Sphere-packing and volume in hyperbolic 3-space
By a theorem of Jørgensen and Thurston [\textit{W. Thurston}, The geometry and topology of 3-manifolds, Princeton Univ., preprint (1978)], the set of volumes of complete hyperbolic 3-manifolds is well-ordered and of order type \(\omega^{\omega}\). In particular, there is a complete hyperbolic 3-manifold of minimum volume \(V_ 1\) among all complete hyperbolic 3-manifolds, and a cusped hyperbolic 3-manifold of minimum volume \(V_{\omega}\). Similar results hold for complete hyperbolic 3- orbifolds, implying the existence of a hyperbolic 3-orbifold of minimum volume \(V_ 1'\), and a cusped hyperbolic 3-orbifold of minimum volume \(V_ c'.\) The paper discusses inequalities for \(V_ 1\), \(V_{\omega}\), \(V_ 1'\) and \(V_ c'\); these are based on sphere packing arguments in hyperbolic 3-space \(H^ 3\). In particular it is shown that the orbifold \(Q_ 1=H^ 3/PGL_ 2({\mathcal O}_ 3)\) has minimum volume \(V_ c'=V/12\) among all orientable cusped hyperbolic 3-orbifolds; here, \({\mathcal O}_ 3\) is the ring of integers in \({\mathbb{Q}}(\sqrt{-3})\), and V is the volume of the ideal regular tetrahedron in \(H^ 3\).
cusped hyperbolic 3-manifold of minimum volume
cusped hyperbolic 3-orbifold of minimum volume
scientific article; zbMATH DE number 3988273
scientific article