The uniqueness of polynomial crystallographic actions

Abstract.

Let $\Gamma$ be a polycyclic-by-finite group. It is proved in [8] that $\Gamma$ admits a polynomial action of bounded degree on $\mathbb{R}^n$ which is properly discontinuous and such that the quotient $\Gamma\backslash \mathbb{R}^n$ is compact. We prove here that such an action is unique up to conjugation by a polynomial transformation of $\mathbb{R}^n$.