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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.
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