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

We show that the 1-cusped quotient of hyperbolic space H 3 \mathbb {H}^3 by tetrahedral Coxeter group alttext="normal Gamma Subscript asterisk Baseline equals left-bracket 5 comma 3 6 right-bracket"> mathvariant="normal">Γ ∗ = [ 5 , 6 stretchy="false">] encoding="application/x-tex">\Gamma _*=[5,3,6] has minimal volume among all non-arithmetic cusped 3-orbifolds, and as such it is uniquely determined. Furthermore, lattice asterisk"> _* incommensurable to any Gromov-Piatetski-Shapiro type lattice. Our methods have their origin in work C. Adams [J. Differential Geom. 34 (1991), pp. 115–141; Noncompact 3-orbifolds small volume, de Gruyter, Berlin, 1992]. extend considerably this approach via geometry underlying horoball configuration induced a cusp.