Portraits of Groups on Bordered Surfaces
Inhaltsverzeichnis
Reference
Jay Zimmerman: Portraits of Groups on Bordered Surfaces. In: Bridges 2016, Pages 241–246.
DOI
Abstract
This paper looks at representing a group G as a group of transformations of an orientable compact bordered Klein surface. We construct visual representations (portraits) of three groups S4, Z2 S3 and a group L* of order 32. These groups have real genus 3, 2 and 5 respectively. The first two groups are M*-groups; which means that they act on surfaces with maximal symmetry. They are also the only solvable M* - simple groups.
Extended Abstract
Bibtex
@inproceedings{bridges2016:241,
author = {Jay Zimmerman},
title = {Portraits of Groups on Bordered Surfaces},
pages = {241--246},
booktitle = {Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture},
year = {2016},
editor = {Eve Torrence, Bruce Torrence, Carlo S\'equin, Douglas McKenna, Krist\'of Fenyvesi and Reza Sarhangi},
isbn = {978-1-938664-19-9},
issn = {1099-6702},
publisher = {Tessellations Publishing},
address = {Phoenix, Arizona},
url = {http://de.evo-art.org/index.php?title=Portraits_of_Groups_on_Bordered_Surfaces },
note = {Available online at \url{http://archive.bridgesmathart.org/2016/bridges2016-241.html}}
}
Used References
1. W. Burnside, Theory of groups of finite order, (Cambridge University Press 1911).
2. E. Bujalance, J.J. Etayo, J.M. Gamboa and G. Gromadzki, Automorphism Groups of Compact Bordered Klein Surfaces, A Combinatorial Approach, Lecture Notes in Mathematics 1439, Springer-Verlag, Berlin, Heidelberg, 1990.
3. Coxeter, H.S.M. and Moser, W.O.J., Generators and Relations for Discrete Group, Reprint of Fourth Ed., Springer-Verlag, New York 1984.
4. N. Greenleaf and C. L. May, Bordered Klein surfaces with maximal symmetry, Trans. Amer. Math. Soc. 274 (1982), 265 – 283.
5. C. L. May, Groups of small real genus, Houston J. Math. 20, No. 3, (1994), 393 – 408.
6. J. J. van Wijk, Symmetric Tilings of Closed Surfaces: Visualization of Regular Maps, ACM Transactions on Graphics, vol. 28, no. 3, Article 49 (2009) Proceedings ACM SIGGRAPH’09.
7. J. Zimmerman, Portraits of Groups, Conference Proceedings 2006, Bridges London: Mathematical Connections in Art, Music and Science, London, England, 131 - 134.
8. J. Zimmerman, A Portrait of a Quadrilateral Group, Conference Proceedings 2011, Bridges Coimbra: Mathematical Connections in Art, Music and Science, Coimbra, Portugal, 505 - 508.
Links
Full Text
http://archive.bridgesmathart.org/2016/bridges2016-241.pdf
