We relate the completed cohomology groups of SLN(O-F), where O-F is the ring of integers of a number field, to K-theory and Galois cohomology. Various consequences include showing that Borel's stable classes become infinitely p-divisible up the p-congruence tower if and only if a certain p-adic zeta value is nonzero. We use our results to compute H-2(Gamma(N)(p),F-p) (for sufficiently large N), where Gamma(N)(p) is the full level-p congruence subgroup of SLN(Z).
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