The covolume of discrete subgroups of Iso(H2m)(H2m)

Let X2mX2m be one of the spaces of constant curvature and Γ be a discrete and finitely presented subgroup of Iso(X2m)(X2m). 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.