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 (2019), Cohomology of Fuchsian Groups and Non-Euclidean Crystallographic Groups, arXiv:1910.00519, Bibcode:2019arXiv191000519H.