Organizing volumes of right-angled hyperbolic polyhedra

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. 51M10, 57M50; 52B99

The smallest Haken hyperbolic polyhedra

“…If P contained no prismatic 4-circuits, then P ⊥ admits a structure as a compact right-angled hyperbolic polyhedron such that Vol(P ⊥ ) ≤ Vol(P). The smallest volume compact, right-angled polyhedron is the right-angled dodecahedron [14]. But the volume of this polyhedron is 4.306207..., which is considerably larger than Vol(C).…”

“…The idea behind the proof of Theorem 1.1 is to use Proposition 3.6 in conjunction with techniques established by Atkinson [5,6] and Inoue [14] to restrict the possible combinatorial types of polyhedra with small volumes. All polyhedral volumes in the proof are calculated using known formulae and the computational software Orb, developed by Heard [13].…”

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

“…The following result was obtained by Inoue in [38] (see also the survey paper [63]): Inoue's theorem was strengthened in [12]. We denote by P B the set of r-barrels Q r with r 5 and consider the class P ⊥ B = P \ P B .…”

Virtually Fibering Right-Angled Coxeter Groups

“…We refer to graphs satisfying the conditions in Theorem 5.7 as Pogorelov graphs. Inoue gave a recursive construction of all (duals of) Pogorelov graphs [Ino08]. Expressing his result in terms of duals, the base case consists of the dual Löbell graphs.…”