Volume formula for a Bianchi orbifold

Statement

Let $K$ be an imaginary quadratic field with discriminant $D_K$ . For the Bianchi orbifold with curvature $-1$ ,

\operatorname{vol}\bigl(\operatorname{PSL}_2(\mathcal O_K)\backslash\mathbb H^3\bigr) =\frac{|D_K|^{3/2}}{4\pi^2}\,\zeta_K(2),

where $\zeta_K$ is the Dedekind zeta function.

Normalization checks

The formula uses the full ring of integers, the projective SL group, and the curvature- $-1$ metric. For $K=\mathbb Q(i)$ , $|D_K|=4$ , so the coefficient becomes $2/\pi^2$ .

Passing to a subgroup of index $m$ multiplies the quotient volume by $m$ . Replacing PSL by a larger projective group therefore requires a separate index calculation; the formula cannot be transferred unchanged.

Interpretation

The finite positive zeta value yields finite positive hyperbolic volume even though the quotient has cusp ends and is noncompact.

References

F. Paulin, Regards croisés sur les séries de Poincaré et leurs applications, Theorem 6, p. 12. Author’s text