Pointed Hopf algebras, the Dixmier-Moeglin Equivalence and Noetherian group algebras

This paper addresses the interactions between three properties that a group algebra or more generally a pointed Hopf algebra may possess: being noetherian, having finite Gelfand-Kirillov dimension, and satisfying the Dixmier-Moeglin equivalence. First it is shown that the second and third of these properties are equivalent for group algebras $kG$ of polycyclic-by-finite groups, and are, in turn, equivalent to G being nilpotent-by-finite. In characteristic 0, this enables us to extend this equivalence to certain cocommutative Hopf algebras. In the second and third parts of the paper finiteness conditions for group algebras are studied. In the second section we examine when a group algebra satisfies the Goldie conditions, while in the final section we discuss what can be said about a minimal counterexample to the conjecture that if $kG$ is noetherian then G is polycyclic-by-finite.