Bianchi orbifold - Knowlpedia

The Bianchi orbifold attached to $K=\mathbb Q(\sqrt{-d})$ , with $d>0$ square-free, is

M_d=\operatorname{PSL}_2(\mathcal O_K)\backslash\mathbb H^3,

equipped with the hyperbolic orbifold structure induced by the Bianchi group and its isometric action. The backslash denotes orbits for the left group action.

Features

It is complete, orientable, noncompact, and of finite volume. Its cusps encode the ideal class group, while its volume involves a Dedekind zeta value.

Why retain the orbifold structure?

The class of $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ has order two in every Bianchi group: its square is $-I$ . Its hyperbolic action has fixed points. Thus a Bianchi quotient has nontrivial orbifold isotropy; one cannot simply apply a free-action manifold theorem.

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