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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Polynomial structures on polycyclic groups


Authors: Karel Dekimpe and Paul Igodt
Journal: Trans. Amer. Math. Soc. 349 (1997), 3597-3610
MSC (1991): Primary 57S30, 20F34, 20F38
MathSciNet review: 1422895
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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 $G$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01924-7
PII: S 0002-9947(97)01924-7
Keywords: Semi-simple splitting, affine structures
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.)
Article copyright: © Copyright 1997 American Mathematical Society