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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 for nilpotent groups


Authors: Karel Dekimpe, Paul Igodt and Kyung Bai Lee
Journal: Trans. Amer. Math. Soc. 348 (1996), 77-97
MSC (1991): Primary 57S30, 22E40, 22E25
MathSciNet review: 1327254
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Abstract: If a polycyclic-by-finite rank-$K$ group $\Gamma$ admits a faithful affine representation making it acting on $\Bbb R^{K}$ properly discontinuously and with compact quotient, we say that $\Gamma$ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups $\Gamma$. Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent groups $N$, affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Mal$'$cev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the ``affine defect number''. We prove that the known counterexamples to Milnor's question have the smallest possible affine defect, i.e. affine defect number equal to one.


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Additional Information

Paul Igodt
Affiliation: Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium
Email: Paul.Igodt@kulak.ac.be

Kyung Bai Lee
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: kblee@dstn3.math.uoknor.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01513-9
PII: S 0002-9947(96)01513-9
Keywords: Lattices in nilpotent Lie groups, complete affinely flat structures
Received by editor(s): May 8, 1994
Additional Notes: The first author is Research Assistant of the National Fund For Scientific Research (Belgium)
Article copyright: © Copyright 1996 American Mathematical Society