The mod 2 cohomology rings of SL_2 of the imaginary quadratic integers

Ethan Berkove (1) , Alexander D. Rahm (2)

1 Mathematics Department, Lafayette College
2 NUIG - Department of Mathematics

When a group acts cellularly on a space, the torsion subcomplex contains all of the homological information at a given prime number. The recently introduced technique of torsion subcomplex reduction has been successfully applied under certain hypotheses to determine the Farrell cohomology of discrete groups. In this paper, we extend the torsion subcomplex reduction technique to the case where the kernel of the group action is nontrivial. This is accomplished by algebraically excising the torsion subcomplex from the equivariant spectral sequence. We demonstrate the power of this "découpage in equivariant cohomology" by establishing general dimension formulae for the second page of the equivariant spectral sequence of the action of the SL_2 groups over imaginary quadratic integers on their associated symmetric space. Then, we show how to obtain the mod 2 cohomology rings of our groups from this information.

Mots clés: Cohomology of arithmetic groups

Domaines: K-théorie et homologie [math.KT]

Ethan Berkove, Alexander D. Rahm. The mod 2 cohomology rings of SL_2 of the imaginary quadratic integers. 2013. ⟨hal-00769261v2⟩