A refinement of a conjecture of Quillen

Abstract: We present some new results on the cohomology of a large scope of SL_2-groups in degrees above the virtual cohomological dimension; yielding some partial positive results for the Quillen conjecture in rank one. We combine these results with the known partial positive results and the known types of counterexamples to the Quillen conjecture, in order to formulate a refined variant of the conjecture.