Intersection Number, Length, and Systole on Compact Hyperbolic Surfaces

The interaction strength I(X) of a compact hyperbolic surface X is the best upper bound for the intersection number of two closed geodesics divided by the product of their lengths. Let $M_g$ be the moduli space of compact hyperbolic surfaces of genus g and sys(X) the length of a shortest closed geodesic on $X\in M_g$. We determine the asymptotic behavior of I(X), as $X\to \mathrm{\infty }$ in $M_g$, in terms of sys(X). We also determine the approximate behavior of the minimum of I(X) over $M_g$, as $g\to \mathrm{\infty }$.