Open Access Open Access  Restricted Access Subscription Access

A classification of $\mathbb{C}$-Fuchsian subgroups of Picard modular groups

Jouni Parkkonen, Frédéric Paulin


Given an imaginary quadratic extension $K$ of $\mathbb{Q}$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal{O}_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb{H}^2_\mathbb{C}$. Complementing work of Holzapfel, Chinburg-Stover and M\"oller-Toledo, we show that these maximal $\mathbb{C}$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline\mathbb{Q}\end{array} \!\Big)$ for some explicit $D\in\mathbb{N}-\{0\}$ and $D_K$ the discriminant of $K$. We thus prove the existence of infinitely many orbits of $K$-arithmetic chains in the hypersphere of $\mathbb{P}_2(\mathbb{C})$.

Full Text:



Borel, A., Density and maximality of arithmetic subgroups, J. Reine Angew. Math. 224 (1966), 78–89.

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

Cartan, E., Sur le groupe de la géométrie hypersphérique, Comment. Math. Helv. 4 (1932), no. 1, 158–171.

Chinburg, T. and Stover, M., Fuchsian subgroups of lattices acting on hermitian symmetric spaces, preprint arxiv:1105.1154v3.

Chinburg, T. and Stover, M., Geodesic curves on Shimura surfaces, preprint arxiv:1506.03299.

Falbel, E., Francsics, G., and Parker, J. R., The geometry of the Gauss-Picard modular group, Math. Ann. 349 (2011), no. 2, 459–508.

Falbel, E. and Parker, J. R., The moduli space of the modular group in complex hyperbolic geometry, Invent. Math. 152 (2003), no. 1, 57–88.

Falbel, E. and Parker, J. R., The geometry of the Eisenstein-Picard modular group, Duke Math. J. 131 (2006), no. 2, 249–289.

Fine, B., Algebraic theory of the Bianchi groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 129, Marcel Dekker, Inc., New York, 1989.

Goldman, W. M., Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 1999.

Gromov, M., Hyperbolic groups, Math. Sci. Res. Inst. Publ., vol. 8, pp. 75--263, Springer-Verlag, New York, 1987.

Holzapfel, R.-P., Arithmetic curves on ball quotient surfaces, Ann. Global Anal. Geom. 1 (1983), no. 2, 21–90.

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

Katok, S., Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.

Knapp, A. W., Lie groups beyond an introduction, Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996.

Kudla, S. S., Intersection numbers for quotients of the complex $2$-ball and Hilbert modular forms, Invent. Math. 47 (1978), no. 2, 189–208.

Maclachlan, C., 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., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 305--311.

Maclachlan, C. and Reid, A. W., Parametrizing Fuchsian subgroups of the Bianchi groups, Canad. J. Math. 43 (1991), no. 1, 158–181.

Maclachlan, C. and Reid, A. W., The arithmetic of hyperbolic $3$-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003.

Möller, M. and Toledo, D., Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces, Algebra Number Theory 9 (2015), no. 4, 897–912.

Parker, J. R., Notes on complex hyperbolic geometry,, 2003.

Parkkonen, J. and Paulin, F., Prescribing the behaviour of geodesics in negative curvature, Geom. Topol. 14 (2010), no. 1, 277–392.

Parkkonen, J. and Paulin, F., Counting and equidistribution in Heisenberg groups, Math. Ann. 367 (2017), no. 1-2, 81–119.

Samuel, P., Théorie algébrique des nombres, Hermann, Paris, 1967.

Serre, J.-P., Lie algebras and Lie groups, second ed., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992.

Stover, M., Volumes of Picard modular surfaces, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3045–3056.

Takeuchi, K., A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), no. 4, 600–612.

Vignéras, M.-F., Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer-Verlag, Berlin, 1980.

Zimmer, R. J., Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984.



  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.

ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library