, Volume 322, Issue 3, pp 563-571

The uniqueness of polynomial crystallographic actions

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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 \(\mathbb{R}^n\) .

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