The lowest volume 3–orbifolds with high torsion

Abstract

Abstract. For each natural number n ≥ 4, we determine the unique lowest volume hyperbolic 3-orbifold whose torsion orders are bounded below by n. This lowest volume orbifold has base space the 3-sphere and singular locus the figure-8 knot, marked n. We apply this result to give sharp lower bounds on the volume of a hyperbolic manifold in terms of the order of elements in its symmetry group.

On Gehring–martin–tan Groups With an Elliptic Generator

“…Denote by F (n) the orbifold with the underlying space S 3 and singular set the figure-eight knot F with singularity index n, n ≥ 4. According to [1], the orbifolds F (n) are extreme in the following sense: let L n denote the set of all orientable hyperbolic 3-orbifolds with nonempty singular set and with all torsion orders bounded below by n. Therefore,…”