The non-arithmetic cusped hyperbolic 3-orbifold of minimal volume

TL;DR

This paper proves that a specific 1-cusped hyperbolic 3-orbifold has the smallest volume among all non-arithmetic cusped hyperbolic 3-orbifolds, establishing its uniqueness and incommensurability with certain lattices.

Contribution

It identifies the minimal volume non-arithmetic cusped hyperbolic 3-orbifold and extends geometric methods based on horoball configurations for such classifications.

Findings

The 1-cusped quotient by the tetrahedral Coxeter group has minimal volume.

This orbifold is uniquely determined among non-arithmetic cusped hyperbolic 3-orbifolds.

The lattice is incommensurable with Gromov-Piatetski-Shapiro type lattices.

Abstract

We show that the 1-cusped quotient of the hyperbolic space $\mathbb{H}^3$ by the tetrahedral Coxeter group $\Gamma_*=[5,3,6]$ has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined. Furthermore, the lattice $\Gamma_*$ is incommensurable to any Gromov-Piatetski-Shapiro type lattice. Our methods have their origin in the work of C. Adams. We extend considerably this approach via the geometry of the underlying horoball configuration induced by a cusp.