Veech surfaces and simple closed curves (with Robert Tang, Jing Tao)
Israel Journal of Mathematics 223 (2018), 323-342

We study the SL(2,R)-infimal lengths of simple closed curves on half-translation 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.