Jørgensen number and arithmeticity

Callahan, Jason

doi:10.1090/S1088-4173-09-00196-9

Jørgensen number and arithmeticity

Author: Jason Callahan
Journal: Conform. Geom. Dyn. 13 (2009), 160-186
MSC (2000): Primary 30F40; Secondary 57M05, 57M25, 57M50
DOI: https://doi.org/10.1090/S1088-4173-09-00196-9
Published electronically: July 23, 2009
MathSciNet review: 2525101
Full-text PDF Free Access

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.

1. Colin C. Adams, Waist size for cusps in hyperbolic 3-manifolds, Topology 41 (2002), no. 2, 257–270. MR 1876890, https://doi.org/10.1016/S0040-9383(00)00034-3
2. M. D. Baker and A. W. Reid, Arithmetic knots in closed 3-manifolds, J. Knot Theory Ramifications 11 (2002), no. 6, 903–920. Knots 2000 Korea, Vol. 3 (Yongpyong). MR 1936242, https://doi.org/10.1142/S0218216502002049
3. Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777
4. Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, https://doi.org/10.1006/jsco.1996.0125
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 (2002), no. 2, 161–179. MR 1895609, https://doi.org/10.7146/math.scand.a-14368
7. F. W. Gehring, J. P. Gilman, and G. J. Martin, Kleinian groups with real parameters, Commun. Contemp. Math. 3 (2001), no. 2, 163–186. MR 1831927, https://doi.org/10.1142/S0219199701000342
8. F. W. Gehring, C. Maclachlan, and G. J. Martin, Two-generator arithmetic Kleinian groups. II, Bull. London Math. Soc. 30 (1998), no. 3, 258–266. MR 1608106, https://doi.org/10.1112/S0024609397004359
9. F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3611–3643. MR 1433117, https://doi.org/10.1090/S0002-9947-97-01989-2
10. F. W. Gehring and G. J. Martin, Stability and extremality in Jørgensen’s inequality, Complex Variables Theory Appl. 12 (1989), no. 1-4, 277–282. MR 1040927, https://doi.org/10.1080/17476938908814372
11. F. González-Acuña and A. Ramırez.
Jørgensen subgroups of the Picard group.
Osaka J. Math., 44(2):471-482, 2007.
12. Leon Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573. MR 148620, https://doi.org/10.1090/S0002-9904-1963-11001-0
13. Fritz Grunewald and Joachim Schwermer, Subgroups of Bianchi groups and arithmetic quotients of hyperbolic 3-space, Trans. Amer. Math. Soc. 335 (1993), no. 1, 47–78. MR 1020042, https://doi.org/10.1090/S0002-9947-1993-1020042-6
14. Craig D. Hodgson and Jeffrey R. Weeks, Symmetries, isometries and length spectra of closed hyperbolic three-manifolds, Experiment. Math. 3 (1994), no. 4, 261–274. MR 1341719
15. Troels Jørgensen, On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), no. 3, 739–749. MR 427627, https://doi.org/10.2307/2373814
16. Troels Jørgensen and Maire Kiikka, Some extreme discrete groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 245–248. MR 0399452
17. Changjun Li, Makito Oichi, and Hiroki Sato, Jørgensen groups of parabolic type II (countably infinite case), Osaka J. Math. 41 (2004), no. 3, 491–506. MR 2107659
18. Changjun Li, Makito Oichi, and Hiroki Sato, Jørgensen groups of parabolic type I (finite case), Comput. Methods Funct. Theory 5 (2005), no. 2, 409–430. MR 2205423, https://doi.org/10.1007/BF03321107
19. Changjun Li, Makito Oichi, and Hiroki Sato, Jørgensen groups of parabolic type III (uncountably infinite case), Kodai Math. J. 28 (2005), no. 2, 248–264. MR 2153913, https://doi.org/10.2996/kmj/1123767006
20. C. Maclachlan and G. J. Martin, 2-generator arithmetic Kleinian groups, J. Reine Angew. Math. 511 (1999), 95–117. MR 1695792, https://doi.org/10.1515/crll.1999.511.95
21. C. Maclachlan and A. W. Reid, Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 251–257. MR 898145, https://doi.org/10.1017/S030500410006727X
22. Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957
23. Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
24. Robert Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3) 24 (1972), 217–242. MR 0300267, https://doi.org/10.1112/plms/s3-24.2.217
25. Robert Riley, A quadratic parabolic group, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281–288. MR 412416, https://doi.org/10.1017/S0305004100051094
26. Hiroki Sato, One-parameter families of extreme discrete groups for Jørgensen’s inequality, In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) Contemp. Math., vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 271–287. MR 1759686, https://doi.org/10.1090/conm/256/04013
27. Hiroki Sato, The Jørgensen number of the Whitehead link group, Bol. Soc. Mat. Mexicana (3) 10 (2004), no. Special Issue, 495–502. MR 2199365
28. Hiroki Sato and Ryuji Yamada, Some extreme Kleinian groups for Jørgensen’s inequality, Rep. Fac. Sci. Shizuoka Univ. 27 (1993), 1–8. MR 1217933
29. Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1–77 (1971). MR 284516, https://doi.org/10.1016/0001-8708(71)90027-2
30. Kisao Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), no. 1, 91–106. MR 429744, https://doi.org/10.2969/jmsj/02910091
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.