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Crystallographic groups with trivial center and outer automorphism group

Published online by Cambridge University Press:  06 March 2017

RAFAŁ LUTOWSKI  and
ANDRZEJ SZCZEPAŃSKI
RAFAŁ LUTOWSKI
Affiliation:
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland. e-mail: [email protected], [email protected]
ANDRZEJ SZCZEPAŃSKI
Affiliation:
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland. e-mail: [email protected], [email protected]
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Abstract

Let Γ be a crystallographic group of dimension n, i.e. a discrete, cocompact subgroup of Isom(ℝn) = O(n) ⋉ ℝn. For any n ⩾ 2, we construct a crystallographic group with a trivial center and trivial outer automorphism group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 363 - 368
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Belolipetsky, M. and Lubotzky, A. Finite groups and hyperbolic manifolds. Invent. Math. 162 (2005), no. 3, 459–472.Google Scholar
[2] Charlap, L.S. Bieberbach Groups and Flat Manifolds. Universitext (Springer-Verlag. New York, 1986).Google Scholar
[3] Lutowski, R. On symmetry of flat manifolds. Exp. Math. 18 (2009), no. 3, 201–204.Google Scholar
[4] Lutowski, R. Finite outer automorphism groups of crystallographic groups. Exp. Math. 22 (2013), no. 4, 456–464.Google Scholar
[5] Serre, J.P. Linear Representations of Finite Groups (Springer-Verlag, New York-Heidelberg, 1977).CrossRefGoogle Scholar
[6] Szczepański, A. Problems on Bieberbach groups and flat manifolds. Geom. Dedicata 120 (2006), 111–118.Google Scholar
[7] Szczepański, A. Geometry of the crystallographic groups . Algebra and Discrete Mathematics, vol. 4 (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012).Google Scholar
[8] Waldmüller, R. A flat manifold with no symmetries. Exp. Math. 12 (2003), no. 1, 71–77.Google Scholar