Measuring pants
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- by Nhat Minh Doan, Hugo Parlier and Ser Peow Tan PDF
- Trans. Amer. Math. Soc. 376 (2023), 5281-5306 Request permission
Abstract:
We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum “dominates” another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface.References
- Ara Basmajian, The orthogonal spectrum of a hyperbolic manifold, Amer. J. Math. 115 (1993), no. 5, 1139–1159. MR 1246187, DOI 10.2307/2375068
- Martin Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geom. Topol. 15 (2011), no. 2, 707–733. MR 2800364, DOI 10.2140/gt.2011.15.707
- Martin Bridgeman and Ser Peow Tan, Identities on hyperbolic manifolds, Handbook of Teichmüller theory. Vol. V, IRMA Lect. Math. Theor. Phys., vol. 26, Eur. Math. Soc., Zürich, 2016, pp. 19–53. MR 3497292
- Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
- Virginie Charette and William M. Goldman, McShane-type identities for affine deformations, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 5, 2029–2041 (English, with English and French summaries). MR 3732683, DOI 10.5802/aif.3128
- Feng Luo and Ser Peow Tan, A dilogarithm identity on moduli spaces of curves, J. Differential Geom. 97 (2014), no. 2, 255–274. MR 3263507
- Hengnan Hu and Ser Peow Tan, New identities for small hyperbolic surfaces, Bull. Lond. Math. Soc. 46 (2014), no. 5, 1021–1031. MR 3262203, DOI 10.1112/blms/bdu056
- Greg McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math. 132 (1998), no. 3, 607–632. MR 1625712, DOI 10.1007/s002220050235
- Maryam Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. (2) 168 (2008), no. 1, 97–125. MR 2415399, DOI 10.4007/annals.2008.168.97
- William P. Thurston, Minimal stretch maps between hyperbolic surfaces, Preprint, math.GT (1986) [math/9801039v1].
Additional Information
- Nhat Minh Doan
- Affiliation: Institute of Mathematics, Vietnam Academy of Science and Technology, Vietnam
- Email: dnminh@math.ac.vn
- Hugo Parlier
- Affiliation: Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg
- MR Author ID: 767561
- ORCID: 0000-0001-5618-509X
- Email: hugo.parlier@uni.lu
- Ser Peow Tan
- Affiliation: Department of Mathematics, National University of Singapore, Singapore
- MR Author ID: 331663
- ORCID: 0000-0002-2382-0656
- Email: mattansp@nus.edu.sg
- Received by editor(s): February 26, 2020
- Received by editor(s) in revised form: September 15, 2020, and December 9, 2020
- Published electronically: May 17, 2023
- Additional Notes: Research of the first author was supported by FNR PRIDE15/10949314/GSM. Research of the second author was partially supported by ANR/FNR project SoS, INTER/ANR/16/11554412/SoS, ANR-17-CE40-0033. Research of the third author was partially supported by R146-000-289-114
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 5281-5306
- MSC (2020): Primary 32G15, 57K20, 37D20; Secondary 30F10, 30F60, 53C23, 57M50
- DOI: https://doi.org/10.1090/tran/8893