Bianchi groups are nonuniform lattices

Statement

Every Bianchi group $\Gamma_d=\operatorname{PSL}_2(\mathcal O_{\mathbb Q(\sqrt{-d})})$ is a nonuniform lattice in $\operatorname{PSL}_2(\mathbb C)$ . Thus it is discrete, and its hyperbolic quotient is noncompact with finite volume.

Discreteness directly

In the chosen complex embedding, $\mathcal O_K$ is generated over $\mathbb Z$ by two real-linearly independent complex numbers. It is therefore discrete in $\mathbb C$ . Matrices with these entries form a discrete subset of $M_2(\mathbb C)$ ; passing to the finite central quotient preserves discreteness.

What needs more than discreteness

Finite covolume requires reduction theory; it does not follow merely from integral entries. Noncompactness is witnessed by cusps, including the end associated to the boundary point $\infty$ . The volume formula and cusp correspondence give more precise statements.

References

T. Church, B. Farb, and A. Putman, Integrality in the Steinberg module and the top-dimensional cohomology of SL_n O_K, Example 5.6, p. 31 of the linked version. Author-hosted paper
F. Paulin, Regards croisés sur les séries de Poincaré et leurs applications, Theorem 6, p. 12. Author’s text