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

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

2011-04-18

arxiv.org/abs/1104.3437v1

Math. Notes 89:1 (2011), 31-36

Mathematics

Geometric Topology

Scientific paper

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 $\operatorname{vert} (R) \leqslant 56$.

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