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The Quarterly Journal of Mathematics Advance Access first published online on November 9, 2006
This version published online on December 4, 2006
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hal019
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© 2006. Published by Oxford University Press. All rights reserved

Conformal measures associated to ends of hyperbolic n-manifolds

James W. Anderson1,{dagger}, Kurt Falk2 {ddagger} ¶ and Pekka Tukia3 §

1 School of Mathematics, University of Southampton, Southampton SO17 1BJ
2 Mathematics Department, NUI Maynooth, Co. Kildare, Ireland
3 Department of Mathematics and Statistics, PO Box 68 (Gustaf Hällströminkatu 2B), FI-00014 University of Helsinki, Finland

{dagger} Corresponding author. E-mail: j.w.anderson{at}maths.soton.ac.uk


  Abstract

Let {Gamma} be a non-elementary Kleinian group acting on the closed n-dimensional unit ball and assume that its Poincaré series converges at the exponent {alpha}. Let M{Gamma} be the {Gamma}-quotient of the open unit ball. We consider certain families E = {E1, ..., Ep} of open subsets of M{Gamma} such that M{Gamma}\({cup}E isin EE) is compact. The sets Ei are the ends of M{Gamma} and E is a complete collection of ends for M{Gamma}. We associate to each end E isin E an {alpha}-conformal measure such that the measures corresponding to different ends are mutually singular if non-trivial. Additionally, each {alpha}-conformal measure for {Gamma} on the limit set {Lambda}({Gamma}) of {Gamma} can be written as a sum of such conformal measures associated to ends E isin E. In dimension 3, our results overlap with some results of Bishop and Jones (The law of the iterated logarithm for Kleinian groups, Cont. Math. 211 (1997), 17–50.).


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