, Volume 322, Issue 3, pp 563-571

The uniqueness of polynomial crystallographic actions

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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 $\mathbb{R}^n$ .

Received: 30 October 2000 / Revised version: 12 July 2001 / Published online: 18 January 2002