Arithmeticity, discreteness and volume

Gehring, F.; Maclachlan, C.; Martin, G.; Reid, A.

doi:10.1090/S0002-9947-97-01989-2

Arithmeticity, discreteness and volume
HTML articles powered by AMS MathViewer

by F. W. Gehring, C. Maclachlan, G. J. Martin and A. W. Reid PDF

Trans. Amer. Math. Soc. 349 (1997), 3611-3643 Request permission

Abstract:

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of ${PSL}(2,\mathbf {c})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple. We then list all “small” discrete groups generated by elliptics of order $2$ and $n$, $n=3,4,5,6,7$.

References

Colin C. Adams, The noncompact hyperbolic $3$-manifold of minimal volume, Proc. Amer. Math. Soc. 100 (1987), no. 4, 601–606. MR 894423, DOI 10.1090/S0002-9939-1987-0894423-8
Colin C. Adams, Limit volumes of hyperbolic three-orbifolds, J. Differential Geom. 34 (1991), no. 1, 115–141. MR 1114455
Hyman Bass, Groups of integral representation type, Pacific J. Math. 86 (1980), no. 1, 15–51. MR 586867
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
A. Borel, Commensurability classes and volumes of hyperbolic $3$-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 1–33. MR 616899
L. A. Best, On torsion-free discrete subgroups of $\textrm {PSL} (2,\textbf {C})$ with compact orbit space, Canadian J. Math. 23 (1971), 451–460. MR 284542, DOI 10.4153/CJM-1971-048-9
Chun Cao, On three generator Möbius groups, New Zealand J. Math. 23 (1994), no. 2, 111–120. MR 1313446
Ted Chinburg, A small arithmetic hyperbolic three-manifold, Proc. Amer. Math. Soc. 100 (1987), no. 1, 140–144. MR 883417, DOI 10.1090/S0002-9939-1987-0883417-4
Ted Chinburg and Eduardo Friedman, The smallest arithmetic hyperbolic three-orbifold, Invent. Math. 86 (1986), no. 3, 507–527. MR 860679, DOI 10.1007/BF01389265
T. Chinburg, E. Friedman, K. N. Jones and A. W. Reid, The smallest volume arithmetic hyperbolic 3-manifold, Preprint.
Marc Culler and Peter B. Shalen, Paradoxical decompositions, $2$-generator Kleinian groups, and volumes of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 231–288. MR 1135928, DOI 10.1090/S0894-0347-1992-1135928-4
D. A. Derevnin and A. D. Mednykh, Geometric properties of discrete groups acting with fixed points in a Lobachevskiĭ space, Dokl. Akad. Nauk SSSR 300 (1988), no. 1, 27–30 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 3, 614–617. MR 948799
D. J. H. Garling, A course in Galois theory, Cambridge University Press, Cambridge, 1986. MR 876676
D. Gabai, On the geometric and topological rigidity of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 10 (1997), 37–74.
D. Gabai, R. Meyerhoff and N. Thurston, Personal Communication.
F. W. Gehring and G. J. Martin, Axial distances in discrete Möbius groups, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 6, 1999–2001. MR 1154948, DOI 10.1073/pnas.89.6.1999
F. W. Gehring and G. J. Martin, $6$-torsion and hyperbolic volume, Proc. Amer. Math. Soc. 117 (1993), no. 3, 727–735. MR 1116260, DOI 10.1090/S0002-9939-1993-1116260-4
F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups, J. Anal. Math. 63 (1994), 175–219. MR 1269219, DOI 10.1007/BF03008423
F. W. Gehring and G. J. Martin, On the minimal volume hyperbolic $3$-orbifold, Math. Res. Lett. 1 (1994), no. 1, 107–114. MR 1258496, DOI 10.4310/MRL.1994.v1.n1.a13
F. W. Gehring and G. J. Martin, On the Margulis constant for Kleinian groups, I. MSRI preprint $\#$ 038–95
F. W. Gehring and G. J. Martin, Commutator spectra for discrete groups with an elliptic generator (in preparation).
F. W. Gehring and G. J. Martin, Tetrahedral, octahedral and icosahedral subgroups of a Kleinian group (in preparation).
F. W. Gehring and G. J. Martin, The volume of hyperbolic $3$–folds with $p$–torsion, $p\geq 6$. Centre for Mathematics and its Applications, Preprint (1995).
F. W. Gehring and G. J. Martin, Precisely invariant collars and the volumes of hyperbolic $3$–folds. Centre for Mathematics and its Applications, Preprint (1995).
F. W. Gehring and G. J. Martin, Torsion and volume in hyperbolic 3-folds (to appear).
H. J. Godwin, On quartic fields of signature one with small discriminant, Quart. J. Math. Oxford Ser. (2) 8 (1957), 214–222. MR 97375, DOI 10.1093/qmath/8.1.214
Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia, A characterization of arithmetic subgroups of $\textrm {SL}(2,\textbf {R})$ and $\textrm {SL}(2,\textbf {C})$, Math. Nachr. 159 (1992), 245–270. MR 1237113, DOI 10.1002/mana.19921590117
Robert D. Horowitz, Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math. 25 (1972), 635–649. MR 314993, DOI 10.1002/cpa.3160250602
Kerry N. Jones and Alan W. Reid, Non-simple geodesics in hyperbolic $3$-manifolds, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 2, 339–351. MR 1281551, DOI 10.1017/S0305004100072625
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, DOI 10.1017/S030500410006727X
C. Maclachlan and A. W. Reid, The arithmetic structure of tetrahedral groups of hyperbolic isometries, Mathematika 36 (1989), no. 2, 221–240 (1990). MR 1045784, DOI 10.1112/S0025579300013097
Albert Marden, The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383–462. MR 349992, DOI 10.2307/1971059
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
Robert Meyerhoff, The cusped hyperbolic $3$-orbifold of minimum volume, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 154–156. MR 799800, DOI 10.1090/S0273-0979-1985-15401-1
Walter D. Neumann and Alan W. Reid, Arithmetic of hyperbolic manifolds, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 273–310. MR 1184416
Michael Pohst and Hans Zassenhaus, On effective computation of fundamental units. I, Math. Comp. 38 (1982), no. 157, 275–291. MR 637307, DOI 10.1090/S0025-5718-1982-0637307-6
A. W. Reid, Ph. D Thesis, University of Aberdeen, 1987.
Alan W. Reid, A note on trace-fields of Kleinian groups, Bull. London Math. Soc. 22 (1990), no. 4, 349–352. MR 1058310, DOI 10.1112/blms/22.4.349
Alan W. Reid, Arithmeticity of knot complements, J. London Math. Soc. (2) 43 (1991), no. 1, 171–184. MR 1099096, DOI 10.1112/jlms/s2-43.1.171
Kisao Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), no. 4, 600–612. MR 398991, DOI 10.2969/jmsj/02740600
W.P. Thurston The geometry and topology of $3$–manifolds Princeton Lecture Notes, 1977.
Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
È. B. Vinberg, Rings of definition of dense subgroups of semisimple linear groups. , Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 45–55 (Russian). MR 0279206