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Geometrically finite groups, Patterson-Sullivan measures and Ratner's ridigity theorem

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The rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated. Ratner's theorem generalizes to these foliations with respect to the Patterson-Sullivan measure. In the spirit of Mostow, we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups. Finally, we show that the frame flow on geometrically finite hyperbolic manifolds is Bernoulli.

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Authors and Affiliations

  1. Department of Mathematics, California Institute of Technology, 91125, Pasadena, CA, USA

    L. Flaminio

  2. Department of Mathematics, SUNY at Stony Brook, 11794, NY, USA

    R. J. Spatzier

Authors
  1. L. Flaminio
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  2. R. J. Spatzier
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partially supported by NSF Grant No. DMS-820-04024

partially supported by NSF Grant No. DMS-85-02319

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Flaminio, L., Spatzier, R.J. Geometrically finite groups, Patterson-Sullivan measures and Ratner's ridigity theorem. Invent Math 99, 601–626 (1990). https://doi.org/10.1007/BF01234433

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