Geometry & Topology

The stable homology of congruence subgroups

Frank Calegari

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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().

Article information

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

Dates
Received: 7 November 2013
Revised: 27 December 2014
Accepted: 26 January 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
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

Subjects
Primary: 11F75: Cohomology of arithmetic groups 19F99: None of the above, but in this section
Secondary: 11F80: Galois representations

Keywords
arithmetic groups stable homology completed homology $K$–theory

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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