Algebraic intersections on Bouw-M\"oller surfaces, and more general convex polygons

TL;DR

This paper investigates the intersection properties of closed curves on translation surfaces, extending previous work to a broader class including Bouw-M"oller surfaces, and provides sharp estimates for a geometric invariant called KVol.

Contribution

It extends intersection analysis to Bouw-M"oller surfaces and provides sharp estimates for KVol based on geometric constraints.

Findings

Sharp estimate for KVol on Bouw-M"oller surfaces with a single singularity.

Extension of intersection results to a large family of translation surfaces.

Ability to compute KVol on the $SL_2( eal)$-orbit of these surfaces.

Abstract

This paper focuses on intersection of closed curves on translation surfaces. Namely, we investigate the question of determining the intersection of two closed curves of a given length on such surfaces. This question has been investigated in several papers and this paper complement the work of Boulanger, Lanneau and Massart done for double regular polygons, and extend the results to a large family of surfaces which includes in particular Bouw-M\"oller surfaces. Namely, we give an estimate for KVol on surfaces based on geometric constraints (angles and indentifications of sides). This estimate is sharp in the case of Bouw-M\"oller surfaces with a unique singularity, and it allows to compute KVol on the $SL_2(\mathbb{R})$-orbit of such surfaces.