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Ford and Dirichlet Regions for Fuchsian Groups

Published online by Cambridge University Press:  20 November 2018

A. F. Beardon  and
P. J. Nicholls
A. F. Beardon
Affiliation:
University of Cambridge, Cambridge, Great Britain
P. J. Nicholls
Affiliation:
Northern Illinois University, Dekalb, Illinois
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There has recently been some interest in a class of limit points for Fuchsian groups now known as Garnett points [5], [8]. In this paper we show that such points are intimately connected with the structure of Dirichlet regions and the same ideas serve to show that the Ford and Dirichlet regions are merely examples of one single construction which also yields fundamental regions based at limit points (and which properly lies in the subject of inversive geometry). We examine in the general case how the region varies continuously with the construction. Finally, we consider the linear measure of the set of Garnett points.

2. Hyperbolic space. Let Δ be any open disc (or half-plane) in the extended complex plane C∞: usually Δ will be the unit disc or the upper half-plane. We may regard Δ as the hyperbolic plane in the usual way and the conformai isometries of Δ are simply the Moebius transformations of Δ onto itself.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 4 , 01 August 1982 , pp. 806 - 815
Copyright
Copyright © Canadian Mathematical Society 1982

References

References>

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