Veech surfaces and simple closed curves

Max Forester; Robert Tang; Jing Tao

doi:10.1007/s11856-017-1617-5

Veech surfaces and simple closed curves

Max Forester^*, Robert Tang, Jing Tao

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	323-342
Number of pages	20
Journal	Israel Journal of Mathematics
Volume	223
Issue number	1
DOIs	https://doi.org/10.1007/s11856-017-1617-5
Publication status	Published - 1 Feb 2018
Externally published	Yes

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Forester, M., Tang, R., & Tao, J. (2018). Veech surfaces and simple closed curves. Israel Journal of Mathematics, 223(1), 323-342. https://doi.org/10.1007/s11856-017-1617-5

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abstract = "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.",

author = "Max Forester and Robert Tang and Jing Tao",

note = "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: {\textcopyright} 2018, Hebrew University of Jerusalem.",

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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.

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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.

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DO - 10.1007/s11856-017-1617-5

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JO - Israel Journal of Mathematics

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Veech surfaces and simple closed curves

Abstract

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