Algebraic intersections on Bouw-Möller surfaces, and more general convex polygons

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.