TY - JOUR
T1 - Veech surfaces and simple closed curves
AU - Forester, Max
AU - Tang, Robert
AU - Tao, Jing
N1 - Funding Information:
Acknowledgements. The authors thank Alex Wright for many helpful conversations regarding SL(2, R)-orbit closures. We also thank Jenya Sapir for encouraging us to prove continuity of SL(2, R)-infimal length, and the referee for providing helpful feedback. Forester was partially supported by NSF award DMS-1105765, and Tao by NSF award DMS-1311834.
Publisher Copyright:
© 2018, Hebrew University of Jerusalem.
PY - 2018/2/1
Y1 - 2018/2/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85035093853&partnerID=8YFLogxK
U2 - 10.1007/s11856-017-1617-5
DO - 10.1007/s11856-017-1617-5
M3 - Article
AN - SCOPUS:85035093853
SN - 0021-2172
VL - 223
SP - 323
EP - 342
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -