Polynomial structures on polycyclic groups
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- by Karel Dekimpe and Paul Igodt PDF
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Abstract:
We know, by recent work of Benoist and of Burde & Grunewald, that there exist polycyclic–by–finite groups $G$, of rank $h$ (the examples given were in fact nilpotent), admitting no properly discontinuous affine action on $\mathbb {R}^h$. On the other hand, for such $G$, it is always possible to construct a properly discontinuous smooth action of $G$ on $\mathbb {R}^h$. Our main result is that any polycyclic–by–finite group $G$ of rank $h$ contains a subgroup of finite index acting properly discontinuously and by polynomial diffeomorphisms of bounded degree on $\mathbb {R}^h$. Moreover, these polynomial representations always appear to contain pure translations and are extendable to a smooth action of the whole group $G$.References
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Additional Information
- Karel Dekimpe
- Affiliation: Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium
- Email: Karel.Dekimpe@kulak.ac.be
- Paul Igodt
- Affiliation: Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium
- Email: Paul.Igodt@kulak.ac.be
- Received by editor(s): January 2, 1996
- Additional Notes: The first author is Postdoctoral Fellow of the Fund for Scientific Research-Flanders (F.W.O.)
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 3597-3610
- MSC (1991): Primary 57S30, 20F34, 20F38
- DOI: https://doi.org/10.1090/S0002-9947-97-01924-7
- MathSciNet review: 1422895