On Farrell-Tate cohomology of SL\_2 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.