On the minimal volume hyperbolic 3-orbifold
New results concerning the minimal volume 3-dimensional hyperbolic orbifold are announced. A hyperbolic 3-orbifold \(\Omega\) is the orbit space of a Kleinian group \(\Gamma\). That is \(\Omega= \mathbb{H}/\Gamma\), where \(\Gamma\) is a discrete nonelementary orientation preserving group of isometries of hyperbolic 3-space \(\mathbb{H}^3\). The first main result of the paper is the following. The unique Kleinian group of minimal \(\infty\)-volume which contains a torsion element of order \(p> 4\) is a \(\mathbb{Z}_2\)-extension \(\Gamma_{353}\) of the orientation preserving index 2 subgroup of the group generated by reflections in the faces of the hyperbolic tetrahedron with Coxeter diagram 3-5-3. In particular, \(\Gamma_{353}\) is arithmetic and has co-volume \[ \text{Vol}(\mathbb{H}^3/\Gamma_{353})= 0.039050\dots. \] Recall that the finite subgroup of a Kleinian group are either cyclic, dihedral or one of the symmetry groups of a regular polyhedron, that is the tetrahedral, octahedral or icosahedral groups. It was shown by \textit{D. Derevnin} and the reviewer [Sov. Math., Dokl. 37, No. 3, 614-617 (1988; Zbl 0713.30044)] that the minimal distance between fixed points of icosahedral subgroups of a discrete group is not less then \(\text{Arccosh}({2+ \sqrt 5\over 2})= 1.3825\dots\) and the extremal case is realized for the group \(\Gamma_{353}\). The second main result states that the unique Kleinian group of minimal co-volume which contains a tetrahedral, octahedral or icosahedral subgroup is the group \(\Gamma_{353}\).
scientific article; zbMATH DE number 794218