Geometry of infinitely generated Veech groups

Hubert, Pascal; Schmidt, Thomas

doi:10.1090/S1088-4173-06-00120-2

Geometry of infinitely generated Veech groups
HTML articles powered by AMS MathViewer

by Pascal Hubert and Thomas A. Schmidt

Conform. Geom. Dyn. 10 (2006), 1-20

DOI: https://doi.org/10.1090/S1088-4173-06-00120-2

Published electronically: January 10, 2006

PDF | Request permission

Abstract:

Veech groups uniformize Teichmüller geodesics in Riemann moduli space. We gave examples of infinitely generated Veech groups; see Duke Math. J. 123 (2004), 49–69. Here we show that the associated Teichmüller geodesics can even have both infinitely many cusps and infinitely many infinite ends.

References

José Bertin, Compactification des schémas de Hurwitz, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 11, 1063–1066 (French, with English and French summaries). MR 1396641
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
Pierre Dèbes, Méthodes topologiques et analytiques en théorie inverse de Galois: théorème d’existence de Riemann, Arithmétique de revêtements algébriques (Saint-Étienne, 2000) Sémin. Congr., vol. 5, Soc. Math. France, Paris, 2001, pp. 27–41 (French, with English and French summaries). MR 1924915
Clifford J. Earle and Frederick P. Gardiner, Teichmüller disks and Veech’s $\scr F$-structures, Extremal Riemann surfaces (San Francisco, CA, 1995) Contemp. Math., vol. 201, Amer. Math. Soc., Providence, RI, 1997, pp. 165–189. MR 1429199, DOI 10.1090/conm/201/02618

Unipotent flows and branched covers of Veech surfaces

Automorphic Functions

E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR 871067, DOI 10.1007/978-3-642-68649-8
Eugene Gutkin, Pascal Hubert, and Thomas A. Schmidt, Affine diffeomorphisms of translation surfaces: periodic points, Fuchsian groups, and arithmeticity, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 6, 847–866 (2004) (English, with English and French summaries). MR 2032528, DOI 10.1016/j.ansens.2003.05.001
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
Andrew Haas, Dirichlet points, Garnett points, and infinite ends of hyperbolic surfaces. I, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 1, 3–29. MR 1375503
Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
Pascal Hubert and Thomas A. Schmidt, Veech groups and polygonal coverings, J. Geom. Phys. 35 (2000), no. 1, 75–91. MR 1767943, DOI 10.1016/S0393-0440(99)00080-7
P. Hubert and T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 2, 461–495. MR 1824961, DOI 10.5802/aif.1829
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
John Hamal Hubbard, Sur les sections analytiques de la courbe universelle de Teichmüller, Mem. Amer. Math. Soc. 4 (1976), no. 166, ix+137. MR 430321, DOI 10.1090/memo/0166
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
A. N. Zemljakov and A. B. Katok, Topological transitivity of billiards in polygons, Mat. Zametki 18 (1975), no. 2, 291–300 (Russian). MR 399423
Eduard Looijenga, Correspondences between moduli spaces of curves, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 131–150. MR 1722542
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

Dynamics of $\text {SL}(2,\mathbb R)$ over moduli space in genus two

Teichmüller curves in genus two: Discriminant and spin

K. Magaard, T. Shaska, S. Shpectorov, and H. Völklein, The locus of curves with prescribed automorphism group, Sūrikaisekikenkyūsho K\B{o}kyūroku 1267 (2002), 112–141. Communications in arithmetic fundamental groups (Kyoto, 1999/2001). MR 1954371
C. Maclachlan and W. J. Harvey, On mapping-class groups and Teichmüller spaces. 4, Proc. London Math. Soc. (3) 30 (1975), no. part, 496–512. MR 374414, DOI 10.1112/plms/s3-30.4.496
Howard Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169–200. MR 644018, DOI 10.2307/1971341
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
Martin Möller, Teichmüller curves, Galois actions and $\widehat {GT}$-relations, Math. Nachr. 278 (2005), no. 9, 1061–1077. MR 2150378, DOI 10.1002/mana.200310292

Variations of Hodge structures of a Teichmüller curve

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
P. J. Nicholls, Garnett points for Fuchsian groups, Bull. London Math. Soc. 12 (1980), no. 3, 216–218. MR 572105, DOI 10.1112/blms/12.3.216
Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575, DOI 10.1017/CBO9780511600678
H. Popp, On a conjecture of H. Rauch on theta constants and Riemann surfaces with many automorphisms, J. Reine Angew. Math. 253 (1972), 66–77. MR 472842, DOI 10.1515/crll.1972.253.66
Harry E. Rauch, Theta constants on a Riemann surface with many automorphisms, Symposia Mathematica, Vol. III (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 305–323. MR 0260996
Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496. MR 624833
William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975, DOI 10.1515/9781400865321
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
William A. Veech, Geometric realizations of hyperelliptic curves, Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992) Plenum, New York, 1995, pp. 217–226. MR 1402493
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

Construction of Hurwitz Spaces