Verification of the Quillen conjecture in the rank 2 imaginary quadratic case

Abstract : We confirm a conjecture of Quillen in the case of the mod 2 coho-mology of arithmetic groups SL2(O Q(√ −m) [ 1 2 ]), where O Q(√ −m) is an imaginary quadratic ring of integers. To make explicit the free module structure on the cohomology ring conjectured by Quillen, we compute the mod 2 cohomology of SL2(Z[ √ −2 ][ 1 2 ]) via the amalgamated decomposition of the latter group.