The stable homology of congruence subgroups

Abstract

We relate the completed cohomology groups of ${SL}_N\left({\mathcal{O}}_F\right)$, where ${\mathcal{O}}_F$ 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 $H_2\left({\Gamma }_N\left(p\right),{\mathbb{F}}_p\right)$ (for sufficiently large $N$), where ${\Gamma }_N\left(p\right)$ is the full level-$p$ congruence subgroup of ${SL}_N\left(\mathbb{Z}\right)$.