Billiards, quadrilaterals and moduli spaces

Eskin, Alex; McMullen, Curtis; Mukamel, Ronen; Wright, Alex

doi:10.1090/jams/950

Billiards, quadrilaterals and moduli spaces
HTML articles powered by AMS MathViewer

by Alex Eskin, Curtis T. McMullen, Ronen E. Mukamel and Alex Wright;

J. Amer. Math. Soc. 33 (2020), 1039-1086

DOI: https://doi.org/10.1090/jams/950

Published electronically: September 25, 2020

HTML | PDF | Request permission

Uncorrected version: Original version posted September 25, 2020
Corrected version: This version corrects a publisher error appearing in the title.

References

P. Apisa, Periodic points in genus two: Holomorphic sections over Hilbert modular varieties, Teichmüller dynamics, and billiards, preprint, 2017.
David Aulicino, Duc-Manh Nguyen, and Alex Wright, Classification of higher rank orbit closures in $\mathcal H^{\rm {odd}}(4)$, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 8, 1855–1872. MR 3518480, DOI 10.4171/JEMS/631
Herbert Lange and Christina Birkenhake, Complex abelian varieties, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR 1217487, DOI 10.1007/978-3-662-02788-2
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
Fabrizio Catanese, Michael Lönne, and Fabio Perroni, Irreducibility of the space of dihedral covers of the projective line of a given numerical type, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22 (2011), no. 3, 291–309. MR 2847474, DOI 10.4171/RLM/601
C. Chevalley and A. Weil, Über das Verhalten der Integral erster Gattung bei Automorphismen des Funktionenkörpers, A. Weil, Oeuvres Scient., volume I, pages 68–71. Springer-Verlag, 1980.
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
Jordan S. Ellenberg, Endomorphism algebras of Jacobians, Adv. Math. 162 (2001), no. 2, 243–271. MR 1859248, DOI 10.1006/aima.2001.1994
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, Maryam Mirzakhani, and Amir Mohammadi, Isolation, equidistribution, and orbit closures for the $\textrm {SL}(2,\Bbb R)$ action on moduli space, Ann. of Math. (2) 182 (2015), no. 2, 673–721. MR 3418528, DOI 10.4007/annals.2015.182.2.7
Gerard van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. MR 930101, DOI 10.1007/978-3-642-61553-5
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
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
Maxim Kontsevich and Anton Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), no. 3, 631–678. MR 2000471, DOI 10.1007/s00222-003-0303-x
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
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
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 geodesics of infinite complexity, Acta Math. 191 (2003), no. 2, 191–223. MR 2051398, DOI 10.1007/BF02392964
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, 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, Navigating moduli space with complex twists, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1223–1243. MR 3055760, DOI 10.4171/JEMS/390
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
J.-F. Mestre, Familles de courbes hyperelliptiques à multiplications réelles, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 193–208 (French). MR 1085260, DOI 10.1007/978-1-4612-0457-2_{9}
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
William P. Thurston, Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 511–549. MR 1668340, DOI 10.2140/gtm.1998.1.511
William A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2) 124 (1986), no. 3, 441–530. MR 866707, DOI 10.2307/2007091
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
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
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, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015), no. 1, 413–438. MR 3318755, DOI 10.2140/gt.2015.19.413
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