A classification of -Fuchsian subgroups of Picard modular groups
Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 75-92.

Given an imaginary quadratic extension K of , we classify the maximal nonelementary subgroups of the Picard modular group PU(1,2;𝒪 K ) preserving a totally real totally geodesic plane in the complex hyperbolic plane 2 . We prove that these maximal -Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius Δ of the corresponding -circle lies in -{0}, then the stabiliser arises from the quaternion algebra Δ,|D K | . We thus prove the existence of infinitely many orbits of K-arithmetic -circles in the hypersphere of 2 ().

Reçu le : 2018-12-01
Révisé le : 2018-09-02
Accepté le : 2018-12-02
Publié le : 2019-03-04
DOI : https://doi.org/10.5802/cml.51
Classification : 11F06,  11R52,  20H10,  20G20,  53C17,  53C55
Mots clés: Picard modular group, ball quotient, arithmetic Fuchsian groups, Heisenberg group, quaternion algebra, complex hyperbolic geometry, -circle, hypersphere
@article{CML_2018__10_2_75_0,
     author = {Jouni Parkkonen and Fr\'ed\'eric Paulin},
     title = {A classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groups},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {10},
     number = {2},
     year = {2018},
     pages = {75-92},
     doi = {10.5802/cml.51},
     language = {en},
     url = {cml.centre-mersenne.org/item/CML_2018__10_2_75_0/}
}
Parkkonen, Jouni; Paulin, Frédéric. A classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groups. Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 75-92. doi : 10.5802/cml.51. https://cml.centre-mersenne.org/item/CML_2018__10_2_75_0/

[1] A. Borel. Density and maximality of arithmetic subgroups. J. reine angew. Math., 224:78–89, 1966.

[2] A. Borel and Harish-Chandra. Arithmetic subgroups of algebraic groups. Annals of Mathematics, 75:485–535, 1962.

[3] T. Chinburg and M. Stover. Fuchsian subgroups of lattices acting on hermitian symmetric spaces. [arXiv:1105.1154v3].

[4] T. Chinburg and M. Stover. Geodesic curves on Shimura surfaces. Topology Proc., 52:113–121, 2018.

[5] W. M. Goldman. Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford University Press, 1999.

[6] R.-P. Holzapfel. Ball and surface arithmetics. Aspects of Mathematics, E29. Friedr. Vieweg & Sohn, 1998.

[7] H. Jacobowitz. An introduction to CR structures, Mathematical Surveys and Monographs, 32. American Mathematical Society, 1990.

[8] S. Katok. Fuchsian groups. Chicago Lectures in Mathematics. University of Chicago Press, 1992.

[9] C. Maclachlan. Fuchsian subgroups of the groups PSL 2 (O d ). In Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., 112, pages 305–311. Cambridge Univ. Press, 1986.

[10] C. Maclachlan and A. W. Reid. The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, 219. Springer-Verlag, 2003.

[11] M. Möller and D. Toledo. Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces. Algebra Number Theory, 9(4):897–912, 2015.

[12] G. D. Mostow. Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78. Princeton University Press, 1973.

[13] J. R. Parker. Traces in complex hyperbolic geometry. In Geometry, topology and dynamics of character varieties, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 32, pages 191–245. World Sci. Publ., 2012.

[14] J. R. Parker. Complex hyperbolic Kleinian groups. Cambridge University Press, to appear.

[15] J. Parkkonen and F. Paulin. Prescribing the behaviour of geodesics in negative curvature. Geom. Topol., 14(1):277–392, 2010.

[16] J. Parkkonen and F. Paulin. A classification of -Fuchsian subgroups of Picard modular groups. Math. Scand., 121(1):57–74, 2017.

[17] J. Parkkonen and F. Paulin. Counting and equidistribution in Heisenberg groups. Math. Ann., 367:81–119, 2017.

[18] P. Samuel. Théorie algébrique des nombres. Hermann, 1967.

[19] J.-P. Serre. Cours d’arithmétique. PUF, 1970.

[20] M. Stover. Volumes of Picard modular surfaces. Proc. Amer. Math. Soc., 139(9):3045–3056, 2011.

[21] K. Takeuchi. A characterization of arithmetic Fuchsian groups. J. Math. Soc. Japan, 27(4):600–612, 1975.

[22] M.-F. Vignéras. Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800. Springer-Verlag, 1980.

[23] R. J. Zimmer. Ergodic theory and semisimple groups, Monographs in Mathematics, 81. Birkhäuser Verlag, 1984.