Intersection Numbers of Geodesic Arcs

Abstract

For a compact surface $S$ with constant negative curvature $-κ$ (for some $κ>0$) and genus $g\geq2$, we show that the tails of the distribution of $i(α,β)/l(α)l(β)$ (where $i(α,β)$ is the intersection number of the closed geodesics 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 $κT^2/(2π^2(g-1))$ is also estimated by a decreasing exponential function. And, as a corollary of the latter, we obtain a result of S. Lalley which states that most of the closed geodesics $α$ on $S$ with $l(α)\leq T$ have roughly $κl(α)^2/(2π^2(g-1))$ self-intersections, when $T$ is large.