Intersection Number, Length, and Systole on Compact Hyperbolic Surfaces

TL;DR

The paper determines the precise asymptotic growth of the interaction strength $I(X)=\sup_{\gamma_1,\gamma_2} \frac{i(\gamma_1,\gamma_2)}{\ell_X(\gamma_1)\ell_X(\gamma_2)}$ for compact hyperbolic surfaces in the moduli space $\mathcal{M}_g$, showing $I(X)\sim \frac{1}{2\,\mathrm{sys}(X)\log(1/\mathrm{sys}(X))}$ as $X$ goes to infinity in $\mathcal{M}_g$ (with $g\ge 2$). It also establishes that $\min_{X\in\mathcal{M}_g} I(X) \asymp \frac{1}{(\log g)^2}$ and extends the analysis to finite-volume surfaces with cusps, where $I(X_r)\sim \max\left( \frac{1}{2s\log(1/s)}, \frac{1}{r(\log(1/r))^2}\right)$ for $s=\min(\mathrm{sys}(X),1/2)$. The work also connects $I(X)$ to geodesic currents via $I_{\Delta}(X) \in [I(X)/2,I(X)]$ and shows $I_{simple}(X)\asymp_g I(X)$, while proving the expected value of $I(X)$ under the Weil–Petersson measure is finite. The approach blends thin/thick decompositions, collar geometry, and current-based arguments to yield sharp asymptotics and uniform moduli-space bounds with implications for intersection theory on hyperbolic surfaces.

Abstract

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 $X \to \infty$ in $M_g$, in terms of sys(X). We also determine the approximate behavior of the minimum of I(X) over $M_g$, as $g \to \infty$.

Figures (11)

• Figure 1:
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (38)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4