Sparse curve systems have intermediate growth type

TL;DR

The paper studies sparse curve systems on a closed orientable surface $\Sigma_g$ by defining $sysp(g)$ as the maximal size of a finite family of pairwise non-isotopic simple closed curves with average pairwise intersection number at most $1$, and generalizing to $sysp_f(g)$ for $f(g)$-sparsity. It proves a sharp growth-type result: for $f(g)=g^\alpha$ with $\alpha\in(-1,1]$, the maximal size satisfies a lower bound of order roughly $[2g^{(1-\alpha)/2}]\,2^{g^{(1+\alpha)/2}}$ and an upper bound of order $2g\,e^{\sqrt{128}\,g^{(1+\alpha)/2}}$, implying $sysp(g)$ itself grows like $c^{\sqrt{g}}$ with $c$ between $2$ and $81938$. The lower bound comes from a necklace-constructed family yielding many non-isotopic curves with controlled intersections, while the upper bound follows from the Hubard-Parlier crossing-number inequality applied to curves (simple or not). These results illuminate the intermediate growth behavior between linear and exponential in $g$ and relate to broader questions about curve counting on surfaces.

Abstract

A system of simple closed curves on a surface of genus $g$ is said to be sparse if their average pairwise intersection number does not exceed one. We show that the maximal size of a sparse curve systems grows roughly like a function of type $c^{\sqrt{g}}$, with $c$ between $2$ and $81938$.

Figures (3)

• Figure 1: The four arcs on $\Sigma_{1,2}$.
• Figure 2: The genus $h$ surface $N$ consists of $h-1$ copies of $\Sigma_{1,2}$ and $h-1$ annuli.
• Figure 3: The surface $\Sigma_g$ consists of $h'$ copies of $N$ attached to the surface $B$.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Remark