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

DOI:
https://doi.org/10.1090/S0002-9947-98-02266-1

MathSciNet review:
1633056

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Abstract: A. Borel proved that, if the fundamental group of an aspherical manifold is centerless and the outer automorphism group of is torsion-free, then admits no periodic maps, or equivalently, there are no non-trivial finite groups of homeomorphisms acting effectively on . 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 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**

Email:
Wim.Malfait@kulak.ac.be

DOI:
https://doi.org/10.1090/S0002-9947-98-02266-1

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