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Kleinian groups with unbounded limit sets

Published online by Cambridge University Press:  18 May 2009

A. F. Beardon
A. F. Beardon
Affiliation:
St. Catharine's College, Cambridge
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The easiest way to construct automorphic functions is by means of the Poincaré series. If G is a Kleinian group with ∞ an ordinary point of G and if k ≧ 4, then

where Vz=(az+b)/(cz+d) and ad-bc=1. The convergence of this series is the crucial step in showing that the Poincaré series converges and is an automorphic form on G If ∞ is a limit point of ∞ the series in (1) may diverge and one can derive automorphic forms on ∞ from the Poincaré series of some conjugate group. These constructions are described in greater detail in /3, pp. 155–165].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 13 , Issue 1 , March 1972 , pp. 24 - 28
Copyright
Copyright © Glasgow Mathematical Journal Trust 1972

References

REFERENCES

1.Akaza, T., Poincaré theta series and singular sets of Schottky groups, Nagoya Math. J. 24 (1964), 43–65.CrossRefGoogle Scholar
2.Beardon, A. F., The Hausdorff dimension of singular sets of properly discontinuous groups, Amer. J. Math. 88 (1966), 722–736.Google Scholar
3.Lehner, J., Discontinuous groups and automorphic functions, Amer. Math. Soc. Math. Surveys, No 8 (Providence, R.I., 1964).CrossRefGoogle Scholar