  In a previous paper, the author (together with Matthew Emerton) proved that
the completed cohomology groups of SL_N(Z) are stable in fixed degree as N goes
to infinity (Z may be replaced by the ring O_F of integers of any number
field). In this paper, we relate these completed cohomology groups 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 non-zero. 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 SL_N(Z).
