Non-Euclidean crystallographic group

In mathematics, a non-Euclidean crystallographic group, NEC group or N.E.C. group is a discrete group of isometries of the hyperbolic plane. These symmetry groups correspond to the wallpaper groups in euclidean geometry. A NEC group which contains only orientation-preserving elements is called a Fuchsian group, and any non-Fuchsian NEC group has an index 2 Fuchsian subgroup of orientation-preserving elements.

The hyperbolic triangle groups are notable NEC groups. Others are listed in Orbifold notation here.

See also

• Non-Euclidean geometry
• Isometry group
• Fuchsian group
• Uniform tilings in hyperbolic plane

References

• H.C. Wilkie, On non-Euclidean crystallographic groups,
  Mathematische Zeitschrift, Volume 91, April 1966, Pages 87 – 102, ISSN 0025-5874
• Emilio Bujalance, Automorphism groups of compact planar Klein surfaces,
  manuscripta mathematica, Volume 56, March 1986, Pages 105 - 124, ISSN 0025-2611