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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Jørgensen number and arithmeticity
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by Jason Callahan
Conform. Geom. Dyn. 13 (2009), 160-186
Published electronically: July 23, 2009


The Jørgensen number of a rank-two non-elementary Kleinian group $\Gamma$ is \[ J(\Gamma ) = \inf \{|\mathrm {tr}^2 X - 4| + |\mathrm {tr} [X, Y] - 2| : \langle X, Y \rangle = \Gamma \}. \] Jørgensen’s Inequality guarantees $J(\Gamma ) \geq 1$, and $\Gamma$ is a Jørgensen group if $J(\Gamma ) = 1$. This paper shows that the only torsion-free Jørgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jørgensen groups, and establishes a characterization of cocompact arithmetic Jørgensen groups. The paper concludes with computations of $J(\Gamma )$ for several non-cocompact Kleinian groups including some two-bridge knot and link groups.
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Bibliographic Information
  • Jason Callahan
  • Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712 and Department of Mathematics, St. Edward’s University, 3001 South Congress Avenue, Austin, Texas 78704
  • MR Author ID: 877083
  • Email:;
  • Received by editor(s): May 14, 2009
  • Published electronically: July 23, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 13 (2009), 160-186
  • MSC (2000): Primary 30F40; Secondary 57M05, 57M25, 57M50
  • DOI:
  • MathSciNet review: 2525101