Geometriae Dedicata

, Volume 177, Issue 1, pp 43–60 | Cite as

On the isometry group of a compact flat orbifold

  • John G. Ratcliffe
  • Steven T. Tschantz
Original Paper


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 \(\Gamma \).


Flat orbifold Isometry group Lie group Crystallographic group  Nielsen realization problem 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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