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Convex fundamental regions for Fuchsian groups

Published online by Cambridge University Press:  24 October 2008

P. Nicholls  and
R. Zarrow
P. Nicholls
Affiliation:
Northern Illinois University, De Kalb
R. Zarrow
Affiliation:
Northern Illinois University, De Kalb
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Extract

1. Introduction. Let G be a Fuchsian group which acts on the unit disc Δ. A fundamental region D for G acting in Δ is a subset of Δ such that D is open and connected and each point of Δ is G-equivalent to exactly one point in D or at least one point in D̅ (the closure of D in Δ). Throughout this paper we consider only fundamental regions which are (hyperbolically) convex. Beardon has shown (2) that it is possible for a convex fundamental region to have certain undesirable properties. It can happen that a convex region is not locally finite, i.e. there exist points of Δ where infinitely many G images of D accumulate. For a domain D we denote by F the set of such points.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 3 , November 1978 , pp. 507 - 518
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Beardon, A. F. On fundamental regions of Kleinian groups. (Preprint.)Google Scholar
(2)Beardon, A. F.Fundamental domains for Kleinian groups, Discontinuous groups and Riernann surfaces. Annals of Math. Studies, no. 79 (1974), 31–41.Google Scholar
(3)Beardon, A. F.Convex fundamental domains of Fuchsian groups. Proc. Cambridge Philos. Soc. 76 (1974), 511–513.CrossRefGoogle Scholar
(4)Beardon, A. F. and Maskit, B.Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta Math. 132 (1974), 1–12.CrossRefGoogle Scholar
(5)Greenberg, L.Fundamental polygons for Fuchsian groups. J. Analyse Math. 18 (1967), 99–105.CrossRefGoogle Scholar
(6)Marden, A.On finitely generated Fuchsian groups. Comm. Math. Helv. 42 (1967), 81–85.CrossRefGoogle Scholar
(7)Nicholls, P. J.Special limit points for Fuchsian groups and automorphic functions near the limit set. Indiana U. Math. J. 24 (1974), 143–148.CrossRefGoogle Scholar
(8)Purzitsky, N.Canonical generators of Fuchsian groups. Illinois J. Math. 18 (1974), 484–490.CrossRefGoogle Scholar