Veech surfaces and simple closed curves

We study the SL(2, ?)-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.