Everything is illuminated

Abstract

We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of the SL(2,R)-action on the moduli space of translation surfaces. We characterize surfaces with the finite blocking property and bounded blocking property, completing work of the second-named author. Concerning the illumination problem, we also extend results of Hubert-Schmoll-Troubetzkoy, removing the hypothesis that the surface in question is a lattice surface, thus settling a conjecture. Our results crucially rely on the recent breakthrough results of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi, and on related results of Wright.