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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
The stable homology of congruence subgroups

Frank Calegari

Geometry & Topology 19 (2015) 3149–3191
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().

Keywords
arithmetic groups, stable homology, completed homology, $K$–theory
Mathematical Subject Classification 2010
Primary: 11F75, 19F99
Secondary: 11F80
References
Publication
Received: 7 November 2013
Revised: 27 December 2014
Accepted: 26 January 2015
Published: 6 January 2016
Proposed: Benson Farb
Seconded: Danny Calegari, Walter Neumann
Authors
Frank Calegari
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208
USA