A theorem of Ahlfors for hyperbolic spaces

Chen, Su Shing

doi:10.1090/S0002-9947-1978-0496817-0

A theorem of Ahlfors for hyperbolic spaces
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Trans. Amer. Math. Soc. 242 (1978), 401-406 Request permission

Abstract:

L. Ahlfors has proved that if the Dirichlet fundamental polyhedron of a Kleinian group G in the unit ball ${B^3}$ has finitely many sides, then the normalized Lebesgue measure of $L(G)$ is either zero or one. We generalize this theorem and a theorem of Beardon and Maskit to the n-dimensional case.

References

Lars V. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 251–254. MR 194970, DOI 10.1073/pnas.55.2.251
Alan F. Beardon and Bernard Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1–12. MR 333164, DOI 10.1007/BF02392106
S. S. Chen and L. Greenberg, Hyperbolic spaces, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 49–87. MR 0377765

Finiteness theorems for Fuchsian and Kleinian groups

L. Greenberg, Fundamental polyhedra for kleinian groups, Ann. of Math. (2) 84 (1966), 433–441. MR 200446, DOI 10.2307/1970456
Leon Greenberg, Discrete subgroups of the Lorentz group, Math. Scand. 10 (1962), 85–107. MR 141731, DOI 10.7146/math.scand.a-10515
A. W. Knapp, Fatou’s theorem for symmetric spaces. I, Ann. of Math. (2) 88 (1968), 106–127. MR 225938, DOI 10.2307/1970557
Joseph Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964. MR 0164033, DOI 10.1090/surv/008

Discrete Moebius groups, fundamental polyhedra and convergence