The lowest volume 3-orbifolds with high torsion

TL;DR

This paper identifies the hyperbolic 3-orbifold with the smallest volume for each torsion bound n, revealing a unique minimal structure with implications for volume bounds in hyperbolic manifolds.

Contribution

It determines the unique lowest volume hyperbolic 3-orbifold with a given torsion lower bound, linking geometric properties to symmetry group element orders.

Findings

Identifies the minimal volume orbifold for each n ≥ 4

Shows the minimal orbifold has base space S^3 and figure-8 knot singularity

Provides sharp volume bounds based on symmetry group elements

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.