# American Institute of Mathematical Sciences

April  2008, 2(2): 209-248. doi: 10.3934/jmd.2008.2.209

## Algebraically periodic translation surfaces

 1 Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, United States 2 Department of Mathematics, Cornell University, Ithaca, NY 14853, United States

Received  May 2007 Revised  October 2007 Published  January 2008

We develop an algebraic framework for studying translation surfaces. We study the Sah--Arnoux--Fathi-invariant and the collection of directions in which it vanishes. We show that these directions are described by a number field which we call the periodic direction field. We study the $J$-invariant of a translation surface, introduced by Kenyon and Smillie and used by Calta. We relate the $J$-invariant to the periodic direction field. For every number field $K\subset\ \mathbb R$ we show that there is a translation surface for which the periodic direction field is $K$. We study automorphism groups associated to a translation surface and relate them to the $J$-invariant. We relate the existence of decompositions of translation surfaces into squares with the total reality of the periodic direction field.
Citation: Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209
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