Hostname: page-component-54dcc4c588-r5qjk Total loading time: 0 Render date: 2025-10-08T00:45:51.314Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized symmetry

Published online by Cambridge University Press:  17 April 2009

J.B. Wilker
J.B. Wilker
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a set and G a group which acts on X and is generated by two elements α and b. Motivated by a geometric problem of L. Fejes Tóth, we define a subset S ⊂ X to have [α, b]-symmetry if its images under α and b satisfy Sα ∩ Sb = S. The problem of finding all sets with [α, b]- symmetry when an arbitrary 2-generator group G acts on an arbitrary space X is shown to be equivalent to the same problem in the special case when the 2-generator free group acts on itself by right translation. This action is modelled in the hyperbolic plane in a way that helps to reveal the [α, b]- symmetric subsets of the free group.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 18 , Issue 3 , June 1978 , pp. 373 - 393
Copyright
Copyright © Australian Mathematical Society 1978

References

[1]Burnside, W., Theory of groups of finite order (Cambridge University Press, Cambridge, 1897).Google Scholar
[2]Fourneau, René, “Nonclosed simplices and quasi-simplices”, Mathematika 24 (1977), 71–85.Google Scholar
[3]Magnus, Wilhelm, Noneualidean tesselations and their groups (Pure and Applied Mathematics, 61. Academic Press, New York and London, 1974).Google Scholar
[4]Meschkowski, Herbert, Unsolved and unsolvable problems in geometry (translated by Burlak, Jane A.C.. Oliver and Boyd, Edinburgh, London; Frederick Ungar, New York; 1966).Google Scholar
[5]Wilker, J.B., “Sets homothetic to their intersection with a translate”, Canad. Math. Bull. 18 (1975), 739–748.CrossRefGoogle Scholar