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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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  • 1. Auslander, L. On a problem of Philip Hall. Ann. of Math. (2) 86 (1967), 112-116. MR 36:1540
  • 2. Auslander, L. Simply transitive groups of affine motions. Amer. J. Math. 99 (1977), 809-826. MR 56:5782
  • 3. Benoist, Y. Une nilvariété non affine. C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 983-986. MR 93j:20008
  • 4. Benoist, Y. Une nilvariété non affine. J. Differential Geom. 41 (1995), 21-52. MR 96c:53077
  • 5. Burde, D. Affine structures on nilmanifolds. Internat. J. Math. 7 (1996), 599-616. CMP 97:02
  • 6. Burde, D. and Grunewald, F. Modules for certain Lie algebras of maximal class. J. Pure Applied Algebra 99 (1995), 239-254. MR 96d:17007
  • 7. Conner, P. E. and Raymond, F. Deforming homotopy equivalences to homeomorphisms in aspherical manifolds. Bull. A.M.S. 83 (1977), 36-85. MR 57:7629
  • 8. Dekimpe, K. Polynomial structures and the uniqueness of affinely flat infra-nilmanifolds. 1994. Preprint.
  • 9. Dekimpe, K. and Igodt, P. Polycyclic-by-finite groups admit a bounded-degree polynomial structure. Preprint 1996 (to appear in Invent. Math.)
  • 10. Dekimpe, K. and Igodt, P. Polynomial structures for iterated central extensions of abelian-by-nilpotent groups. Proc. Barcelona Conf. Algebraic Topology 1994, Birkhäuser, 1996, pp. 155-166. CMP 96:15
  • 11. Dekimpe, K., Igodt, P., and Lee, K. B. Polynomial structures for nilpotent groups. Trans.
    A. M. S. 348 (1996), 77-97. MR 96e:20051
  • 12. Fried, D. Distality, completeness and affine structures. J. Differential Geom. 24 (1986), 265-273. MR 88i:53060
  • 13. Grunewald, F. J., Pickel, P. F., and Segal, D. Polycyclic groups with isomorphic finite quotients. Ann. of Math. (2) 111 (1980), 155-195. MR 81i:20045
  • 14. Kamishima, Y., Lee, K. B., and Raymond, F. The Seifert construction and its applications to infra-nilmanifolds. Quarterly J. Math. Oxford Ser. (2) 34 (1983), 433-452. MR 85k:57038
  • 15. Lee, K. B. Aspherical manifolds with virtually 3-step nilpotent fundamental group. Amer. J. Math. 105 (1983), 1435-1453. MR 85d:57032
  • 16. Lee, K. B. and Raymond, F. Geometric realization of group extensions by the Seifert construction. Contemporary Math. A. M. S. 33 (1984), 353-411. MR 86h:57043
  • 17. Mal'cev, A. I. On a class of homogeneous spaces. Translations A.M.S. 39 (1951); reprinted in Amer. Math. Soc. Transl. (1) 9 (1962), 276-307. MR 12:589e
  • 18. Milnor, J. On fundamental groups of complete affinely flat manifolds. Adv. Math. 25 (1977), 178-187. MR 56:13130
  • 19. Passman, D. S. The algebraic structure of group rings. Wiley, 1977. MR 81d:16001
  • 20. Scheuneman, J. Affine structures on three-step nilpotent Lie algebras. Proc. A.M.S. 46 (1974), 451-454. MR 54:470
  • 21. Scheuneman, J. Translations in certain groups of affine motions. Proc. Amer. Math. Soc. 47 (1975), 223-228. MR 51:8337
  • 22. Segal, D. Polycyclic groups. Cambridge University Press, 1983. MR 85h:20003

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Additional Information

Karel Dekimpe
Affiliation: Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium

Paul Igodt
Affiliation: Katholieke Universiteit Leuven, Campus Kortrijk, B-8500 Kortrijk, Belgium

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

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