Veech surfaces and simple closed curves

TL;DR

This paper characterizes Veech surfaces through SL(2,R)-infimal lengths of simple closed curves and explores the structure of the virtual triangle area spectrum, revealing a dichotomy in its distribution.

Contribution

It provides a new characterization of Veech surfaces based on curve lengths and analyzes the virtual triangle spectrum's gap or density properties.

Findings

Veech surfaces are characterized by specific length properties.

The virtual triangle area spectrum either has a positive gap or is dense near zero.

The auxiliary polygon approach is key to these results.

Abstract

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.