Bianchi group - Knowlpedia

For $K=\mathbb Q(\sqrt{-d})$ , $d>0$ square-free, the Bianchi group is

\Gamma_d=\operatorname{PSL}_2(\mathcal O_K) =\operatorname{SL}_2(\mathcal O_K)/\{\pm I\},

where $\mathcal O_K$ is the ring of integers of the imaginary quadratic field. We use the ring-valued projective convention and the inclusion $K\subseteq\mathbb C$ to regard $\Gamma_d$ as a subgroup of $\operatorname{PSL}_2(\mathbb C)$ .

Concrete matrices

Its elements are determinant-one matrices $\begin{pmatrix}a&b\\c&e\end{pmatrix}$ with entries in $\mathcal O_K$ , where a matrix and its negative represent the same element. On the boundary sphere the action is $z\mapsto(az+b)/(cz+e)$ .

For $d=1$ , the entries are Gaussian integers $a+bi$ , $a,b\in\mathbb Z$ . The matrix $\begin{pmatrix}1&i\\0&1\end{pmatrix}$ acts on the boundary by $z\mapsto z+i$ .

Geometry and conventions

The hyperbolic action gives the Bianchi orbifold. Some sources call the unprojectivized SL group a Bianchi group; its central kernel acts trivially on hyperbolic space. Replacing PSL by PGL can change the effective group and the quotient volume.

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