#### Volume 19, issue 6 (2015)

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The stable homology of congruence subgroups

### Frank Calegari

Geometry & Topology 19 (2015) 3149–3191
##### Abstract

We relate the completed cohomology groups of ${SL}_{N}\left({\mathsc{O}}_{F}\right)$, where ${\mathsc{O}}_{F}$ 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 ${SL}_{N}\left(ℤ\right)$.

##### Keywords
arithmetic groups, stable homology, completed homology, $K$–theory
##### Mathematical Subject Classification 2010
Primary: 11F75, 19F99
Secondary: 11F80