The stable homology of congruence subgroups

Calegari, Frank

doi:10.2140/gt.2015.19.3149

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The stable homology of congruence subgroups

Frank Calegari

Geometry & Topology 19 (2015) 3149–3191

DOI: 10.2140/gt.2015.19.3149

Abstract

We relate the completed cohomology groups of ${SL}_{N} (O_{F})$ , where $O_{F}$ 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 $H_{2} (Γ_{N} (p), F_{p})$ (for sufficiently large $N$ ), where $Γ_{N} (p)$ is the full level- $p$ congruence subgroup of ${SL}_{N} (ℤ)$ .

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

Volume 19, issue 6 (2015)