Hopf algebras, Steinberg modules, and the unstable cohomology of SLₙ (Z) and GLₙ (Z)

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 SLₙ (Z).