Transactions of the American Mathematical Society

Author(s): F. W. Gehring; C. Maclachlan; G. J. Martin; A. W. Reid
Journal: Trans. Amer. Math. Soc. 349 (1997), 3611-3643.
MSC (1991): Primary 30F40, 20H10, 57N10
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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 $2$

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1.: C. Adams, The noncompact hyperbolic -manifold of minimal volume, Proc. Amer. Math. Soc. 100 (1987) 601-606. MR 88m:57018
2.: C. Adams, Limit volumes of hyperbolic -orbifolds, J. Diff. Geom. 34 (1991) 115-141. MR 92d:57029
3.: H. Bass, Groups of integral representation type, Pacific J. Math. 86 (1980) 15-51. MR 82c:20014
4.: A. F. Beardon, The geometry of discrete groups, Springer-Verlag 1983. MR 85d:22026
5.: A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa 8 (1981) 1-33. MR 82j:22008
6.: L. A. Best, On torsion-free discrete subgroups of $PSL_2(\mathbf {C})$ with compact orbit space, Can. J. Math. 23 (1971) 451-460. MR 44:1767
7.: C. Cao, On three-generator Möbius groups, New Zealand Math. J. 23 (1994) 111-120. MR 96b:30093
8.: T. Chinburg A small arithmetic hyperbolic three-manifold, Proc. Amer. Math. Soc. 100 (1987) 140-144. MR 88b:57043
9.: T. Chinburg and E. Friedman, The smallest arithmetic hyperbolic 3-orbifold, Invent. Math. 86 (1986) 507-527. MR 88a:22022
10.: T. Chinburg, E. Friedman, K. N. Jones and A. W. Reid, The smallest volume arithmetic hyperbolic 3-manifold, Preprint.
11.: M. Culler and P. Shalen, Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds, J. Amer. Math. Soc. 5 (1992) 231-288. MR 93a:57017
12.: D. A. Derevnin and A. D. Mednykh, Geometric properties of discrete groups acting with fixed points in Lobachevsky space, Dokl. Akad. Nauk SSSR 300 (1988) 614-617. MR 90a:30131
13.: D. J. H. Garling, A course in Galois theory, Camb. Univ. Press 1986. MR 88d:12007
14.: D. Gabai, On the geometric and topological rigidity of hyperbolic -manifolds, J. Amer. Math. Soc. 10 (1997), 37-74. CMP 97:02
15.: D. Gabai, R. Meyerhoff and N. Thurston, Personal Communication.
16.: F. W. Gehring and G. J. Martin, Axial distances in discrete Möbius groups, Proc. Natl. Acad. Sci. USA 89 (1992) 1999-2001. MR 92m:30082
17.: F. W. Gehring and G. J. Martin, 6-torsion and hyperbolic volume, Proc. Amer. Math. Soc. 117 (1993) 727-735. MR 93d:30056
18.: F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups, J. d'Analyse Math. 63 (1994) 175-219. MR 96c:30040
19.: F. W. Gehring and G. J. Martin, On the minimal volume hyperbolic 3-orbifold, Math. Res. Letters 1 (1994) 107-114. MR 95b:30072
20.: F. W. Gehring and G. J. Martin, On the Margulis constant for Kleinian groups, I. MSRI preprint $\#$ 038-95
21.: F. W. Gehring and G. J. Martin, Commutator spectra for discrete groups with an elliptic generator (in preparation).
22.: F. W. Gehring and G. J. Martin, Tetrahedral, octahedral and icosahedral subgroups of a Kleinian group (in preparation).
23.: F. W. Gehring and G. J. Martin, The volume of hyperbolic -folds with -torsion, $p\geq 6$ . Centre for Mathematics and its Applications, Preprint (1995).
24.: F. W. Gehring and G. J. Martin, Precisely invariant collars and the volumes of hyperbolic -folds. Centre for Mathematics and its Applications, Preprint (1995).
25.: F. W. Gehring and G. J. Martin, Torsion and volume in hyperbolic 3-folds (to appear).
26.: H. J. Godwin, On quartic fields with signature one and small discriminant, Quart. J. Math. Oxford (2) 8 (1957) 214-222. MR 20:3844
27.: H. M. Hilden, M-T. Lozano and J. M. Montesinos, A characterization of arithmetic subgroups of $\operatorname {SL} (2,\textbf {R})$ and $\operatorname {SL} (2,\textbf {C})$ , Math. Nach. 159 (1992) 245-270. MR 94i:20088
28.: R. Horowitz, Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math. 25 (1972) 635-649. MR 47:3542
29.: K. N. Jones and A. W. Reid, Non-simple geodesics in hyperbolic 3-manifolds, Math. Proc. Camb. Phil. Soc. 116 (1994), 339-351. MR 95e:57025
30.: C. Maclachlan and A. W. Reid, Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups, Math. Proc. Camb. Phil. Soc 102 (1987) 251-257. MR 88j:20040
31.: C. Maclachlan and A.W. Reid, The arithmetic structure of tetrahedral groups of hyperbolic isometries, Mathematika 36 (1989) 221-240. MR 91b:11055
32.: A. Marden, The geometry of finitely generated Kleinian groups, Annals of Math. 99 (1974) 383-462. MR 50:2485
33.: B. Maskit, Kleinian groups, Springer-Verlag 1988. MR 90a:30132
34.: R. Meyerhoff, The cusped hyperbolic 3-orbifold of minimum volume, Bull. Amer. Math. Soc. 13 (1985) 154-156. MR 87b:22022
35.: W. D. Neumann and A. W. Reid, Arithmetic of hyperbolic 3-manifolds. In TOPOLOGY '90, Proceedings of the Research Semester in Low-Dimensional Topology at Ohio State University, 273-310. Editors, B. Apanasov, W. D. Neumann, A. W. Reid and L. Siebenmann, De Gruyter Verlag 1992. MR 94c:57024
36.: M. Pohst, P. Weiler and H. Zassenhaus On effective computation of fundamental units II, Math. Comp. 38 (1982), 293-329. MR 83e:12005b
37.: A. W. Reid, Ph. D Thesis, University of Aberdeen, 1987.
38.: A. W. Reid, A note on trace-fields of Kleinian groups, Bull. London Math. Soc. 22 (1990) 349-352. MR 91d:20056
39.: A. W. Reid, Arithmeticity of Knot Complements, J. London Math. Soc.(2) 43 (1991) 171-184. MR 92a:57011
40.: K. Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975) 600-612. MR 53:2842
41.: W.P. Thurston The geometry and topology of -manifolds Princeton Lecture Notes, 1977.
42.: M-F. Vignéras, Arithmétique des algèbres de quaternions. Lecture Notes in Math., vol. 800, Springer-Verlag 1980. MR 82i:12016
43.: E. B. Vinberg, Rings of definition of dense subgroups of semisimple linear groups, Math. USSR Izvestija 35 (1971) 45-55. MR 43:4929

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 30F40, 20H10, 57N10

F. W. Gehring
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003
Email: fgehring@math.lsa.umich.edu

C. Maclachlan
Affiliation: Department of Mathematics, University of Aberdeen, Aberdeen, Scotland
Email: cmac@math.aberdeen.ac.uk

G. J. Martin
Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand and Australian National University, Canberra, Australia
Email: martin@math.auckland.ac.nz

A. W. Reid
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: areid@math.utexas.edu

DOI: 10.1090/S0002-9947-97-01989-2
PII: S 0002-9947(97)01989-2
Received by editor(s): January 4, 1996
Additional Notes: Research supported in part by grants from the U. S. National Science Foundation, the N.Z. Foundation of Research, Science and Technology, the Australian Research Council, the U.K. Royal Society and the U.K. Scientific and Engineering Research Council. We wish also to thank the University of Texas at Austin and the University of Auckland for their hospitality during part of this work. We are grateful to D. J. Lewis who gave us a proof for Theorem 5.12 and to K. N. Jones who computed the co-volumes that appear in Table 11.
Copyright of article: Copyright 1997, American Mathematical Society

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