For a square-free positive integer \(m\) let \(\mathcal O_{-m}\) be the ring of integers in the imaginary quadratic number field \(\mathbb Q(\sqrt{-m})\) and let \(\Gamma=\text{PSL}_2(\mathcal O_{-m})\) be the corresponding Bianchi group. \(\Gamma\) acts naturally on the real hyperbolic three-space \(\mathcal H^3_{\mathbb R}\) and also its complexification \(\mathcal H^3_{\mathbb C}\) and so one defines the orbifolds \([\mathcal H^3_{\mathbb R}/\Gamma]\) and \([\mathcal H^3_{\mathbb C}/\Gamma]\). The latter is called a complexified Bianchi orbifold.
The paper under review computes the Chen-Ruan orbifold cohomology of the complexified Bianchi orbifolds. In the case that \(\pm 1\) are the only units in \(\mathcal O_{-m}\) this result can be summarized as \[H^d_{\text{CR}}([\mathcal H^3_{\mathbb C}/\Gamma])=H^d(\mathcal H^3_{\mathbb R}/\Gamma;\mathbb Q) \oplus \begin{cases}\mathbb Q^{\lambda_4+2\lambda_6-\lambda_6^*} & d=2\\ \mathbb Q^{\lambda_4-\lambda^*_4+2\lambda_6-\lambda_6^*} & d=3\\ 0 & \text{otherwise,} \end{cases}\] where \(\lambda_{2n}\) is the number of conjugacy classes of cyclic subgroups of order \(n\) in \(\Gamma\) and \(\lambda^*_{2n}\) is the number of conjugacy classes of those of them which are contained in a dihedral subgroup of order \(2n\) in \(\Gamma\). The cohomology of the quotient space \(\mathcal H^3_{\mathbb R}/\Gamma\) has been computed numerically elsewhere for a large scope of Bianchi groups.
The paper under review also finds the Chen-Ruan cup product and proves the cohomological crepant resolution conjecture: if \(Y\to \mathcal H^3_{\mathbb C}/\Gamma\) is a crepant resolution then there is an isomorphism as graded \(\mathbb Q\)-algebras between the Chen-Ruan cohomology ring of \([\mathcal H^3_{\mathbb C}/\Gamma]\) and the the singular cohomology ring of \(Y\).