Abstract: Let $k$ be a field, $R$ a $k$-algebra and $A=R[\theta_1, \theta_2,\ldots,\theta_n]$ a Poincaré-Birkhoff-Witt extension of $R$. If each $\theta_i$ acts locally finitely on $R$, then we show that $GKdim(A)=GKdim(R) +n$. From this we deduce some results concerning incomparability, prime length and Tauvel's height formula in the crossed product $R\star g$ where $g$ is a finite-dimensional Lie algebra acting as derivations on $R$. Similar results are obtained for $R\star G$, where $G$ is a free abelian group of finite rank. As a corollary of the results for $G$, Tauvel's height formula is established in $R\otimes _kP(\lambda)$ where $P(\lambda)$ is the quantum torus.
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