Two-sided bounds for the volume of right-angled hyperbolic polyhedra

Abstract

For a compact right-angled polyhedron $R$ in $\mathbb H^3$ denote by $\operatorname{vol} (R)$ the volume and by $\operatorname{vert} (R)$ the number of vertices. Upper and lower bounds for $\operatorname{vol} (R)$ in terms of $\operatorname{vert} (R)$ were obtained in \cite{A09}. Constructing a 2-parameter family of polyhedra, we show that the asymptotic upper bound $5 v_3 / 8$, where $v_3$ is the volume of the ideal regular tetrahedron in $\mathbb H^3$, is a double limit point for ratios $\operatorname{vol} (R) / \operatorname{vert} (R)$. Moreover, we improve the lower bound in the case $\operatorname{vert} (R) \leqslant 56$.