Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension

Let H be a pointed Hopf algebra. We show that under some mild assumptions H and its associated graded Hopf algebra gr H have the same Gelfand-Kirillov dimension (GK-dimension). As an application, we prove that the GK-dimension of a connected Hopf algebra is either infinity or a positive integer. We also classify connected Hopf algebras of GK-dimension 3 over an algebraically closed field of characteristic 0.