asymptotic to $X\log X$ in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.

22 pages. Comments are welcome!