## Geometry & Topology

### The stable homology of congruence subgroups

Frank Calegari

#### Abstract

We relate the completed cohomology groups of $SLN(OF)$, where $OF$ 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 $H2(ΓN(p), Fp)$ (for sufficiently large $N$), where $ΓN(p)$ is the full level-$p$ congruence subgroup of $SLN(ℤ)$.

#### Article information

Source
Geom. Topol., Volume 19, Number 6 (2015), 3149-3191.

Dates
Revised: 27 December 2014
Accepted: 26 January 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858872

Digital Object Identifier
doi:10.2140/gt.2015.19.3149

Mathematical Reviews number (MathSciNet)
MR3447101

Zentralblatt MATH identifier
1336.11045

#### Citation

Calegari, Frank. The stable homology of congruence subgroups. Geom. Topol. 19 (2015), no. 6, 3149--3191. doi:10.2140/gt.2015.19.3149. https://projecteuclid.org/euclid.gt/1510858872

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