Measuring pants

Doan, Nhat Minh; Parlier, Hugo; Tan, Ser

doi:10.1090/tran/8893

Measuring pants
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by Nhat Minh Doan, Hugo Parlier and Ser Peow Tan;

Trans. Amer. Math. Soc. 376 (2023), 5281-5306

DOI: https://doi.org/10.1090/tran/8893

Published electronically: May 17, 2023

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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].