Spectra of normalized volumes of right-angled hyperbolic polyhedra

Abstract

We consider three-dimensional hyperbolic polyhedra of finite volume with finitely many vertices. The normalized volume of a polyhedron is the ratio of its volume to the number of vertices. Given some set of hyperbolic polyhedra, we can associate with it the set of normalized volumes of the polyhedra belonging to it. We call this set the spectrum of normalized volumes of the set under consideration. We focus on right-angled hyperbolic polyhedra. For the subset of ideal polyhedra, we find bounds for the spectrum of normalized volumes and prove that they are sharp. Moreover, we show that the spectrum splits into discrete and dense parts. For the subset of compact polyhedra, we obtain estimates for the spectrum of normalized volumes, prove that the upper bound is sharp, and also establish numerical intervals on which the spectrum is discrete and dense. The endpoints of the indicated numerical intervals are expressed in terms of the volume of the regular ideal hyperbolic tetrahedron and the volume of the regular ideal hyperbolic octahedron.