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On Farrell-Tate cohomology of $$\mathrm{SL}_2$$ over $$S$$-integers. (English) Zbl 1447.20017
Summary: In this paper, we provide number-theoretic formulas for Farrell-Tate cohomology for $$\mathrm{SL}_2$$ 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 $$\mathrm{SL}_2$$ over number fields.

##### MSC:
 20G10 Cohomology theory for linear algebraic groups 11F75 Cohomology of arithmetic groups 20J05 Homological methods in group theory 20J06 Cohomology of groups
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