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

Abstract

We show that the 1-cusped quotient of the hyperbolic space H 3 \mathbb {H}^3 by the tetrahedral Coxeter group Γ ∗ = [ 5 , 3 , 6 ] \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 Γ ∗ \Gamma _* is incommensurable to any Gromov-Piatetski-Shapiro type lattice. Our methods have their origin in the work of C. Adams [J. Differential Geom. 34 (1991), pp. 115–141; Noncompact hyperbolic 3-orbifolds of small volume, de Gruyter, Berlin, 1992]. We extend considerably this approach via the geometry of the underlying horoball configuration induced by a cusp.