Stable cohomology of congruence subgroups

Abstract

We describe the ${\mathbb{F}}_p$-cohomology of the congruence subgroups $S{L}_n(\mathbb{Z},p^m)$ in degrees $\ast <p-1$, for all large enough $n$, establishing a formula proposed by F. Calegari. Along the way, we also establish a formula for the stable cohomology of $S{L}_n(\mathbb{Z}/p)$ with certain twisted coefficients.

arXiv comment40 pages; v2 accepted version, to appear in Compositio Mathematica