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Inventiones mathematicae

, Volume 127, Issue 1, pp 33–50 | Cite as

Geometric exponents and Kleinian groups

  • Christopher J. Bishop
Article

Abstract.

Suppose \(\Lambda\) is the limit set of an analytically finite Kleinian group and that\(\{\Omega_j\}\) is an enumeration of the components of \(\Omega = S^2 \setminus \Lambda\) . Then
$$\sum\limits_j {diam(\Omega _j )^2 < \infty .} $$
This had been conjectured by Maskit. We also define a number of different geometric critical exponents associated to a compact set in the plane which generalize the index of Besicovitch and Taylor on the line. Although these exponents may differ for general sets, we show that they are all equal when \(\Lambda\) is the limit set of a non-elementary, analytically finite Kleinian group and they agree with the classical Poincaré exponent.

Keywords

Critical Exponent Kleinian Group Finite Kleinian Group Geometric Exponent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Christopher J. Bishop
    • 1
  1. 1.Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651, USA e-mail: bishop@math.sunysb.edu USA

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