Why is a discrete $G$-module the same as a comodule over the coalgebra $\mathbb{Z}G$ ?

@s.barmeier: In lemma 5.2 in the paper of Kobayashi (linked in Ralph's answer), the matrix group is required to be finitely generated in order to be virtually torsionless. In your comment every discrete subgroup of a matrix group is virtually torsionless. Are these two conditions the same ? (I know they are the same if the quotient by $G$ is compact).