The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

Abstract

We establish a dimension formula involving a number of parameters for the mod 2 cohomology of finite index subgroups in the Bianchi groups (SL\(_2\) groups over the ring of integers in an imaginary quadratic number field). The proof of our formula involves an analysis of the equivariant spectral sequence, combined with torsion subcomplex reduction. We also provide an algorithm to compute a Ford domain for congruence subgroups in the Bianchi groups from which the parameters in our formula can be explicitly computed.