On Farrell–Tate cohomology of SL2 over S-integers

In this paper, we provide number-theoretic formulas for Farrell–Tate cohomology for SL 2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups. We expose three applications, to (I) detection questions for the Quillen conjecture, (II) the existence of transfers for the Friedlander–Milnor conjecture, (III) cohomology of SL 2 over number fields.