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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.
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Abstract

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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.

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

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