Algebraically periodic translation surfaces

Abstract

Algebraically periodic directions on translation surfaces were introduced by Calta in her study of genus two translation surfaces. We say that a translation surface with three or more algebraically periodic directions is an algebraically periodic surface. We show that for an algebraically periodic surface the slopes of the algebraically periodic directions are given by a number field which we call the periodic direction field. We show that translation surfaces with pseudo-Anosov automorphisms provide examples. In this case the periodic direction field is the holonomy field. We show that every algebraic field arises as the periodic direction field of a translation surface arising from a right-angled billiard table. The J-invariant of a translation surface was introduced by Kenyon and Smillie. We analyze the $J$ invariants of algebraically periodic surfaces and show that in some cases they are determined by the periodic direction field. We give explicit formulas for $J$ invariants in these cases. The Homological Affine Group was introduced by McMullen in his study of translation surfaces in genus two. We calculate this group for many algebraically periodic surfaces and relate it to the automorphism group of the J-invariant. We show that surfaces which admit certain decompositions into squares have totally real periodic direction field. This is related to a result of Hubert and Lanneau.