Non-Euclidean crystallographic group

This article is about the mathematical concept. For the British state-owned venue management company, see NEC 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.

See also

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

References

• Wilkie, H. C. (1966), "On non-Euclidean crystallographic groups", Mathematische Zeitschrift, 91 (2): 87–102, doi:10.1007/bf01110157, MR 0185013.
• Bujalance, Emilio (1986), "Automorphism groups of compact planar Klein surfaces", Manuscripta Mathematica, 56 (1): 105–124, doi:10.1007/BF01171036, MR 0846989.
• Hughes, Sam (2023), "Cohomology of Fuchsian groups and non-Euclidean crystallographic groups", Manuscripta Mathematica, 170 (3–4): 659–676, arXiv:1910.00519, Bibcode:2019arXiv191000519H, doi:10.1007/s00229-022-01369-z.