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Cusps, triangle groups and hyperbolic 3-folds

Part of: Other groups of matrices Riemann surfaces Topological manifolds

Published online by Cambridge University Press:  09 April 2009

Marston D. E. Conder  and
Gaven J. Martin
Marston D. E. Conder
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019 Auckland, NEW ZEALAND, email: conder@mat.aukuni.ac.nz
Gaven J. Martin
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019 Auckland, NEW ZEALAND, email: martin@mat.aukuni.ac.nz
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Abstract

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We provide a number of explicit examples of small volume hyperbolic 3-manifolds and 3-orbifolds with various geometric properties. These include a sequence of orbifolds with torsion of order q interpolating between the smallest volume cusped orbifold (q = 6) and the smallest volume limit orbifold (q → ∞), hyperbolic 3-manifolds with automorphism groups with large orders in relation to volume and in arithmetic progression, and the smallest volume hyperbolic manifolds with totally geodesic surfaces. In each case we provide a presentation for the associated Kleinian group and exhibit a fundamental domain and an integral formula for the co-volume. We discuss other interesting properties of these groups.

MSC classification

Secondary: 20H10: Fuchsian groups and their generalizations 30F40: Kleinian groups 57N10: Topology of general $3$-manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 2 , October 1993 , pp. 149 - 182
Copyright
Copyright © Australian Mathematical Society 1993

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