Osaka Journal of Mathematics

Jø rgensen subgroups of the Picard group

Francisco González-Acuña and Arturo Ramírez

Full-text: Open access


Let $G$ be a subgroup of rank two of the Möbius group $\textit{PSL}(2, \mathbb{C})$. The Jørgensen number $J(G)$ of $G$ is defined by \[ J(G) = \inf \{|\tr^{2} A - 4| + \mathopen|\tr[A, B]-2|\colon \langle A, B\rangle= G\}. \] We describ e all subgroups $G$ of the Picard group $\textit{PSL}(2, \mathbb{Z} + i\mathbb{Z})$ with $J(G) = 1$.

Article information

Osaka J. Math., Volume 44, Number 2 (2007), 471-482.

First available in Project Euclid: 5 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F40: Kleinian groups [See also 20H10]
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations


González-Acuña, Francisco; Ramírez, Arturo. Jø rgensen subgroups of the Picard group. Osaka J. Math. 44 (2007), no. 2, 471--482.

Export citation


  • A.M. Brunner, M.L. Frame, Y.W. Lee and N.J. Wielenberg: Classifying torsion-free subgroups of the Picard Group, Trans. Amer. Math. Soc. 282 (1984), 205--235.
  • T. Jørgensen: On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), 739--749.
  • T. Jørgensen and M. Kiikka: Some extreme discrete groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), 245--248.
  • T. Jørgensen, A. Lascurain and T. Pignataro: Translation extensions of the classical modular group, Complex Variables Theory Appl. 19 (1992), 205--209.
  • S.L. Krushkal$'$, B.N. Apanasov and N.A. Gusevskiĭ: Kleinian Groups and Uniformization in Examples and Problems, Trans. Math. Monographs 62, Amer. Math. Soc., Providence, RI, 1986.
  • R.C. Lyndon and P.E. Schupp: Combinatorial Group Theory, Springer, Berlin, 1977.
  • W. Magnus, A Karrass and D. Solitar: Combinatorial Group Theory, second revised edition, Dover, New York, 1976.
  • A. Marden: Geometrically finite Kleinian groups and their deformation spaces; in Discrete Groups and Automorphic Functions, (Proc. Conf., Cambridge, 1975), Academic Press, London, 1977, 259--293.
  • M.H.A. Newman and J.H.C. Whitehead: On the group of a certain linkage, Quart. J. Math. Oxford 8 (1937), 14--21.
  • H. Sato: One-parameter families of extreme discrete groups for Jørgensen's inequality; in In the Tradition of Ahlfors and Bers, (Stony Brook, NY, 1998), Contemp. Math. 256, Amer. Math. Soc., Providence, RI., 2000, 271--287
  • H. Sato: The Picard group, the Whitehead link and Jørgensen group; in Progress in Analysis, Vol. I, (Berlin, 2001), World Sci. Publ., River Edge, NJ. 2003, 149--158.
  • H. Sato: The Jørgensen number of the Whitehead link group, Boletin de Soc. Mat. Mex. (to appear).
  • H. Sato and R. Yamada: Some extreme Kleinian groups for Jrgensen's inequality, Rep. Fac. Sci. Shizuoka Univ. 27 (1993), 1--8.
  • R.G. Swan: Generators and relations for certain special linear groups, Advances in Math. 6 (1970), 1--77.
  • N. Wielenberg: The structure of certain subgroups of the Picard group, Math. Proc. Cambridge Philos. Soc. 84 (1978), 427--436.