Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-10T10:03:44.297Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost-Bieberbach groups with (in)finite outer automorphism group

Published online by Cambridge University Press:  18 May 2009

Wim Malfait  and
Andrzej Szczepański
Wim Malfait
Affiliation:
Katholieke Universiteit Leuven Campus KortrijkUniversitaire CampusB-8500 KORTRIJK (Belgium) E-Mail: Wim.Malfait@Kulak.ac.be
Andrzej Szczepański
Affiliation:
Andrzej SzczepańskiInstitute of MathematicsUniversity of GdańskUL. Wita Stwosza 5780-952 GDANSK (Poland) E-Mail: matas@paula.univ.gda.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

If we investigate symmetry of an infra-nilmanifold M, the outer automorphism group of its fundamental group (an almost-Bieberbach group) is known to be a crucial object. In this paper, we characterise algebraically almost-Bieberbach groups E with finite outer automorphism group Out(E). Inspired by the description of Anosov diffeomorphisms on M, we also present an interesting class of infinite order outer automorphisms. Another possible type of infinite order outer automorphisms arises when comparing Out(E) with the outer automorphism group of the underlying crystallographic group of E.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 40 , Issue 1 , March 1998 , pp. 47 - 62
Copyright
Copyright © Glasgow Mathematical Journal Trust 1998

References

REFERENCES

1.Bridson, M. and Gersten, S., Optimal isoperimeric inequality for torus bundles over the circle. Quart. J. Math. Oxford (2) 47 (1996) 1–23.Google Scholar
2.Dekimpe, K., Almost Bieberbach Groups: cohomology, construction and classification (Doctoral Thesis, K. U. Leuven, 1993).Google Scholar
3.Dekimpe, K., Determining the translation part of the fundamental group of an infrasolvmanifold of type (R), to appear in Math. Proc. Camb. Phil. Soc.Google Scholar
4.Dekimpe, K., Igodt, P., Kim, S. and Lee, K. B., Affine structures for closed 3-dimensional manifolds with NIL-geometry, Quart. J. Math. Oxford (2) 46 (1995) 141–167.Google Scholar
5.Dekimpe, K., Igodt, P. and Malfait, W., On the Fitting subgroup of almost crystallographic groups, Tijdschrift van het Belgisch Wiskundig Genootschap, B1 (1993) 35–47.Google Scholar
6.Dekimpe, K. and Malfait, W., Almost-crystallographic groups with many outer automorphisms, Comm. Algebra 23(8) (1995), 3073–3083.Google Scholar
7.Franks, J., Anosov diffeomorphisms, In: Global Analysis: Proceedings of the Symposia in Pure Mathematics 14 (1970) 61–93.CrossRefGoogle Scholar
8.Igodt, P. and Malfait, W., Extensions realising a faithful abstract kernel and their automorphisms, Manuscripta Math. 84 (1994) 135–161.Google Scholar
9.Igodt, P. and Malfait, W., Representing the automorphism group of an almost crystallographic group, Proc. Amer. Math. Soc. 124 (2) (1996) 331–340.Google Scholar
10.Kamishima, Y., Lee, K. B. and Raymond, F., The Seifert construction and its applications to infra-nilmanifolds, Quart. J. Math. Oxford (2) 34 (1983) 433–452.Google Scholar
11.Lee, K. B. and Raymond, F., Geometric realization of group extensions by the Seifert construction, Contemporary Math. A. M. S., 33 (1984) 353–411.Google Scholar
12.Lee, K. B. and Raymond, F., Rigidity of almost crystallographic groups, Contemporary Math. A. M. S. 44 (1985) 73–78.Google Scholar
13.Lane, S. Mac, Homology, volume 114 of Die Grundlehren der Math. Wissenschaften (Springer-Verlag: Berlin, Heidelberg, New York, 1975).Google Scholar
14.Mal'cev, A. I., On a class of homogeneous spaces, Translations , A. M. S. 39 (1951) 1–33.Google Scholar
15.Malfait, W., Symmetry of Infra-Nilmanifolds: An Algebraic Approach (Doctoral Thesis, K. U. Leuven, 1994).Google Scholar
16.Manning, A., There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96(3) (1974) 422–429.Google Scholar
17.Parry, W., ergodic peoperties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969) 757–771.Google Scholar
18.Porteous, H. L., Anosov dffeomorphisms of flat manifolds, Topology, 11 (1972) 307–315.Google Scholar
19.Stewart, I. and Tall, D., Algebraic Number Theory, second edition (Chapman and Hall Mathematics Series, 1987).Google Scholar
20.Szczepaŕiski, A., Outer automorphism groups of Bieberbach groups, to appear in Bull, of Belg. Math. Soc. (Simon Stevin), 1996.CrossRefGoogle Scholar