Electronic Only Electronic Research Announcements
Electronic Research Announcements
ISSN 1088-4173
Previous volume | Table of contents | Next volume
Articles in press | Most recent volume | All volumes
Previous Article | Next Article
     

Jørgensen number and arithmeticity

Author(s): Jason Callahan
Journal: Conform. Geom. Dyn. 13 (2009), 160-186.
MSC (2000): Primary 30F40; Secondary 57M05, 57M25, 57M50
Posted: July 23, 2009
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: The Jørgensen number of a rank-two non-elementary Kleinian group $ \Gamma$ is

$\displaystyle J(\Gamma) = \inf\{\vert\mathrm{tr}^2 X - 4\vert + \vert\mathrm{tr} [X, Y] - 2\vert : \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.

References:

1.
C. Adams.
Waist size for cusps in hyperbolic 3-manifolds.
Topology, 41(2):257-270, 2002. MR 1876890 (2003e:57023)

2.
M. D. Baker and A. W. Reid.
Arithmetic knots in closed 3-manifolds.
J. Knot Theory Ramifications, 11(6):903-920, 2002.
Knots 2000 Korea, Vol. 3 (Yongpyong). MR 1936242 (2004b:57009)

3.
A. Beardon.
The geometry of discrete groups, volume 91 of Graduate Texts in Mathematics.
Springer-Verlag, New York, 1983. MR 698777 (85d:22026)

4.
W. Bosma, J. Cannon, and C. Playoust.
The Magma algebra system. I. The user language.
J. Symbolic Comput., 24(3-4):235-265, 1997.
Computational algebra and number theory (London, 1993). MR 1484478

5.
J. Callahan.
The Arithmetic and Geometry of Two-Generator Kleinian Groups.
PhD thesis, The University of Texas at Austin, 2009.

6.
M. D. E. Conder, C. Maclachlan, G. J. Martin, and E. A. O'Brien.
2-generator arithmetic Kleinian groups. III.
Math. Scand., 90(2):161-179, 2002. MR 1895609 (2003b:20071)

7.
F. W. Gehring, J. P. Gilman, and G. J. Martin.
Kleinian groups with real parameters.
Commun. Contemp. Math., 3(2):163-186, 2001. MR 1831927 (2002m:30057)

8.
F. W. Gehring, C. Maclachlan, and G. J. Martin.
Two-generator arithmetic Kleinian groups. II.
Bull. London Math. Soc., 30(3):258-266, 1998. MR 1608106 (99j:30050)

9.
F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid.
Arithmeticity, discreteness and volume.
Trans. Amer. Math. Soc., 349(9):3611-3643, 1997. MR 1433117 (98d:57022)

10.
F. W. Gehring and G. J. Martin.
Stability and extremality in Jørgensen's inequality.
Complex Variables Theory Appl., 12(1-4):277-282, 1989. MR 1040927 (91h:30067)

11.
F. González-Acuña and A. Ramırez.
Jørgensen subgroups of the Picard group.
Osaka J. Math., 44(2):471-482, 2007.

12.
L. Greenberg.
Maximal Fuchsian groups.
Bull. Amer. Math. Soc., 69:569-573, 1963. MR 0148620 (26:6127)

13.
F. Grunewald and J. Schwermer.
Subgroups of Bianchi groups and arithmetic quotients of hyperbolic $ 3$-space.
Trans. Amer. Math. Soc., 335(1):47-78, 1993. MR 1020042 (93c:11024)

14.
C. Hodgson and J. Weeks.
Symmetries, isometries and length spectra of closed hyperbolic three-manifolds.
Experiment. Math., 3(4):261-274, 1994. MR 1341719 (97a:57013)

15.
T. Jørgensen.
On discrete groups of Möbius transformations.
Amer. J. Math., 98(3):739-749, 1976. MR 0427627 (55:658)

16.
T. Jørgensen and M. Kiikka.
Some extreme discrete groups.
Ann. Acad. Sci. Fenn. Ser. A I Math., 1(2):245-248, 1975. MR 0399452 (53:3296)

17.
C. Li, M. Oichi, and H. Sato.
Jørgensen groups of parabolic type II (countably infinite case).
Osaka J. Math., 41(3):491-506, 2004. MR 2107659 (2005h:30082)

18.
C. Li, M. Oichi, and H. Sato.
Jørgensen groups of parabolic type I (finite case).
Comput. Methods Funct. Theory, 5(2):409-430, 2005. MR 2205423 (2006j:30080)

19.
C. Li, M. Oichi, and H. Sato.
Jørgensen groups of parabolic type III (uncountably infinite case).
Kodai Math. J., 28(2):248-264, 2005. MR 2153913 (2006h:30034)

20.
C. Maclachlan and G. J. Martin.
$ 2$-generator arithmetic Kleinian groups.
J. Reine Angew. Math., 511:95-117, 1999. MR 1695792 (2000m:20081)

21.
C. Maclachlan and A. W. Reid.
Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups.
Math. Proc. Cambridge Philos. Soc., 102(2):251-257, 1987. MR 898145 (88j:20040)

22.
C. Maclachlan and A. W. Reid.
The arithmetic of hyperbolic 3-manifolds, volume 219 of Graduate Texts in Mathematics.
Springer-Verlag, New York, 2003. MR 1937957 (2004i:57021)

23.
B. Maskit.
Kleinian groups, volume 287 of Grundlehren der Mathematischen Wissenschaften.
Springer-Verlag, Berlin, 1988. MR 959135 (90a:30132)

24.
R. Riley.
Parabolic representations of knot groups. I.
Proc. London Math. Soc. (3), 24:217-242, 1972. MR 0300267 (45:9313)

25.
R. Riley.
A quadratic parabolic group.
Math. Proc. Cambridge Philos. Soc., 77:281-288, 1975. MR 0412416 (54:542)

26.
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), volume 256 of Contemp. Math., pages 271-287. Amer. Math. Soc., Providence, RI, 2000. MR 1759686 (2001d:30079)

27.
H. Sato.
The Jørgensen number of the Whitehead link group.
Bol. Soc. Mat. Mexicana (3), 10(Special Issue):495-502, 2004. MR 2199365 (2006j:30081)

28.
H. Sato and R. Yamada.
Some extreme Kleinian groups for Jørgensen's inequality.
Rep. Fac. Sci. Shizuoka Univ., 27:1-8, 1993. MR 1217933 (95d:30082)

29.
R. Swan.
Generators and relations for certain special linear groups.
Advances in Math., 6:1-77 (1971), 1971. MR 0284516 (44:1741)

30.
K. Takeuchi.
Arithmetic triangle groups.
J. Math. Soc. Japan, 29(1):91-106, 1977. MR 0429744 (55:2754)

31.
The PARI Group, Bordeaux. XSPARI/GP, version 2.1.7, 2005. available from http://pari. math.u-bordeaux.fr/.

32.
J. Weeks.
SnapPea: a computer program for creating and studying hyperbolic 3-manifolds.
Available at www.geometrygames.org/SnapPea.

Similar Articles:

Retrieve articles in Conformal Geometry and Dynamics with MSC (2000): 30F40, 57M05, 57M25, 57M50

Retrieve articles in all Journals with MSC (2000): 30F40, 57M05, 57M25, 57M50

Additional Information:

Jason Callahan
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712 {\rm and } Department of Mathematics, St. Edward's University, 3001 South Congress Avenue, Austin, Texas 78704
Email: callahan@math.utexas.edu; jasonc@stedwards.edu

DOI: 10.1090/S1088-4173-09-00196-9
PII: S 1088-4173(09)00196-9
Received by editor(s): May 14, 2009
Posted: July 23, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google