The mod 2 cohomology rings of SL\_2 of the imaginary quadratic integers

TL;DR

This paper computes the mod 2 cohomology rings of SL_2 over imaginary quadratic integers by developing dimension formulas for spectral sequences and extending torsion reduction techniques, providing new tools for understanding these algebraic structures.

Contribution

It introduces generalized dimension formulas for the equivariant spectral sequence and extends torsion subcomplex reduction to cases with nontrivial kernels, advancing cohomology computations for these groups.

Findings

Derived explicit second page dimension formulas for the spectral sequence.

Extended torsion reduction techniques to nontrivial kernel cases.

Demonstrated how to compute mod 2 cohomology rings from spectral sequence data.

Abstract

We establish 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. On the way, 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.