Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T01:45:00.526Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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