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TEICHMÜLLER CURVES IN HYPERELLIPTIC COMPONENTS OF MEROMORPHIC STRATA

Part of: Deformations of analytic structures

Published online by Cambridge University Press:  26 March 2025

Martin Möller Open the ORCID record for Martin Möller [Opens in a new window] ,
Scott Mullane ,
Benjamin Bakker  and
Scott Mullane
Martin Möller*
Affiliation:
Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main, Germany
Scott Mullane
Affiliation:
School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia (mullanes@unimelb.edu.au)
Benjamin Bakker
Affiliation:
Dept. of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, USA (bakker.uic@gmail.com)
Scott Mullane
Affiliation:
School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia (mullanes@unimelb.edu.au)
*
(moeller@math.uni-frankfurt.de)
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Abstract

We provide a complete classification of Teichmüller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichmüller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one, and Hurwitz covers of zero residue loci in strata with projective dimension two.

Keywords

Teichmüller curvesstrata of meromorphic differentialslinear manifolds

MSC classification

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 24 , Issue 4 , July 2025 , pp. 1521 - 1546
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

Research of M.M is supported by the DFG-project MO 1884/2-1 and the Collaborative Research Centre TRR 326 “Geometry and Arithmetic of Uniformized Structures”.

Research of S.M is supported by the Alexander von Humboldt Foundation, ERC Advanced Grant “SYZYGY”, and DECRA Grant DE220100918 from the Australian Research Council

References

Apisa, P and Wright, A (2021) Marked points on translation surfaces. Geom. Topol. 25 (6), 2913–2961.10.2140/gt.2021.25.2913CrossRefGoogle Scholar
Bainbridge, M, Chen, D, Gendron, Q, Grushevsky, S and Möller, M (2019) The moduli space of multi-scale differentials. arXiv preprint 1910.13492.Google Scholar
Benirschke, F, Dozier, B and Grushevsky, S (2022) Equations of linear subvarieties of strata of differentials. Geom. Topol. 26(6), 2773–2830.10.2140/gt.2022.26.2773CrossRefGoogle Scholar
Bainbridge, M, Habegger, P and Möller, M (2016) Teichmüller curves in genus three and just likely intersections in $Gnm\times Gna$ . Publ. Math. Inst. Hautes Études Sci. 124, 1–98.10.1007/s10240-016-0084-6CrossRefGoogle Scholar
Bouw, I and Möller, M (2010) Teichmüller curves, triangle groups, and Lyapunov exponents. Ann. of Math. (2) 172(1), 139–185.10.4007/annals.2010.172.139CrossRefGoogle Scholar
Boissy, C (2015) Connected components of the strata of the moduli space of meromorphic differentials. Comment. Math. Helv. 90(2), 255–286.10.4171/cmh/353CrossRefGoogle Scholar
Calta, K (2004) Veech surfaces and complete periodicity in genus two. J. Amer. Math. Soc. 17(4), 871–908.10.1090/S0894-0347-04-00461-8CrossRefGoogle Scholar
Eskin, A and Mirzakhani, M (2018) Invariant and stationary measures for the $SL\left(2,\mathbb{R}\right)$ action on moduli space. Publ. Math. Inst. Hautes Études Sci. 127, 95–324.10.1007/s10240-018-0099-2CrossRefGoogle Scholar
Eskin, A, Mirzakhani, M and Mohammadi, A (2015) Isolation, equidistribution, and orbit closures for the $SL\left(2,\mathbb{R}\right)$ action on moduli space. Ann. Math. (2) 182(2), 673–721.10.4007/annals.2015.182.2.7CrossRefGoogle Scholar
Eskin, A, McMullen, C, Mukamel, R and Wright, A (2020) Billiards, quadrilaterals, and Moduli spaces. J. Amer. Math. Soc. 33(4), 1039–1086.10.1090/jams/950CrossRefGoogle Scholar
Filip, S (2016) Splitting mixed Hodge structures over affine invariant manifolds. Ann. Math. (2) 183(2), 681–713.10.4007/annals.2016.183.2.5CrossRefGoogle Scholar
Filip, S (2024) Translation surfaces: dynamics and Hodge theory. EMS Surv. Math. Sci. 11(1), 63–151.10.4171/emss/78CrossRefGoogle Scholar
McMullen, C (2003) Billiards and Teichmüller curves on Hilbert modular surfaces. J. Amer. Math. Soc. 16(4) (2003), 857–885.10.1090/S0894-0347-03-00432-6CrossRefGoogle Scholar
McMullen, C (2005) Teichmüller curves in genus two: Discriminant and spin. Math. Ann. 333(1), 87–130.10.1007/s00208-005-0666-yCrossRefGoogle Scholar
McMullen, C (2006) Prym varieties and Teichmüller curves . Duke Math. J. 133(3), 569–590.10.1215/S0012-7094-06-13335-5CrossRefGoogle Scholar
McMullen, C (2006) Teichmüller curves in genus two: torsion divisors and ratios of sines. Invent. Math. 165(3), 651–672.10.1007/s00222-006-0511-2CrossRefGoogle Scholar
McMullen, CT (2023) Billiards and Teichmüller curves. Bull. Amer. Math. Soc. (N.S.) 60(2), 195–250.10.1090/bull/1782CrossRefGoogle Scholar
McMullen, C, Mukamel, R and Wright, A (2017) Cubic curves and totally geodesic subvarieties of moduli space. Ann. of Math. (2) 185(3), 957–990.10.4007/annals.2017.185.3.6CrossRefGoogle Scholar
Möller, M (2006) Variations of Hodge structures of a Teichmüller curve. J. Amer. Math. Soc. 19(2), 327–344.10.1090/S0894-0347-05-00512-6CrossRefGoogle Scholar
Matheus, C and Wright, A (2015) Hodge-Teichmüller planes and finiteness results for Teichmüller curves. Duke Math. J. 164(6), 1041–1077.10.1215/00127094-2885655CrossRefGoogle Scholar
Mirzakhani, M and Wright, A (2017) The boundary of an affine invariant submanifold. Invent. Math. 209(3), 927–984.10.1007/s00222-017-0722-8CrossRefGoogle Scholar
Mirzakhani, M and Wright, A (2018) Full-rank affine invariant submanifolds. Duke Math. J. 167(1), 1–40.10.1215/00127094-2017-0036CrossRefGoogle Scholar
Schwab, J (2023) The multi-scale boundary of the Gothic locus and topological invariants. In preparation.Google Scholar
Smillie, J and Weiss, B (2004) Minimal sets for flows on moduli space. Israel J. Math. 142, 249–260.10.1007/BF02771535CrossRefGoogle Scholar
Tahar, G (2018) Chamber structure of modular curves $X1(N)$ . Arnold Math. J. 4(3-4), 459–481.10.1007/s40598-019-00099-7CrossRefGoogle Scholar
Tahar, G (2020) Veech groups of flat surfaces with poles. Ann. Fac. Sci. Toulouse Math. (6) 29(1), 57–78.10.5802/afst.1623CrossRefGoogle Scholar
Valdez, F (2012) Veech groups, irrational billiards and stable abelian differentials. Discrete Contin. Dyn. Syst. 32(3), 1055–1063.10.3934/dcds.2012.32.1055CrossRefGoogle Scholar
Veech, W (1989) Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3), 553–583.10.1007/BF01388890CrossRefGoogle Scholar
Ward, C (1998) Calculation of Fuchsian groups associated to billiards in a rational triangle. Ergodic Theory Dynam. Systems 18(4), 1019–1042.10.1017/S0143385798117479CrossRefGoogle Scholar
Zorich, A (2006) Flat surfaces. In Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems. Berlin: Springer-Verlag, 439–586.10.1007/3-540-31347-8_13CrossRefGoogle Scholar