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Truncated tetrahedra and their reflection groups

Part of: Special aspects of infinite or finite groups Topological manifolds Riemann surfaces

Published online by Cambridge University Press:  09 April 2009

T. H. Marshall
T. H. Marshall
Affiliation:
Department of Mathematics University of Auckland Private Bag 92019 Auckland New Zealand
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Abstract

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We outline the classification, up to isometry, of all tetrahedra in hyperbolic space with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncations are all π/2, and those remaining are all submultiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups.

For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary. In particular, for each g ≥ 2, we find a sequence of hyperbolic manifolds with totally geodesic boundary of genus g, which we conjecture to be of least volume among such manifolds.

Keywords

Coxeter polytopetruncated tetrahedronco-volumetorsion-free subgrouphyperbolic manifold.

MSC classification

Secondary: 30F40: Kleinian groups 57N10: Topology of general $3$-manifolds 20F55: Reflection and Coxeter groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 1 , February 1998 , pp. 54 - 72
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Andreev, E. M., ‘On convex polyhedra in Lobačevskiľ spaces’, Math USSR-Sb. 10 (1970), 413–440.CrossRefGoogle Scholar
[2]Beardon, A. F., The geometry of discrete groups (Springer, Berlin, 1983).CrossRefGoogle Scholar
[3]Conder, M. D. E. and Martin, G. J., ‘Cusps, triangle groups and hyperbolic 3-folds’, J. Austral. Math. Soc. (Series A) 55 (1993), 149–182.CrossRefGoogle Scholar
[4]Coxeter, H. S. M., ‘Discrete groups generated by reflections’, Ann. of Math. 2 (1934), 588–621.CrossRefGoogle Scholar
[5]Fujii, M., ‘Hyperbolic 3-manifolds with totally geodesic boundary which are decomposed into hyperbolic truncated tetrahedra’, Tokyo J. Math. 13 (1990), 353–373.CrossRefGoogle Scholar
[6]Kojima, S. and Miyamoto, Y., ‘The smallest hyperbolic 3-manifolds with totally geodesic boundary’, J. Differential Geom. 34 (1991), 175–192.CrossRefGoogle Scholar
[7]Lanner, F., ‘On complexes with transitive groups of automophisms’, Comm. Sém. Math. Univ. Lund. 11 (1950), 41.Google Scholar
[8]Marshall, T. H., PhD Thesis (University of Auckland, 1994).Google Scholar
[9]Marshall, T. H., ‘Geometry of pseudospheres II’, New Zealand J. Math. to appear.Google Scholar
[10]Maskit, B., Kleinian groups (Springer, Berlin, 1987).CrossRefGoogle Scholar
[11]Milnor, J., ‘Hyperbolic geometry: the first 150 years’, Bull. Amer Math. Soc. 6 (1982), 9–24.CrossRefGoogle Scholar
[12]Selberg, A., On discontinuous groups in higher-dimensional spaces (Tata Institute, Bombay, 1960).Google Scholar
[13]Singerman, D., ‘Subgroups of Fuchsian groups and finite permutation groups’, Bull. London Math. Soc. 2 (1970), 319–323.CrossRefGoogle Scholar
[14]Vinberg, E. B. (ed.), Geometry II, Encyclopaedia Math. Sci. 29 (Springer, Berlin, 1993).CrossRefGoogle Scholar