Stable cohomology of congruence subgroups

TL;DR

This paper computes the mod p cohomology of congruence subgroups of SL_n for large n, confirming a proposed formula, and also determines the stable cohomology of SL_n over finite fields with twisted coefficients.

Contribution

It provides explicit formulas for the cohomology of congruence subgroups and stable cohomology with twisted coefficients, advancing understanding in algebraic topology and number theory.

Findings

Cohomology of $SL_n(Z,p^m)$ described for degrees less than p-1

Established a formula for stable cohomology of $SL_n(Z/p)$ with twisted coefficients

Confirmed a conjectured formula by F. Calegari

Abstract

We describe the $\mathbb{F}_p$-cohomology of the congruence subgroups $SL_n(\mathbb{Z}, p^m)$ in degrees $* < p-1$, 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 $SL_n(\mathbb{Z}/p)$ with certain twisted coefficients.