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Israel Journal of Mathematics

, Volume 223, Issue 1, pp 323–342 | Cite as

Veech surfaces and simple closed curves

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Abstract

We study the SL(2, ℝ)-infimal lengths of simple closed curves on halftranslation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths.

We also revisit the “no small virtual triangles” theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spectrum of a half-translation surface either has a gap above zero or is dense in a neighborhood of zero.

These results make use of the auxiliary polygon associated to a curve on a half-translation surface, as introduced by Tang and Webb.

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Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of OklahomaNormanUSA
  2. 2.Topology and Geometry of Manifolds UnitOkinawa Institute of Science and Technology Graduate UniversityOkinawaJapan
  3. 3.Mathematics DepartmentUniversity of OklahomaNormanUSA

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