Normalized volume spectra of right-angled hyperbolic polyhedra

Abstract

Let a three-dimensional hyperbolic polyhedron $\mathcal P$ have finite volume $\mathrm{vol}(\mathcal P)$ and a finite number of vertices $\mathrm{ver}(\mathcal P)$. We call its normalized volume the quantity $ω(\mathcal P) = \mathrm{vol}(\mathcal P)/ \mathrm{ver}(\mathcal P)$. If $\mathcal{R}$ is some set of hyperbolic polyhedra, then we assign to it the set of normalized volumes $Ω(\mathcal R) = \{ ω(\mathcal P) \mid \mathcal P \in \mathcal R \}$, which we call the spectrum of normalized volumes of the set $\mathcal R$. In the paper we consider the set $\mathcal R_{comp}$ of compact right-angled hyperbolic polyhedra and the set $\mathcal R_{ideal}$ of ideal right-angled hyperbolic polyhedra. We prove that the spectrum $Ω(\mathcal R_{ideal})$ belongs to the interval $\left[\frac{1}{6} v_{\mathrm{oct}}, \frac{1}{2} v_{\mathrm{oct}} \right]$ and both bounds are sharp. Moreover, the spectrum is discrete in $\left[ \frac{1}{6} v_{\mathrm{oct}}, \frac{1}{4} v_{\mathrm{oct}} \right)$ and everywhere dense in $\left[ \frac{1}{4} v_{\mathrm{oct}}, \frac{1}{2} v_{\mathrm{oct}} \right]$, where $v_{\mathrm{oct}}$ is the volume of the regular ideal hyperbolic octahedron. We also establish that the spectrum $Ω(\mathcal R_{comp})$ belongs to the interval $\left[ \frac{5}{192} v_{\mathrm{oct}}, \frac{5}{8} v_{\mathrm{tet}} \right]$ and the upper bound is sharp. Moreover, on the interval $\left[ \frac{5}{192} v_{\mathrm{oct}}, \frac{1}{32} v_{\mathrm{oct}} \right)$ the spectrum is discrete, while on the interval $\left[ \frac{5}{16} v_{\mathrm{tet}}, \frac{5}{8} v_{\mathrm{tet}} \right]$ it is everywhere dense, where $v_{\mathrm{tet}}$ is the volume of the regular ideal hyperbolic tetrahedron.