Ford and Dirichlet regions for discrete groups of hyperbolic motions

Nicholls, P. J.

doi:10.1090/S0002-9947-1984-0728717-5

Ford and Dirichlet regions for discrete groups of hyperbolic motions
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by P. J. Nicholls

Trans. Amer. Math. Soc. 282 (1984), 355-365

DOI: https://doi.org/10.1090/S0002-9947-1984-0728717-5

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Abstract:

It is shown that for a discrete group of hyperbolic motions of the unit ball of ${\mathbf {R}^n}$, there is a single construction of fundamental regions which gives the Ford and Dirichlet regions as special cases and which also yields fundamental regions based at limit points. It is shown how the region varies continuously with the construction. The construction is connected with a class of limit points called Garnett points. The size of the set of such points is investigated.

References

Lars V. Ahlfors, Hyperbolic motions, Nagoya Math. J. 29 (1967), 163–166. MR 233976
A. F. Beardon and P. J. Nicholls, Ford and Dirichlet regions for Fuchsian groups, Canadian J. Math. 34 (1982), no. 4, 806–815. MR 672677, DOI 10.4153/CJM-1982-056-0
P. J. Nicholls, Garnett points for Fuchsian groups, Bull. London Math. Soc. 12 (1980), no. 3, 216–218. MR 572105, DOI 10.1112/blms/12.3.216
Ch. Pommerenke, On the Green’s function of Fuchsian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 409–427. MR 0466534
Vladimir G. Sprindžuk, Metric theory of Diophantine approximations, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto-London, 1979. Translated from the Russian and edited by Richard A. Silverman; With a foreword by Donald J. Newman. MR 548467
Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496. MR 624833