Lower bound for KVol on the minimal stratum of translation surfaces

Abstract

In this paper we are interested in algebraic intersection of closed curves of a given length on translation surfaces. We study the quantity KVol, defined in Cheboui et al. (Bull Soc Math France 149(4):613–640, 2021) and studied in Cheboui et al. (2021), Cheboui et al. (C R Math Acad Sci Paris 359:65–70, 2021), Boulanger et al. (Ann Henri Lebesgue, 2024), and Boulanger (Algebraic intersection, lengths and Veech surfaces, 2023. arXiv:2309.17165), and 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 \(g \ge 2\) such that KVol is arbitrarily close to the genus of the surface, which is conjectured to be the infimum of KVol on \(\mathcal {H}(2g-2)\).