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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Veech surfaces and complete periodicity in genus two
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by Kariane Calta PDF
J. Amer. Math. Soc. 17 (2004), 871-908 Request permission

Abstract:

We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair $(M, \omega )$ where $M$ is a Riemann surface and $\omega$ is an Abelian differential on $M$. Equivalently, a translation surface is a two-manifold which has transition functions which are translations and a finite number of conical singularities arising from the zeros of $\omega$. A direction $v$ on a translation surface is completely periodic if any trajectory in the direction $v$ is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction $v$. Then, we say that a translation surface is completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of the group $SL(2, \mathbb {R})$ on the space of translation surfaces. A surface which has a lattice stabilizer under this action is said to be Veech. Veech proved that any Veech surface is completely periodic, but the converse is false. In this paper, we use the $J$-invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space ${\mathcal H}(2)$ of genus two translation surfaces with corresponding Abelian differentials which have a single double zero. Furthermore, we obtain a classification of all completely periodic surfaces in genus two.
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Additional Information
  • Kariane Calta
  • Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
  • Email: kcalta@math.cornell.edu
  • Received by editor(s): January 13, 2003
  • Published electronically: August 17, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 17 (2004), 871-908
  • MSC (1991): Primary 37A99; Secondary 37E15, 37D40, 37D50
  • DOI: https://doi.org/10.1090/S0894-0347-04-00461-8
  • MathSciNet review: 2083470