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Ford and Dirichlet Regions for Fuchsian Groups

  • A. F. Beardon (a1) and P. J. Nicholls (a2)

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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.

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References

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1. Beardon, A. F., Fundamental domains for Kleinian groups, in Discontinuous groups and Riemann surfaces, Ann. of Math. Studies 79 (1974), 3141.
2. Beardon, A. F. and Maskit, B., Limit points of Kleinian groups and finite sided fundamental polyhedrat Acta Math. 132 (1974), 112.
3. Ford, L. R., Automorphic functions (Chelsea, New York, 1951).
4. Lehner, J., Discontinuous groups and automorphic functions Math. Survey 8 (Amer. Math. Soc, Providence, RI, 1974).
5. Nicholls, P. J., Garnett points for Fuchsian groups, Bull. London Math. Soc. 12 (1980), 216218.
6. Nicholls, P. and Zarrow, R., Convex fundamental regions for Fuchsian groups, Math. Proc. Camb. Phil. Soc. 84 (1978), 507518.
7. Ch., Pommerenke, On the Green s function of Fuchsian groups, Ann. Acad. Scient. Fenn. A.I. 2 (1976), 409427.
8. Sullivan, D., On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in Riemann surfaces and related topics, Ann. of Math. Studies 97 (1980), 465496.
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Ford and Dirichlet Regions for Fuchsian Groups

  • A. F. Beardon (a1) and P. J. Nicholls (a2)

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