Skip to main content
SpringerLink
Log in

A cyclotomic family of thin hypergeometric monodromy groups in \({\text {Sp}}_4({\mathbb {R}})\)

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We exhibit an infinite family of discrete subgroups of \({{\,\mathrm{\textbf{Sp}}\,}}_4(\mathbb {R})\) which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters \(\left( \tfrac{N-3}{2N},\tfrac{N-1}{2N}, \tfrac{N+1}{2N}, \tfrac{N+3}{2N}\right) \) at infinity and maximal unipotent monodromy at zero, for any integer \(N\ge 4\). Additionally, we relate the cones used for ping-pong in \(\mathbb {R}^4\) with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian. These domains of discontinuity lead to uniformizations of variations of Hodge structure with Hodge numbers (1, 1, 1, 1).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. So far, all known thin examples appear also to be (essentially) injective.

References

  1. Barbot, T., Charette, V., Drumm, T., Goldman, W.M., Melnick, K.: A primer on the \((2+1)\) Einstein universe. In Recent developments in pseudo-Riemannian geometry—ESI Lect. Math. Phys., pp. 179–229. Eur. Math. Soc., Zürich (2008). https://doi.org/10.4171/051-1/6

  2. Burelle, J.-P., Charette, V., Francoeur, D., Goldman, W.M.: Einstein tori and crooked surfaces. Adv. Geometry (2021). https://doi.org/10.1515/advgeom-2020-0023

    Article  MathSciNet  Google Scholar 

  3. Beukers, F., Heckman, G.: Monodromy for the hypergeometric function \(_nF_{n-1}\). Invent. Math. Ser. 95(2), 325–354 (1989). https://doi.org/10.1007/BF01393900

    Article  Google Scholar 

  4. Burelle J.-P., Kassel, F.: Crooked surfaces and symplectic Schottky groups. (in preparation) (2018)

  5. Brav, C., Thomas, H.: Thin monodromy in Sp(4). Compos. Math. series 150(3), 333–343 (2014). https://doi.org/10.1112/S0010437X13007550

    Article  MathSciNet  Google Scholar 

  6. Charette, V., Francoeur, D., Lareau-Dussault, R.: Fundamental domains in the Einstein universe. Topol. Appl. Ser. 174, 62–80 (2014). https://doi.org/10.1016/j.topol.2014.06.011

    Article  MathSciNet  Google Scholar 

  7. Detinko, A., Flannery, D.L., Hulpke, A.: Zariski density and computing in arithmetic groups. Math. Comp. 87(310), 967–986 (2018). https://doi.org/10.1090/mcom/3236

    Article  MathSciNet  Google Scholar 

  8. Danciger, J., Guéritaud, F., Kassel, F.: Geometry and topology of complete Lorentz spacetimes of constant curvature. Ann. Sci. Éc. Norm. Supér.(4) 49(1), 1–56 (2016). https://doi.org/10.24033/asens.2275

  9. Deligne, P., Mostow, G.D.: Monodromy of hypergeometric functions and nonlattice integral monodromy. Inst. Hautes Études Sci. Publ. Math. 63, 5–89 (1986)

    Article  MathSciNet  Google Scholar 

  10. Drumm, T.A.: Fundamental polyhedra for Margulis space-times. Topol. Series 31(4), 677–683 (1992). https://doi.org/10.1016/0040-9383(92)90001-X

    Article  MathSciNet  Google Scholar 

  11. Eskin, A., Kontsevich, M., Möller, M., Zorich, A.: Lower bounds for Lyapunov exponents of flat bundles on curves. Geom. Topol. series 22(4), 2299–2338 (2018). https://doi.org/10.2140/gt.2018.22.2299

    Article  MathSciNet  Google Scholar 

  12. Filip, S., Fougeron, C.: Sagemath worksheet with symbolic computations for the cyclotomic family of thin monodromy groups. https://gitlab.com/fougeroc/notebook-cyclotomic-family

  13. Filip, S.: Semisimplicity and rigidity of the Kontsevich-Zorich cocycle. Invent. Math. series 205(3), 617–670 (2016). https://doi.org/10.1007/s00222-015-0643-3

    Article  MathSciNet  Google Scholar 

  14. Filip, S.: Families of K3 surfaces and Lyapunov exponents. Israel J. Math. Series 226(1), 29–69 (2018). https://doi.org/10.1007/s11856-018-1682-4

    Article  MathSciNet  Google Scholar 

  15. Filip, S.: Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents. (2021). arXiv:2110.07533

  16. Filip, S.: Translation surfaces: dynamics and hodge theory. EMS Surv. Math. Sci. pp. 1–91 (2024) (to appear)

  17. Fuchs, E., Meiri, C., Sarnak, P.: Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions. J. Eur. Math. Soc. (JEMS) 16(8), 1617–1671 (2014). https://doi.org/10.4171/JEMS/471

    Article  MathSciNet  Google Scholar 

  18. Fougeron, C.: Parabolic degrees and Lyapunov exponents for hypergeometric local systems. Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1580632

    Article  MathSciNet  Google Scholar 

  19. Guéritaud, F., Guichard, O., Kassel, F., Wienhard, A.: Anosov representations and proper actions. Geom. Topol. series 21(1), 485–584 (2017). https://doi.org/10.2140/gt.2017.21.485

    Article  MathSciNet  Google Scholar 

  20. Gray, J.J.: Linear differential equations and group theory from Riemann to Poincaré. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston. (2008). https://doi.org/10.1007/978-0-8176-4773-5

  21. Guichard, O., Wienhard, A.: Anosov representations: domains of discontinuity and applications. Invent. Math. Ser. 190(2), 357–438 (2012). https://doi.org/10.1007/s00222-012-0382-7

    Article  MathSciNet  Google Scholar 

  22. Kapovich, M., Leeb, B.: Relativizing characterizations of anosov subgroups, I. (2018) . arXiv:1807.00160

  23. Kapovich, M., Leeb, B., Porti, J.: Dynamics on flag manifolds: domains of proper discontinuity and cocompactness. Geom. Topol. Ser. 22(1), 157–234 (2018). https://doi.org/10.2140/gt.2018.22.157

    Article  MathSciNet  Google Scholar 

  24. Labourie, F.: Anosov flows, surface groups and curves in projective space. Invent. Math. Ser. 165(1), 51–114 (2006). https://doi.org/10.1007/s00222-005-0487-3

    Article  MathSciNet  Google Scholar 

  25. Möller, M.: Variations of Hodge structures of a Teichmüller curve. J. Amer. Math. Soc. Ser. 19(2), 327–344 (2006). https://doi.org/10.1090/S0894-0347-05-00512-6

    Article  Google Scholar 

  26. McMullen, C.T.: Billiards, heights, and the arithmetic of non-arithmetic groups. (2020)

  27. Sage Developers. SageMath, the Sage Mathematics Software System (Version 9.2) (2020). https://www.sagemath.org

  28. Sarnak, P.: Notes on thin matrix groups. In: Thin groups and superstrong approximation, vol. 61 of Math. Sci. Res. Inst. Publ., pp. 343–362. Cambridge Univ. Press, Cambridge (2014). https://mathscinet.ams.org/mathscinet-getitem?mr=3220897

  29. Singh, S., Venkataramana, T.N.: Arithmeticity of certain symplectic hypergeometric groups. Duke Math. J. Ser. 163(3), 591–617 (2014). https://doi.org/10.1215/00127094-2410655

    Article  MathSciNet  Google Scholar 

  30. Yoshida, M.: Hypergeometric functions, my love. Aspects of Mathematics, E32. Friedr. Vieweg & Sohn, Braunschweig (1997). https://doi.org/10.1007/978-3-322-90166-8

  31. Zhu, F.: Relatively dominated representations. arXiv:1912.13152

  32. Zorich, A.: Flat surfaces. In: Frontiers in number theory, physics, and geometry. I. pp. 437–583. Springer, Berlin (2006). https://doi.org/10.1007/978-3-540-31347-2_13

Download references

Acknowledgements

This work was supported by the Agence Nationale de la Recherche through the project Codys (ANR 18-CE40-0007). This material is based upon work supported by the National Science Foundation under Grant No. DMS-2005470, DMS-2305394 (SF) and DMS-1638352 (at the IAS).This research was partially conducted during the period the first-named author served as a Clay Research Fellow. SF also gratefully acknowledges support from the Institute for Advanced Study. Part of this work was conducted while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2019 semester, with support from U.S. National Science Foundation grants DMS-1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network), as well as Grant No. DMS-1440140 (MSRI).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL, 60637, USA

    Simion Filip

  2. LAGA - Université Sorbonne Paris Nord, 99 Av. Jean Baptiste Clément, 93430, Villetaneuse, France

    Charles Fougeron

Authors
  1. Simion Filip
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charles Fougeron
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Simion Filip.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Filip, S., Fougeron, C. A cyclotomic family of thin hypergeometric monodromy groups in \({\text {Sp}}_4({\mathbb {R}})\). Geom Dedicata 218, 44 (2024). https://doi.org/10.1007/s10711-024-00893-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10711-024-00893-4

Mathematics Subject Classification

Search

Navigation