Hopf algebras, Steinberg modules, and the unstable cohomology of $SL_n(\mathbb Z)$ and $GL_n(\mathbb Z)$

Abstract

We prove that the direct sum of all homology groups of the integral general linear groups with Steinberg module coefficients form a commutative Hopf algebra, in particular a free graded commutative algebra. We use this to construct new infinite families of unstable cohomology classes of $SL_n(\mathbb Z)$.