Abstract
We develop an algebraic framework for studying translation surfaces. Westudy the Sah--Arnoux--Fathi-invariant and the collection of directions inwhich it vanishes. We show that these directions are described by a numberfield which we call the periodic direction field. We study the$J$-invariant of a translation surface, introduced by Kenyon and Smillieand used by Calta. We relate the $J$-invariant to the periodic directionfield. For every number field $K\subset\ \mathbb R$ we show that there is atranslation surface for which the periodic direction field is $K$. We studyautomorphism groups associated to a translation surface and relate them tothe $J$-invariant. We relate the existence of decompositions of translationsurfaces into squares with the total reality of the periodic directionfield.
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