Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-12T11:56:54.725Z Has data issue: false hasContentIssue false
  • English
  • Français

The arithmetic structure of tetrahedral groups of hyperbolic isometries

Part of: Lie groups

Published online by Cambridge University Press:  26 February 2010

C. Maclachlan  and
A. W. Reid
C. Maclachlan
Affiliation:
Department of Mathematics, University of Aberdeen, The Edward Wright Building, Dunbar Street. Aberdeen, AB9 2TY.
A. W. Reid
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th AvenueColumbus, Ohio, 43210-1174, U.S.A..
Get access

Extract

Introduction. Polyhedra in 3-dimensional hyperbolic space which give rise to discrete groups generated by reflections in their faces have been investigated in [14], [17], [29] and in the case of tetrahedra there are precisely nine compact non-congruent ones with dihedral angles integral submultiples of π [14]. These polyhedral groups give rise to hyperbolic 3-orbifolds and examples of these have been studied, for example, in [3], [15], [18], [24], [25].

MSC classification

Secondary: 22E40: Discrete subgroups of Lie groups
Type
Research Article
Information
Mathematika , Volume 36 , Issue 2 , December 1989 , pp. 221 - 240
Copyright
Copyright © University College London 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Baskan, T. and Macbeath, A. M.. Centralizers of reflections in crystallographic groups. Math. Proc. Camb. Phil. Soc, 92 (1982), 79–91.CrossRefGoogle Scholar
2.Beardon, A. F.. The geometry of discrete groups. Graduate texts in Maths. 91 (Springer-Verlag, 1983).CrossRefGoogle Scholar
3.Best, L. A.. On torsion-free discrete subgroups of PSL2 () with compact orbit space. Can. J. Math., 23 (1971), 451–460.Google Scholar
4.Borel, A.. Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Sc. Norm. Sup. Pisa, 8 (1981), 1–33.Google Scholar
5.Borel, A. and Harish-Chandra, . Arithmetic subgroups of algebraic groups. Ann. of Math., 75 (1962), 485–534.Google Scholar
6.Chinburg, T.. Volumes of hyperbolic manifolds. J. Dig. Geom., 8 (1983), 783–789.Google Scholar
7.Chinburg, T. and Friedman, E.. The smallest arithmetic hyperbolic three-orbifold. Invent. Math., 86 (1986), 507–527.CrossRefGoogle Scholar
8.Chowla, S. and Selberg, A.. On Epstein's zeta-function. j Reine Angew. Math., 227 (1967), 86–110.Google Scholar
9.Cohn, H.. A classical invitation to algebraic numbers and class fields. Universitext (Springer-Verlag, 1978).CrossRefGoogle Scholar
10.Godwin, H. J.. On quartic fields of signature one and small discriminant. Quart. J. Math. Oxford (2), 8 (1957), 214–222.CrossRefGoogle Scholar
11.Gromov, M.. Hyperbolic manifolds according to Thurson and Jørgenson. Sém. Bourbaki 546, L.N.M. 842, 40–53 (Springer-Verlag, 1981).Google Scholar
12.Lam, T. Y.. Algebraic theory of quadratic forms. Mathematics lecture note series (Benjamin, 1973).Google Scholar
13.Lang, S.. Algebraic number theory (Reading, Mass.: Addison-Wesley, 1970).Google Scholar
14.Lanner, F.. On complexes with transitive groups of automorphisms. Comm. Sem. Math., Univ. Lund (1950).Google Scholar
15.Macbeath, A. M.. Commensurability of cocompact three dimensional hyperbolic groups. Duke Math. J., 50 (1983), 1245–1253.CrossRefGoogle Scholar
16.MacLachlan, C. and Reid, A. W.. Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups. Math. Proc. Camb. Phil. Soc, 102 (1987), 251–257.Google Scholar
17.Makarov, V. S.. A class of decomposition of Lobachevskii space. Soviet Math. Dokl, 161 (1965), pp. 227–278.Google Scholar
18.Mednykh, A. D.. Automorphism groups of three-dimensional hyperbolic manifolds. Soviet Math. Dokl., 32 (1985), 633–636.Google Scholar
19.Meyerhoff, R.. Appendix to: A lower bound for the volume of hyperbolic 3-orbifolds. Preprint.Google Scholar
20.Meyerhoff, R.. Sphere-packing and volume in hyperbolic 3-space. Comment. Math. Helv., 61 (1986), 271–278.Google Scholar
21.Nikulin, V. V.. On arithmetic groups generated by reflections in Lobachevskii spaces. Math. U.S.S.R. Izvest., 16 (1981), 573–601.Google Scholar
22.Pohst, M., Weiler, P. and Zassenhaus, H.. On effective computation of fundamental units II. Math. Comp., 38 (1982), 293–329.CrossRefGoogle Scholar
23.Reid, A. W.. Ph.D. Thesis (Aberdeen, 1987).Google Scholar
24.Scott, G. P.. Subgroups of surface groups are almost geometric. J. London Math. Soc. (2), 17 (1978), 555–565.CrossRefGoogle Scholar
25.Thurston, W. P.. The geometry and topology of 3-manifolds. Mimeographed lecture notes (Princeton, 1978).Google Scholar
26.Thurston, W. P.. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Prov. Sym. Pure Maths., 39 (1983), Part I, 87–111.CrossRefGoogle Scholar
27.Tits, J.. Classification of algebraic semi-simple groups. Proc. of Symposia in Pure Maths. Vol. IX (Providence, 1966), 33–62.Google Scholar
28.Vigneras, M. F.. Arithmétique des algèbres de quaternions. L.N.M. 800 (Springer-Verlag, 1980).CrossRefGoogle Scholar
29.Vinberg, E. B.. Discrete groups generated by reflections in Lobachevskii space. Math. Sb., 114 (1967), 429–444.CrossRefGoogle Scholar
30.Zagier, D.. Hyperbolic 3-manifolds and special values of Dedekind zeta-functions. Invent. Math., 83 (1986), 285–301.Google Scholar