With cedram.org Confluentes Mathematici Search for an article Search within the site English français Table of contents for this issue | Previous article | Next article Jouni Parkkonen; Frédéric PaulinA classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groupsConfluentes Mathematici, 10 no. 2 (2018), p. 75-92, doi: 10.5802/cml.51 Article PDF Class. Math.: 11F06, 11R52, 20H10, 20G20, 53C17, 53C55Keywords: Picard modular group, ball quotient, arithmetic Fuchsian groups, Heisenberg group, quaternion algebra, complex hyperbolic geometry, $\protect \mathbb{R}$-circle, hypersphere Résumé - AbstractGiven an imaginary quadratic extension $K$ of $\mathbb{Q}$, we classify the maximal nonelementary subgroups of the Picard modular group $\operatorname{PU}(1,2;{\mathcal{O}}_K)$ preserving a totally real totally geodesic plane in the complex hyperbolic plane $\mathbb{H}^2_\mathbb{C}$. We prove that these maximal $\mathbb{R}$-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius $\Delta$ of the corresponding $\mathbb{R}$-circle lies in $\mathbb{N}-\lbrace 0\rbrace$, then the stabiliser arises from the quaternion algebra $\Big (\!\frac{\Delta \,,\, |D_K|}{\mathbb{Q}}\!\Big )$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb{R}$-circles in the hypersphere of $\mathbb{P}_2(\mathbb{C})$. Bibliography[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.  MR 1695450[6] R.-P. Holzapfel. Ball and surface arithmetics. Aspects of Mathematics, E29. Friedr. Vieweg & Sohn, 1998.  MR 1685419[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 ${\rm PSL}_2(O_d)$. 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.  MR 1937957[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 $\mathbb{C}$-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.  MR 215808[19] J.-P. Serre. Cours dâarithmÃ©tique. PUF, 1970.  MR 255476[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.  MR 580949[23] R. J. Zimmer. Ergodic theory and semisimple groups, Monographs in Mathematics, 81. Birkhäuser Verlag, 1984.  MR 776417 © 2012Confluentes Mathematici Published by the Institut Camille Jordan CNRS - UMR 5208 and the Unité de Mathématiques Pures et Appliquées CNRS - UMR 5669 of Université de Lyon eISSN : 1793-7434