Billiards and Teichmüller curves

McMullen, Curtis

doi:10.1090/bull/1782

Billiards and Teichmüller curves
HTML articles powered by AMS MathViewer

by Curtis T. McMullen HTML | PDF

Bull. Amer. Math. Soc. 60 (2023), 195-250

Abstract:

A Teichmüller curve $V \subset \mathcal {M}_g$ is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems.

References

Paul Apisa, $\textrm {GL}_2\Bbb R$ orbit closures in hyperelliptic components of strata, Duke Math. J. 167 (2018), no. 4, 679–742. MR 3769676, DOI 10.1215/00127094-2017-0043
Enrico Arbarello, Maurizio Cornalba, and Phillip A. Griffiths, Geometry of algebraic curves. Volume II, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 268, Springer, Heidelberg, 2011. With a contribution by Joseph Daniel Harris. MR 2807457, DOI 10.1007/978-3-540-69392-5
Pierre Arnoux and Thomas A. Schmidt, Veech surfaces with nonperiodic directions in the trace field, J. Mod. Dyn. 3 (2009), no. 4, 611–629. MR 2587089, DOI 10.3934/jmd.2009.3.611
David Aulicino and Duc-Manh Nguyen, Rank 2 affine manifolds in genus 3, J. Differential Geom. 116 (2020), no. 2, 205–280. MR 4168204, DOI 10.4310/jdg/1603936812
Matt Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol. 11 (2007), 1887–2073. MR 2350471, DOI 10.2140/gt.2007.11.1887
Matt Bainbridge and Martin Möller, The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3, Acta Math. 208 (2012), no. 1, 1–92. MR 2910796, DOI 10.1007/s11511-012-0074-6
J. Boulanger, Central points of the double heptagon translation surface are not connexion points, arXiv:2009.01748, 2020.
N. Bourbaki, Groupes et algèbres de Lie, Ch. IV–VI, Actualites Scientifiques et Industrielles, 1337, Hermann, Paris, 1968; Masson, Paris, 1981.
Irene I. Bouw and Martin Möller, Differential equations associated with nonarithmetic Fuchsian groups, J. Lond. Math. Soc. (2) 81 (2010), no. 1, 65–90. MR 2580454, DOI 10.1112/jlms/jdp059
Irene I. Bouw and Martin Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2) 172 (2010), no. 1, 139–185. MR 2680418, DOI 10.4007/annals.2010.172.139
Kariane Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004), no. 4, 871–908. MR 2083470, DOI 10.1090/S0894-0347-04-00461-8
Matteo Costantini and André Kappes, The equation of the Kenyon-Smillie $(2,3,4)$-Teichmüller curve, J. Mod. Dyn. 11 (2017), 17–41. MR 3588522, DOI 10.3934/jmd.2017002
D. Davis and S. Lelièvre, Periodic paths on the pentagon, double pentagon and golden L, arXiv:1810.11310, 2018.
Laura DeMarco, The conformal geometry of billiards, Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 1, 33–52. MR 2738905, DOI 10.1090/S0273-0979-2010-01322-7
Eduard Duryev, Teichmuller Curves in Genus Two: Square-tiled Surfaces and Modular Curves, ProQuest LLC, Ann Arbor, MI, 2018. Thesis (Ph.D.)–Harvard University. MR 4187609
Alex Eskin, Simion Filip, and Alex Wright, The algebraic hull of the Kontsevich-Zorich cocycle, Ann. of Math. (2) 188 (2018), no. 1, 281–313. MR 3815463, DOI 10.4007/annals.2018.188.1.5
Alex Eskin, Curtis T. McMullen, Ronen E. Mukamel, and Alex Wright, Billiards, quadrilaterals and moduli spaces, J. Amer. Math. Soc. 33 (2020), no. 4, 1039–1086. MR 4155219, DOI 10.1090/jams/950
Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 903027
E. Goujard, Sous–variétés totalement géodésiques des espaces de modules de Riemann, Astérisque 430 (2021), 407–424.
Eugene Gutkin and Chris Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000), no. 2, 191–213. MR 1760625, DOI 10.1215/S0012-7094-00-10321-3
Elise Hanson, Adam Merberg, Christopher Towse, and Elena Yudovina, Generalized continued fractions and orbits under the action of Hecke triangle groups, Acta Arith. 134 (2008), no. 4, 337–348. MR 2449157, DOI 10.4064/aa134-4-4
W. Patrick Hooper, Another Veech triangle, Proc. Amer. Math. Soc. 141 (2013), no. 3, 857–865. MR 3003678, DOI 10.1090/S0002-9939-2012-11379-6
W. Patrick Hooper, Grid graphs and lattice surfaces, Int. Math. Res. Not. IMRN 12 (2013), 2657–2698. MR 3071661, DOI 10.1093/imrn/rns124
John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 2, Matrix Editions, Ithaca, NY, 2016. Surface homeomorphisms and rational functions. MR 3675959
Pascal Hubert and Erwan Lanneau, Veech groups without parabolic elements, Duke Math. J. 133 (2006), no. 2, 335–346. MR 2225696, DOI 10.1215/S0012-7094-06-13326-4
Pascal Hubert and Thomas A. Schmidt, Infinitely generated Veech groups, Duke Math. J. 123 (2004), no. 1, 49–69. MR 2060022, DOI 10.1215/S0012-7094-04-12312-8
Pascal Hubert and Thomas A. Schmidt, An introduction to Veech surfaces, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 501–526. MR 2186246, DOI 10.1016/S1874-575X(06)80031-7
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Richard Kenyon and John Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000), no. 1, 65–108. MR 1760496, DOI 10.1007/s000140050113
Steven Kerckhoff, Howard Masur, and John Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. (2) 124 (1986), no. 2, 293–311. MR 855297, DOI 10.2307/1971280
Abhinav Kumar and Ronen E. Mukamel, Real multiplication through explicit correspondences, LMS J. Comput. Math. 19 (2016), no. suppl. A, 29–42. MR 3540944, DOI 10.1112/S1461157016000188
Abhinav Kumar and Ronen E. Mukamel, Algebraic models and arithmetic geometry of Teichmüller curves in genus two, Int. Math. Res. Not. IMRN 22 (2017), 6894–6942. MR 3737325, DOI 10.1093/imrn/rnw193
E. Lanneau and M. Möller. Non–existence and finiteness results for Teichmüller curves in Prym loci, Exp. Math., to appear.
Erwan Lanneau and Duc-Manh Nguyen, Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4, J. Topol. 7 (2014), no. 2, 475–522. MR 3217628, DOI 10.1112/jtopol/jtt036
Erwan Lanneau and Duc-Manh Nguyen, Weierstrass Prym eigenforms in genus four, J. Inst. Math. Jussieu 19 (2020), no. 6, 2045–2085. MR 4167002, DOI 10.1017/s1474748019000057
Anne Larsen, Chaya Norton, and Bradley Zykoski, Strongly obtuse rational lattice triangles, Trans. Amer. Math. Soc. 374 (2021), no. 10, 7119–7142. MR 4315599, DOI 10.1090/tran/8415
Christopher J. Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer’s number, Geom. Topol. 8 (2004), 1301–1359. MR 2119298, DOI 10.2140/gt.2004.8.1301
Pierre Lochak, On arithmetic curves in the moduli spaces of curves, J. Inst. Math. Jussieu 4 (2005), no. 3, 443–508. MR 2197065, DOI 10.1017/S1474748005000101
Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957, DOI 10.1007/978-1-4757-6720-9
Howard Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J. 53 (1986), no. 2, 307–314. MR 850537, DOI 10.1215/S0012-7094-86-05319-6
Howard Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Math. J. 66 (1992), no. 3, 387–442. MR 1167101, DOI 10.1215/S0012-7094-92-06613-0
Howard Masur, Ergodic theory of translation surfaces, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 527–547. MR 2186247, DOI 10.1016/S1874-575X(06)80032-9
Howard Masur and Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015–1089. MR 1928530, DOI 10.1016/S1874-575X(02)80015-7
Curtis T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857–885. MR 1992827, DOI 10.1090/S0894-0347-03-00432-6
Curtis T. McMullen, Teichmüller curves in genus two: discriminant and spin, Math. Ann. 333 (2005), no. 1, 87–130. MR 2169830, DOI 10.1007/s00208-005-0666-y
Curtis T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J. 133 (2006), no. 3, 569–590. MR 2228463, DOI 10.1215/S0012-7094-06-13335-5
Curtis T. McMullen, Teichmüller curves in genus two: torsion divisors and ratios of sines, Invent. Math. 165 (2006), no. 3, 651–672. MR 2242630, DOI 10.1007/s00222-006-0511-2
Curtis T. McMullen, Dynamics of $\textrm {SL}_2(\Bbb R)$ over moduli space in genus two, Ann. of Math. (2) 165 (2007), no. 2, 397–456. MR 2299738, DOI 10.4007/annals.2007.165.397
Curtis T. McMullen, Braid groups and Hodge theory, Math. Ann. 355 (2013), no. 3, 893–946. MR 3020148, DOI 10.1007/s00208-012-0804-2
Curtis T. McMullen, Diophantine and ergodic foliations on surfaces, J. Topol. 6 (2013), no. 2, 349–360. MR 3065179, DOI 10.1112/jtopol/jts033
Curtis T. McMullen, Teichmüller dynamics and unique ergodicity via currents and Hodge theory, J. Reine Angew. Math. 768 (2020), 39–54. MR 4168686, DOI 10.1515/crelle-2019-0037
Curtis T. McMullen, Modular symbols for Teichmüller curves, J. Reine Angew. Math. 777 (2021), 89–125. MR 4292865, DOI 10.1515/crelle-2021-0019
Curtis T. McMullen, Billiards, heights, and the arithmetic of non-arithmetic groups, Invent. Math. 228 (2022), no. 3, 1309–1351. MR 4419633, DOI 10.1007/s00222-022-01101-4
C. McMullen, Letter to Leininger, et al., 13 July 2003.
Curtis T. McMullen, Ronen E. Mukamel, and Alex Wright, Cubic curves and totally geodesic subvarieties of moduli space, Ann. of Math. (2) 185 (2017), no. 3, 957–990. MR 3664815, DOI 10.4007/annals.2017.185.3.6
Martin Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math. 165 (2006), no. 3, 633–649. MR 2242629, DOI 10.1007/s00222-006-0510-3
Martin Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), no. 2, 327–344. MR 2188128, DOI 10.1090/S0894-0347-05-00512-6
Martin Möller, Affine groups of flat surfaces, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, 2009, pp. 369–387. MR 2497782, DOI 10.4171/055-1/11
Martin Möller, Shimura and Teichmüller curves, J. Mod. Dyn. 5 (2011), no. 1, 1–32. MR 2787595, DOI 10.3934/jmd.2011.5.1
Martin Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces, Amer. J. Math. 136 (2014), no. 4, 995–1021. MR 3245185, DOI 10.1353/ajm.2014.0026
Martin Möller and David Torres-Teigell, Euler characteristics of Gothic Teichmüller curves, Geom. Topol. 24 (2020), no. 3, 1149–1210. MR 4157552, DOI 10.2140/gt.2020.24.1149
Martin Möller and Don Zagier, Modular embeddings of Teichmüller curves, Compos. Math. 152 (2016), no. 11, 2269–2349. MR 3577896, DOI 10.1112/S0010437X16007636
Ronen E. Mukamel, Orbifold points on Teichmüller curves and Jacobians with complex multiplication, Geom. Topol. 18 (2014), no. 2, 779–829. MR 3180485, DOI 10.2140/gt.2014.18.779
Ronen E. Mukamel, Polynomials defining Teichmüller curves and their factorizations $\textrm {mod}\,p$, Exp. Math. 30 (2021), no. 1, 19–31. MR 4223280, DOI 10.1080/10586458.2018.1488156
Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
Jan-Christoph Puchta, On triangular billiards, Comment. Math. Helv. 76 (2001), no. 3, 501–505. MR 1854695, DOI 10.1007/PL00013215
J.-C. Puchta, Addendum to “On triangular billiards”, Preprint, 2021.
G. Salmon, Higher Plane Curves, Hodges, Foster and Figgis, Dublin, 1879.
Richard Evan Schwartz, Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories, Experiment. Math. 18 (2009), no. 2, 137–171. MR 2549685, DOI 10.1080/10586458.2009.10128891
R. Schwartz, Billiards from the square to the stadium, ICM Proceedings, 2022, to appear.
Kisao Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), no. 1, 91–106. MR 429744, DOI 10.2969/jmsj/02910091
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
David Torres-Teigell and Jonathan Zachhuber, Orbifold points on Prym-Teichmüller curves in genus 3, Int. Math. Res. Not. IMRN 4 (2018), 1228–1280. MR 3801461, DOI 10.1093/imrn/rnw277
David Torres-Teigell and Jonathan Zachhuber, Orbifold points on Prym-Teichmüller curves in genus 4, J. Inst. Math. Jussieu 18 (2019), no. 4, 673–706. MR 3963516, DOI 10.1017/s1474748017000196
W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583. MR 1005006, DOI 10.1007/BF01388890
William A. Veech, Geometric realizations of hyperelliptic curves, Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992) Plenum, New York, 1995, pp. 217–226. MR 1402493
M. Viana, Dynamics of interval exchange transformations and Teichmüller flows, Preprint, 2008.
Ya. B. Vorobets, Plane structures and billiards in rational polygons: the Veech alternative, Uspekhi Mat. Nauk 51 (1996), no. 5(311), 3–42 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 5, 779–817. MR 1436653, DOI 10.1070/RM1996v051n05ABEH002993
Ya. B. Vorobets, Plane structures and billiards in rational polyhedra, Uspekhi Mat. Nauk 51 (1996), no. 1(307), 145–146 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 1, 177–178. MR 1392678, DOI 10.1070/RM1996v051n01ABEH002769
Clayton C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems 18 (1998), no. 4, 1019–1042. MR 1645350, DOI 10.1017/S0143385798117479
Alex Wright, Schwarz triangle mappings and Teichmüller curves: the Veech-Ward-Bouw-Möller curves, Geom. Funct. Anal. 23 (2013), no. 2, 776–809. MR 3053761, DOI 10.1007/s00039-013-0221-z
Alex Wright, Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surv. Math. Sci. 2 (2015), no. 1, 63–108. MR 3354955, DOI 10.4171/EMSS/9
Alex Wright, From rational billiards to dynamics on moduli spaces, Bull. Amer. Math. Soc. (N.S.) 53 (2016), no. 1, 41–56. MR 3403080, DOI 10.1090/bull/1513
Alex Wright, Totally geodesic submanifolds of Teichmüller space, J. Differential Geom. 115 (2020), no. 3, 565–575. MR 4120819, DOI 10.4310/jdg/1594260019
Jean-Christophe Yoccoz, Interval exchange maps and translation surfaces, Homogeneous flows, moduli spaces and arithmetic, Clay Math. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 2010, pp. 1–69. MR 2648692
Jonathan Zachhuber, The Galois action and a spin invariant for Prym-Teichmüller curves in genus 3, Bull. Soc. Math. France 146 (2018), no. 3, 427–439 (English, with English and French summaries). MR 3936530, DOI 10.24033/bsmf.2766
Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437–583. MR 2261104, DOI 10.1007/978-3-540-31347-2_13