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

Abstract

We show that the 1-cusped quotient of the hyperbolic space ${\mathbb{H}}^3$ by the tetrahedral Coxeter group ${\Gamma }_{\ast }=[5,3,6]$ has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined. Furthermore, the lattice ${\Gamma }_{\ast }$ 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.

50 pages, 31 figures, submitted for publication