Abstract
Let \(\Gamma \) be an \(n\)-dimensional crystallographic group. We prove that the group \(\mathrm{Isom}(E^n/\Gamma )\) of isometries of the flat orbifold \(E^n/\Gamma \) is a compact Lie group whose component of the identity is a torus of dimension equal to the first Betti number of the group \(\Gamma \). This implies that \(\mathrm{Isom}(E^n/\Gamma )\) is finite if and only if \(\Gamma /[\Gamma , \Gamma ]\) is finite. We also generalize known results on the Nielsen realization problem for torsion-free \(\Gamma \) to arbitrary \(\Gamma \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, J.F.: Lectures on Lie Groups. W. A. Benjamin, New York (1969)
Brown, H., Neubüser, J., Zassenhaus, H.: On integral groups. III: normalizers. Math. Comp. 27, 167–182 (1973)
Charlap, L.S., Vasquez, A.T.: Compact flat Riemannian manifolds III: the group of affinities. Am. J. Math. 95, 471–494 (1973)
Farkas, D.R.: Crystallographic groups and their mathematics. Rocky Mountain J. Math. 11, 511–551 (1981)
Gubler, M.: Normalizer groups and automorphism groups of symmetry groups. Z. Krist. 158, 1–26 (1982)
Holt, D.F.: An interpretation of the cohomology groups \(H^n(G, M)\). J. Algebra 60, 307–320 (1979)
Lee, K.B.: Geometric realization of \(\pi _0\varepsilon (M)\). Proc. Am. Math. Soc. 86, 353–357 (1982)
Lee, K.B., Raymond, F.: Topological, affine and isometric actions on flat Riemannian manifolds I. J. Differ. Geom. 16, 255–269 (1981)
Lee, K.B., Raymond, F.: Topological, affine and isometric actions on flat Riemannian manifolds II. Topol. Appl. 13, 295–310 (1982)
Mac Lane, S.: Homology. Springer, New York (1967)
Mac Lane, S., Whitehead, J.H.C.: On the 3-type of a complex. Proc. Nat. Acad. Sci. USA 30, 41–48 (1956)
Ratcliffe, J.G.: Crossed extensions. Trans. Am. Math. Soc. 257, 73–89 (1980)
Ratcliffe, J.G.: Ratcliffe Foundations of Hyperbolic Manifolds, Second Edition, Graduate Texts in Math., vol. 149. Springer, Berlin (2006)
Ratcliffe, J.G., Tschantz, S.T.: Abelianization of space groups. Acta Cryst. A65, 18–27 (2009)
Ratcliffe, J.G., Tschantz, S.T.: Fibered orbifolds and crystallographic groups. Algebr. Geom. Topol. 10, 1627–1664 (2010)
Szczepański, A.: Holonomy groups of crystallographic groups with finite outer automorphism groups, Recent advances in group theory and low dimensional topology. In: Mennicke, J., Cho, J.R. (eds.) Proceedings of the German-Korea Workshop, Pusan 2000, Res. Exp. Math. 27. Heldermann Verlag, Lemgo, pp. 163–165 (2003)
Szczepański, A.: Geometry of Crystallographic Groups, Algebra and Discrete Math, vol. 4. Word Scientific, Singapore (2012)
Zieschang, H., Zimmermann, B.: Endliche Gruppen von Abbildungsklassen gefaserter 3-Mannigfal tigkeiten. Math. Ann. 240, 41–62 (1979)
Zimmermann, B.: Über Gruppen von Homöomorphismen Seifertscher Faserräume und Flacher Mannigfaltigkeiten. Manuscripta Math. 30, 361–373 (1980)
Zimmermann, B.: Nielsensche Realisierungssätze für Seifertfaserungen. Manuscripta Math. 51, 225–242 (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ratcliffe, J.G., Tschantz, S.T. On the isometry group of a compact flat orbifold. Geom Dedicata 177, 43–60 (2015). https://doi.org/10.1007/s10711-014-9976-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-014-9976-0