Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Kleinian groups of divergence type


Author: P. J. Nicholls
Journal: Proc. Amer. Math. Soc. 83 (1981), 319-324
MSC: Primary 30F40; Secondary 20H10
MathSciNet review: 624922
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a Kleinian group $ \Gamma $ acting in the unit ball $ B$ we consider the series $ {\Sigma _{\gamma \in \Gamma }}{(1 - \left\vert {\gamma (0)} \right\vert)^2}$. If the series diverges, $ \Gamma $ is said to be of divergence type. From the point of view of the ergodic properties of the group action it is essential to know whether or not $ \Gamma $ is of divergence type. If $ \Gamma $ is geometrically finite then $ \Gamma $ is of divergence type if and only if it is of the first kind. However in the nongeometrically finite case it is not known whether there are any groups of divergence type.

In this paper we give a geometric criterion which is sufficient to ensure divergence type and use this to construct an example of a nongeometrically finite Kleinian group of divergence type.


References [Enhancements On Off] (What's this?)

  • [1] Lars V. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 251–254. MR 0194970
  • [2] A. F. Beardon and P. J. Nicholls, On classical series associated with Kleinian groups, J. London Math. Soc. (2) 5 (1972), 645–655. MR 0320305
  • [3] Bernard Maskit, On Poincaré’s theorem for fundamental polygons, Advances in Math. 7 (1971), 219–230. MR 0297997
  • [4] Peter J. Nicholls, Fundamental regions and the type problem for a Riemann surface, Math. Z. 174 (1980), no. 2, 187–196. MR 592914, 10.1007/BF01293537
  • [5] Burton Rodin and Leo Sario, Principal functions, In collaboration with Mitsuru Nakai, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR 0229812
  • [6] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. MR 0264064
  • [7] 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30F40, 20H10

Retrieve articles in all journals with MSC: 30F40, 20H10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0624922-X
Keywords: Kleinian group, geometrically finite, Dirichlet region
Article copyright: © Copyright 1981 American Mathematical Society