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Mirror stabilizers for lattice complex hyperbolic triangle groups

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Abstract

For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizers of (reprentatives of each group orbit of) mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.

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References

  1. Barthel, G., Hirzebruch, F., Höfer, T.: Geradenkonfigurationen und algebraische Flächen. Aspects of Mathematics, vol. D4. Springer, Wiesbaden (1987)

  2. Beardon, A.F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics, vol. 91. Springer, New York (1983)

    Book  Google Scholar 

  3. Borel, A., Harish-Chandra: Arithmetic subgroups of algebraic groups. Ann. Math. 2(75), 485–535 (1962)

    Article  MathSciNet  Google Scholar 

  4. Chen, S.S., Greenberg, L.: Hyperbolic Spaces. Contribution to Analysis, Collection of Papers Dedicated to Lipman Bers, pp. 49–87 (1974)

  5. Couwenberg, W., Heckman, G., Looijenga, E.: Geometric structures on the complement of a projective arrangement. Publ. Math. Inst. Hautes Étud. Sci. 101, 69–161 (2005)

    Article  MathSciNet  Google Scholar 

  6. de Rham, G.: Sur les polygones générateurs de groupes fuchsiens. Enseign. Math. 2(17), 49–61 (1971)

    Google Scholar 

  7. Deligne, P., Mostow, G.D.: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. Inst. Hautes Étud. Sci. 63, 5–89 (1986)

    Article  Google Scholar 

  8. Deraux, M.: Spocheck (2019). https://hal.archives-ouvertes.fr/hal-02398953

  9. Deraux, M.: gitlab project neat subgroups. https://plmlab.math.cnrs.fr/deraux/neat-subs

  10. Deraux, M.: Non-arithmetic ball quotients from a configuration of elliptic curves in an Abelian surface. Comment. Math. Helv. 93(3), 533–554 (2018)

    Article  MathSciNet  Google Scholar 

  11. Deraux, M.: Non-arithmetic lattices and the Klein quartic. J. Reine Angew. Math. 754, 253–279 (2019)

    Article  MathSciNet  Google Scholar 

  12. Deraux, M., Parker, J.R., Paupert, J.: Census of the complex hyperbolic sporadic triangle groups. Exp. Math. 20(4), 467–486 (2011)

    Article  MathSciNet  Google Scholar 

  13. Deraux, M., Parker, J.R., Paupert, J.: New nonarithmetic complex hyperbolic lattices. Invent. Math. 203(3), 681–771 (2016)

    Article  MathSciNet  Google Scholar 

  14. Deraux, M., Parker, J.R., Paupert, J.: New nonarithmetic complex hyperbolic lattices II. Mich. Math. J. 70(1), 133–205 (2021)

    Article  MathSciNet  Google Scholar 

  15. Erlandsson, V., Parkkonen, J., Paulin, F., Souto, J.: Counting and equidistribution of reciprocal closed geodesic in negative curvature (in preparation)

  16. Falbel, E., Pasquinelli, I.: Complex hyperbolic orbifolds and Lefschetz fibrations. Preprint (2020). arXiv:2003.06466

  17. Goldman, W.M.: Complex Hyperbolic Geometry. Clarendon Press, Oxford (1999)

    Book  Google Scholar 

  18. Jiang, Y., Wang, J., Yang, F.: Arithmeticity of \(\mathbb{C}\)-fuchsian subgroups of some non-arithmetic lattices. Preprint (2022). https://doi.org/10.21203/rs.3.rs-2366924/v1

  19. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vol. II, Volume 15 of Interscience. Pure and Applied Mathematics. Interscience Publishers, New York (1969)

  20. Mostow, G.D.: On a remarkable class of polyhedra in complex hyperbolic space. Pac. J. Math. 86, 171–276 (1980)

    Article  MathSciNet  Google Scholar 

  21. Sun, L.: C-Fuchsian subgroups of some non-arithmetic lattices. Korean J. Math. 30(2), 315–333 (2022)

    MathSciNet  Google Scholar 

  22. Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein Birthday Schrift Dedicated to David Epstein on the Occasion of his 60th Birthday, pp. 511–549. Institute of Mathematics, University of Warwick, Warwick (1998)

  23. Wells, J.: Non-arithmetic hybrid lattices in \(\text{ PU }(2,1)\). Geom. Dedic. 208, 1–11 (2020)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Univ. Grenoble Alpes, CNRS, IF, 38000, Grenoble, France

    Martin Deraux

  2. Sorbonne Université, Paris, France

    Martin Deraux

  3. CNRS, INRIA, IMJ-PRG, Ouragan, Université de Paris, 75005, Paris, France

    Martin Deraux

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  1. Martin Deraux
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Correspondence to Martin Deraux.

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Deraux, M. Mirror stabilizers for lattice complex hyperbolic triangle groups. Geom Dedicata 218, 58 (2024). https://doi.org/10.1007/s10711-024-00910-6

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