×

Arithmetic Veech sublattices of SL\((2,\mathbf {Z})\). (English) Zbl 1244.32009

A Teichmüller curve in the moduli space \(\mathcal{M}_{g,[n]}\) of genus \(g\) Riemann surfaces with \(n\) punctures \(S_{g,n}\) is a holomorphic curve \(f: C\to \mathcal{M}_{g,[n]}\) which is generically one-to-one and a local isometry with respect to the Kobayashi metric. In 2009, C. T. McMullen [Math. Res. Lett. 16, No. 4, 647–649 (2009; Zbl 1187.32010)] has proved that every Teichmüller curve has a model as an algebraic curve over \(\overline{\mathbb{Q}}\); in this paper the authors prove that, conversely, every algebraic curve over \(\overline{\mathbb{Q}}\) is birationally equivalent over \(\mathbb{C}\) to a Teichmüller curve (that can even be chosen in \(\mathcal{M}_{g'}\) for \(g'\) large enough, where \(\mathcal{M}_{g'}\) is the moduli space of genus \(g'\) Riemann surfaces with no punctures).
This result is a consequence of the main (group-theoretical) theorem of this paper. A Veech group is the stabilizer in the mapping class group \(\mathrm{Mod}(S_{1,1})\cong SL(2,\mathbb{Z})\) of the conjugacy class of a finite-index subgroup in the fundamental group of \(S_{1,1}\), the torus with one puncture. Then the authors prove that every finite-index subgroup of the level 2 principal congruence subgroup \(\Gamma(2)\subset SL(2,\mathbb{Z})\) containing \(\{\pm1\}\) is a Veech group.
Reviewer: Marco Abate (Pisa)

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Citations:

Zbl 1187.32010
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves , J. Pure Appl. Algebra 159 (2001), 123-147. · Zbl 1045.14013 · doi:10.1016/S0022-4049(00)00056-6
[2] J. S. Birman and H. M. Hilden, On isotopies of homeomorphisms of Riemann surfaces , Ann. of Math. (2) 97 (1973), 424-439. · Zbl 0237.57001 · doi:10.2307/1970830
[3] M. Boggi, The congruence subgroup property for the hyperelliptic modular group: The open surface case , Hiroshima Math. J. 39 (2009), 351-362. · Zbl 1209.14023
[4] I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents , Ann. of Math. (2) 172 (2010), 139-185. · Zbl 1203.37049 · doi:10.4007/annals.2010.172.139
[5] K.-U. Bux, M. Ershov, and A. S. Rapinchuk, The congruence subgroup property for Aut( F 2 ) : A group-theoretic proof of Asada’s theorem , Groups Geom. Dyn. 5 (2011), 327-353. · Zbl 1251.20035 · doi:10.4171/GGD/130
[6] S. Diaz, R. Donagi, and D. Harbater, Every curve is a Hurwitz space , Duke Math. J. 59 (1989), 737-746. · Zbl 0712.14013 · doi:10.1215/S0012-7094-89-05933-4
[7] B. Farb and D. Margalit, A Primer on Mapping Class Groups , Princeton Univ. Press, Princeton, 2011. · Zbl 1245.57002
[8] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic , Duke Math. J. 103 (2000), 191-213. · Zbl 0965.30019 · doi:10.1215/S0012-7094-00-10321-3
[9] P. Hubert and S. Lelievre, Noncongruence subgroups in \(\mathcal{H}(2)\) , Int. Math. Res. Not. IMRN 2005 , no. 1, 47-64. · Zbl 1069.30074 · doi:10.1155/IMRN.2005.47
[10] P. Hubert, H. Masur, T. Schmidt, and A. Zorich, “Problems on billiards, flat surfaces and translation surfaces” in Problems on Mapping Class Groups and Related Topics , Proc. Sympos. Pure Math. 74 , Amer. Math. Soc., Providence, 2006. · Zbl 1307.37019
[11] P. Lochak, On arithmetic curves in the moduli spaces of curves , J. Inst. Math. Jussieu 4 (2005), 443-508. · Zbl 1094.14018 · doi:10.1017/S1474748005000101
[12] H. Masur, Closed trajectories for quadratic differentials with an application to billiards , Duke Math. J. 53 (1986), 307-314. · Zbl 0616.30044 · doi:10.1215/S0012-7094-86-05319-6
[13] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces , J. Amer. Math. Soc. 16 (2003), 857-885. · Zbl 1030.32012 · doi:10.1090/S0894-0347-03-00432-6
[14] C. T. McMullen, Rigidity of Teichmüller curves , Math. Res. Lett. 16 (2009), 647-649. · Zbl 1187.32010 · doi:10.4310/MRL.2009.v16.n4.a7
[15] D. B. McReynolds, The congruence subgroup problem for braid groups: Thurston’s proof , preprint, [math.GT] · Zbl 1331.20046
[16] M. Möller, Teichmüller curves, Galois actions and GT-relations , Math. Nachr. 278 (2005), 1061-1077. · Zbl 1081.14039 · doi:10.1002/mana.200310292
[17] M. Möller, Variations of Hodge structures of a Teichmüller curve , J. Amer. Math. Soc. 19 (2006), 327-344. · Zbl 1090.32004 · doi:10.1090/S0894-0347-05-00512-6
[18] G. Schmithüsen, An algorithm for finding the Veech group of an origami , Experiment. Math. 13 (2004), 459-472. · Zbl 1078.14036 · doi:10.1080/10586458.2004.10504555
[19] G. Schmithüsen, Veech groups of origamis , Ph.D. dissertation, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2005. · Zbl 1099.14015
[20] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , Invent. Math. 97 (1989), 553-583; Correction , Invent. Math. 103 (1991), 447. · Zbl 0676.32006 · doi:10.1007/BF01388890
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.