Organizing Volumes of Right-Angled Hyperbolic Polyhedra

TL;DR

This paper introduces combinatorial operations called decomposition and edge surgery on right-angled hyperbolic polyhedra, simplifying their structure while decreasing volume, enabling the organization and identification of minimal volume polyhedra.

Contribution

It defines new combinatorial operations with geometric volume-decreasing properties, facilitating the classification of right-angled hyperbolic polyhedra by volume.

Findings

Operations simplify polyhedra to Lobell polyhedra

Volume decreases under these operations

Identifies polyhedra with smallest volumes

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 L\"obell 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.