Model aspherical manifolds with no periodic maps
HTML articles powered by AMS MathViewer
- by Wim Malfait PDF
- Trans. Amer. Math. Soc. 350 (1998), 4693-4708 Request permission
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.References
- Louis Auslander, Bieberbach’s theorems on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. (2) 71 (1960), 579–590. MR 121423, DOI 10.2307/1969945
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- P. E. Conner and Frank Raymond, Manifolds with few periodic homeomorphisms, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 299, Springer, Berlin, 1972, pp. 1–75. MR 0358835
- P. E. Conner, Frank Raymond, and Peter J. Weinberger, Manifolds with no periodic maps, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 299, Springer, Berlin, 1972, pp. 81–108. MR 0358836
- Dekimpe, K. Almost–Bieberbach Groups: Affine and Polynomial Structures. Lect. Notes in Math. 1639, Springer–Verlag, 1996.
- Dekimpe, K. Determining the translation part of the fundamental group of an infra-solvmanifold of type (R). Math. Proc. Camb. Phil. Soc. 122 (1997), 515–524.
- Karel Dekimpe, Paul Igodt, and Wim Malfait, On the Fitting subgroup of almost crystallographic groups, Bull. Soc. Math. Belg. Sér. B 45 (1993), no. 1, 35–47. MR 1314931
- V. V. Gorbacevič, Discrete subgroups of solvable Lie groups of type $(\textrm {R})$, Mat. Sb. (N.S.) 85 (127) (1972), 238–255 (Russian). MR 0289711
- V. V. Gorbacevič, Lattices in solvable Lie groups, and deformations of homogeneous spaces, Mat. Sb. (N.S.) 91(133) (1973), 234–252, 288 (Russian). MR 0352329
- I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 81
- Hurewicz, W. Beiträge zur Topologie der Deformationen. IV. Asphärische Räume. Nederl. Akad. Wetensch. Proc., 1936, 39, pp. 215–224.
- Paul Igodt and Wim Malfait, Extensions realising a faithful abstract kernel and their automorphisms, Manuscripta Math. 84 (1994), no. 2, 135–161. MR 1285953, DOI 10.1007/BF02567450
- Paul Igodt and Wim Malfait, Representing the automorphism group of an almost crystallographic group, Proc. Amer. Math. Soc. 124 (1996), no. 2, 331–340. MR 1301030, DOI 10.1090/S0002-9939-96-03141-3
- Yoshinobu Kamishima, Kyung Bai Lee, and Frank Raymond, The Seifert construction and its applications to infranilmanifolds, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 136, 433–452. MR 723280, DOI 10.1093/qmath/34.4.433
- Kyung Bai Lee, Infra-solvmanifolds of type (R), Quart. J. Math. Oxford Ser. (2) 46 (1995), no. 182, 185–195. MR 1333830, DOI 10.1093/qmath/46.2.185
- K. B. Lee and Frank Raymond, Topological, affine and isometric actions on flat Riemannian manifolds, J. Differential Geometry 16 (1981), no. 2, 255–269. MR 638791
- Kyung Bai Lee and Frank Raymond, Geometric realization of group extensions by the Seifert construction, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 353–411. MR 767121, DOI 10.1090/conm/033/767121
- Kyung Bai Lee and Frank Raymond, Seifert manifolds modelled on principal bundles, Transformation groups (Osaka, 1987) Lecture Notes in Math., vol. 1375, Springer, Berlin, 1989, pp. 207–215. MR 1006694, DOI 10.1007/BFb0085611
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- Wim Malfait, Nielsen’s theorem for model aspherical manifolds, Manuscripta Math. 90 (1996), no. 1, 63–83. MR 1387755, DOI 10.1007/BF02568294
- William Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771. MR 260975, DOI 10.2307/2373350
- Frank Raymond and Jeffrey L. Tollefson, Closed $3$-manifolds with no periodic maps, Trans. Amer. Math. Soc. 221 (1976), no. 2, 403–418. MR 415620, DOI 10.1090/S0002-9947-1976-0415620-9
- Derek J. S. Robinson, Infinite soluble groups with no outer automorphisms, Rend. Sem. Mat. Univ. Padova 62 (1980), 281–294. MR 582957
- Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR 713786, DOI 10.1017/CBO9780511565953
- Andrzej Szczepański, Five-dimensional Bieberbach groups with trivial centre, Manuscripta Math. 68 (1990), no. 2, 191–208. MR 1063225, DOI 10.1007/BF02568759
Additional Information
- Wim Malfait
- Email: Wim.Malfait@kulak.ac.be
- 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.)
- © Copyright 1998 American Mathematical Society
- 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