Lower bound for KVol on the minimal stratum of translation surfaces

TL;DR

This paper investigates the minimal intersection properties of closed curves on translation surfaces, establishing lower bounds for the KVol measure in the minimal stratum of the moduli space, and constructing surfaces approaching this bound.

Contribution

It constructs families of translation surfaces in each connected component of the minimal stratum with KVol arbitrarily close to the genus, supporting a conjecture about the infimum of KVol.

Findings

KVol can be made arbitrarily close to the genus of the surface.

Constructed families of translation surfaces in each component of the minimal stratum.

Supports conjecture that the genus is the infimum of KVol.

Abstract

We are interested in the algebraic intersection of closed curves of a given length on translation surfaces. Namely, we study the quantity KVol which measures how many times can two closed curves of a given length intersect. In this paper, we construct families of translation surfaces in each connected component of the minimal stratum $\mathcal{H}(2g-2)$ of the moduli space of translation surfaces of genus $g \geq 2$ such that KVol is arbitrarily close to the genus of the surface, which is conjectured to be the infimum of KVol on $\mathcal{H}(2g-2)$.