From Wikipedia, the free encyclopedia
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
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.