Hopf algebras, Steinberg modules, and the unstable cohomology of $S{L}_n(\mathbb{Z})$ and $G{L}_n(\mathbb{Z})$

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 $S{L}_n(\mathbb{Z})$.