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Truncated tetrahedra and their reflection groups

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

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

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