A  Bianchi group is a discrete subgroup G of PSL_2( K ), where K is the field of complex numbers. It acts naturally on the complex hyperbolic three-space H^3_K, such that the quotient H^3_K/G is an analytic space with Gorenstein singularities.
      In the seminar I will report on a joint work with Alexander Rahm (University of Luxembourg), where we prove that the Chen-Ruan orbifold cohomology ring of the associated orbifold [H^3_K/G] is isomorphic to the cohomology ring of any crepant resolution Y of H^3_K/G. To this aim, following the classical McKay correspondence, we establish a bijection between conjugacy classes of elements of finite order of G / {1} and exceptional prime divisors of Y.
      This result, together with the computation of certain Gromov-Witten invariants of Y, confirms the validity of Ruan’s conjecture.