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The norm of a Möbius transformation

Published online by Cambridge University Press:  24 October 2008

A. F. Beardon  and
J. B. Wilker
A. F. Beardon
Affiliation:
St Catharine's College, University of Cambridge
J. B. Wilker
Affiliation:
Scarborough College, University of Toronto
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Extract

A Möbius transformation z → (az + b)/(cz + d), ad — bc = 1, acts as a conformal transformation of the Riemann sphere , and its Poincaré extension acts as an isometry of hyperbolic 3-space modelled in the ball < 1. The size of this transformation can be measured by the matrix norm

or by the hyperbolic distance ρ through which its extension moves the point (0, 0, 0).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 2 , September 1984 , pp. 301 - 308
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Beardon, A. F.. The Geometry of Discrete Groups (Springer-Verlag, 1983).CrossRefGoogle Scholar
[2]Beardon, A. F. and Nicholls, P. J.. On classical series associated with Kleinan groups. J. London Math. Soc. (2), 5 (1972), 645–655.CrossRefGoogle Scholar
[3]Wilker, J. B.. Inversive geometry. The Geometric Vein (Springer-Verlag, 1982), 379–442.Google Scholar