Organizing volumes of right-angled hyperbolic polyhedra

Inoue, Taiyo

doi:10.2140/agt.2008.8.1523

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Organizing volumes of right-angled hyperbolic polyhedra

Taiyo Inoue

Algebraic & Geometric Topology 8 (2008) 1523–1565

DOI: 10.2140/agt.2008.8.1523

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

Volume 8, issue 3 (2008)