Normalized volume spectra of right-angled hyperbolic polyhedra

TL;DR

This paper investigates the set of normalized volumes of right-angled hyperbolic polyhedra, establishing bounds, discreteness, and density properties of their spectra for both compact and ideal cases.

Contribution

It provides the first detailed analysis of the spectra of normalized volumes for right-angled hyperbolic polyhedra, including sharp bounds and the nature of their distribution.

Findings

Spectrum of ideal polyhedra lies within [v_oct/6, v_oct/2], with bounds sharp.

Spectrum is discrete below v_oct/4 and dense above v_oct/4 for ideal polyhedra.

Spectrum of compact polyhedra is within [5v_oct/192, 5v_tet/8], with specific density and discreteness intervals.

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 $\omega(\mathcal P) = \mathrm{vol}(\mathcal P)/ \mathrm{ver}(\mathcal P)$. If $\mathcal{R}$ is some set of hyperbolic polyhedra, then we assign to it the set of normalized volumes $\Omega (\mathcal R) = \{ \omega(\mathcal P) \mid \mathcal P \in \mathcal R \}$, 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 $\Omega (\mathcal R_{ideal})$ 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 $\Omega (\mathcal R_{comp})$ 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.