The covolume of discrete subgroups of Iso

Let X 2 m be one of the spaces of constant curvature and Γ be a discrete and finitely presented subgroup of Iso ( X 2 m ) . We combine Schläfli’s reduction formula and Poincaré’s formula with the cycle condition to get a formula for the covolume of Γ which only depends on the combinatorics of a fundamental polytope for Γ and the orders of certain stabilizer subgroups of Γ .