On Farrell-Tate cohomology of SL\_2 over S-integers

TL;DR

This paper derives explicit number-theoretic formulas for Farrell-Tate cohomology of SL_2 over rings of S-integers, enabling new insights into the homology of linear groups and related conjectures.

Contribution

It introduces novel formulas for Farrell-Tate cohomology in the context of S-integers, with applications to key conjectures in algebraic K-theory and group cohomology.

Findings

Formulas for Farrell-Tate cohomology above the virtual cohomological dimension.

Applications to Quillen conjecture detection questions.

Results on transfers related to Friedlander--Milnor conjecture.

Abstract

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.