Veech surfaces and simple closed curves
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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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