The Stable Homology of Congruence Subgroups
Algebraic Topology
2015-02-03 v2 Geometric Topology
Number Theory
Abstract
In a previous paper, the author (together with Matthew Emerton) proved that the completed cohomology groups of SL_N(Z) are stable in fixed degree as N goes to infinity (Z may be replaced by the ring O_F of integers of any number field). In this paper, we relate these completed cohomology groups 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 non-zero. We use our results to compute H_2(Gamma_N(p),F_p) (for sufficiently large N) where Gamma_N(p) is the full level-p congruence subgroup of SL_N(Z).
Keywords
Cite
@article{arxiv.1311.5190,
title = {The Stable Homology of Congruence Subgroups},
author = {Frank Calegari},
journal= {arXiv preprint arXiv:1311.5190},
year = {2015}
}
Comments
To appear in Geometry & Topology; updated to incorporate comments of the referee