The mod 2 cohomology rings of SL2SL2 of the imaginary quadratic integers

We establish general dimension formulae for the second page of the equivariant spectral sequence of the action of the SL2SL2 groups over imaginary quadratic integers on their associated symmetric space. By way of doing this, we extend the torsion subcomplex reduction technique to cases where the kernel of the group action is nontrivial. Using the equivariant and Lyndon–Hochschild–Serre spectral sequences, we investigate the second page differentials and show how to obtain the mod 2 cohomology rings of our groups from this information.