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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Model aspherical manifolds
with no periodic maps

Author: Wim Malfait
Journal: Trans. Amer. Math. Soc. 350 (1998), 4693-4708
MSC (1991): Primary 57S25, 20F34, 20H15
MathSciNet review: 1633056
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Abstract: A. Borel proved that, if the fundamental group $E$ of an aspherical manifold $M$ is centerless and the outer automorphism group of $E$ is torsion-free, then $M$ admits no periodic maps, or equivalently, there are no non-trivial finite groups of homeomorphisms acting effectively on $M$. In the literature, taking off from this result, several examples of (rather complex) aspherical manifolds exhibiting this total lack of periodic maps have been presented.

In this paper, we investigate to what extent the converse of Borel's result holds for aspherical manifolds $M$ arising from Seifert fiber space constructions. In particular, for e.g. flat Riemannian manifolds, infra-nilmanifolds and infra-solvmanifolds of type (R), it turns out that having a centerless fundamental group with torsion-free outer automorphism group is also necessary to conclude that all finite groups of affine diffeomorphisms acting effectively on the manifold are trivial. Finally, we discuss the problem of finding (less complex) examples of such aspherical manifolds with no periodic maps.

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Additional Information

Wim Malfait

Keywords: Aspherical manifold, periodic map, Seifert fiber space construction, infra-nil- and infra-solvmanifold of type (R)
Received by editor(s): December 19, 1996
Additional Notes: The author is a Postdoctoral Fellow of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.)
Article copyright: © Copyright 1998 American Mathematical Society