On volumes of compact and non-compact right-angled hyperbolic polyhedra

There is a recent progress in study of Platonic tessellations of a hyperbolic 3-space and related hyperbolic 3-manifolds by algorithmic topology methods [1, 2]. It is known that some of hyperbolic Platonic solids are right-angled. There we are interested in a class of hyperbolic 3-manifolds which can be decomposed into right-angled hyperbolic polyhedra. Necessary and sufficient conditions for a polyhedron of a given combinatorial type to be realized as a compact right-angled polyhedron in a hyperbolic 3-space were described by Pogorelov in 1967 in the very first issue of “Matematicheskie Zametki” (Mathematical Notes) [3]. The simplest compact right-angled hyperbolic polyhedron is a dodecahedron.
The universal method to construct a hyperbolic 3-manifold from few copies of an arbitrary right-angled hyperbolic polyhedron was given in [4]. This motivates the study of the census of right-angled hyperbolic polyhedra.
Recently, Inoue [5] presented 825 smallest compact right-angled hyperbolic polyhedra. We will discuss a census of non-compact right-angled hyperbolic polyhedra. For compact and non-compact cases both we will present results of numerical computations.
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References:

• B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004), no. 1-3, 253–263.
• M. Goerner, A census of hyperbolic Platonic manifolds and augmented knotted trivalent graphs, arxiv1602.02208.
• A.V. Pogorelov, Regular decomposition of Lobachevskii space, Mat. Zametki 1, No. 1, 3–8 (1967).
• A.Yu. Vesnin, Three-dimensional hyperbolic manifolds of Loebell type, Siberian Math. J. 28, No. 5, 731–734 (1987).
• T. Inoue, The 825 smallest right-angled hyperbolic polyhedra, arxiv:1512.01761.