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Divergence type of some subgroups of finitely generated Fuchsian groups

Published online by Cambridge University Press:  19 September 2008

Mary Rees*
Affiliation:
From the Institut des Hautes Etudes Scientifiques, Bures-sur-Yuette, France
*
Address for correspondence: Dr Mary Rees, Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France.
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Abstract

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Let Г be a finitely generated discrete subgroup of the isometries of the hyperbolic plane H2 with at least one parabolic element. We prove that, if Г1 is a subgroup of Г with Г/Г1 abelian, the ‘critical exponent’ of Г1 is the same as that of Г. We give necessary and sufficient conditions-in terms of the rank of Г/Г1, the critical exponent of Г, and the image of parabolic elements of Г in Г/Г1 - for the Poincaré series of Г1 to diverge at the critical exponent.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1981

References

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