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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 sameGelfand-Kirillov dimension. As an application, we prove that theGelfand-Kirillov dimension of a connected Hopf algebra is either infinity or apositive integer. We also classify connected Hopf algebras of GK-dimensionthree over an algebraically closed field of characteristic zero.

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