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. 51M10, 57M50; 52B99
An algorithm for determining the list of smallest volume rightangled hyperbolic polyhedra in dimension 3 is described. This algorithm has been implemented on computer using the program Orb to compute volumes, and the first 825 polyhedra in the list have been determined.
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