Intersection Numbers in a Hyperbolic Surface

Abstract

For a compact surface $S$ with constant negative curvature $-\kappa$ (for some $\kappa>0$) and genus $\mathfrak g\geq2$, we show that the tails of the distribution of $i(\alpha,\beta)/l(\alpha)l(\beta)$ (where $i(\alpha,\beta)$ is the intersection number of the closed geodesics $\alpha$ and $\beta$ and $l(\cdot)$ denotes the geometric length) are estimated by a decreasing exponential function. As a consequence, we find the asymptotic normalized average of the intersection numbers of pairs of closed geodesics on $S$. In addition, we prove that the size of the sets of geodesics whose $T$-self-intersection number is not close to $\kappa T^2/(2\pi^2(\mathfrak g-1))$ is also estimated by a decreasing exponential function. And, as a corollary of the latter, we obtain a result of Lalley which states that most of the closed geodesics $\gamma$ on $S$ with $l(\gamma)\leq T$ have roughly $\kappa l(\gamma)^2/(2\pi^2(\mathfrak g-1))$ self-intersections, when $T$ is large.