Volume 8, issue 3 (2008)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Organizing volumes of right-angled hyperbolic polyhedra

Taiyo Inoue

Algebraic & Geometric Topology 8 (2008) 1523–1565
Abstract

This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Löbell polyhedra, a class of polyhedra which generalizes the dodecahedron. Furthermore, these combinatorial operations are shown to have geometric realizations which are volume decreasing. This allows for an organization of the volumes of right-angled hyperbolic polyhedra and allows, in particular, the determination of the polyhedra with smallest and second smallest volumes.

Keywords
hyperbolic, geometry, right-angled, polyhedra
Mathematical Subject Classification 2000
Primary: 51M10, 57M50
Secondary: 52B99
References
Publication
Received: 15 August 2007
Revised: 12 March 2008
Accepted: 9 July 2008
Published: 15 September 2008
Authors
Taiyo Inoue
Department of Mathematics
University of California
Berkeley, CA 94720
USA
http://www.math.berkeley.edu/~inoue