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

DOI10.1007/BF01234433zbMath0667.57014OpenAlexW2119461448WikidataQ125924547 ScholiaQ125924547MaRDI QIDQ1117506

Ralf J. Spatzier, Livio Flaminio

Publication date: 1990

Published in: Inventiones Mathematicae (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/143772

zbMATH Keywords

Patterson-Sullivan measureframe flownonexistence of invariant measurable distributions on the boundary of hyperbolic spacerigidity of the horospherical foliations of geometrically finite hyperbolic manifolds

Mathematics Subject Classification ID

Dynamics induced by flows and semiflows (37C10) Discrete subgroups of Lie groups (22E40) Discontinuous groups of transformations (57S30) Foliations in differential topology; geometric theory (57R30)

Related Items (15)

Closed geodesics and holonomies for Kleinian manifolds ⋮ Classification of joinings for Kleinian groups ⋮ Mixing of frame flow for rank one locally symmetric spaces and measure classification ⋮ Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps ⋮ Torus counting and self-joinings of Kleinian groups ⋮ A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds ⋮ Equidistribution and counting for orbits of geometrically finite hyperbolic groups ⋮ Apollonian circle packings: dynamics and number theory ⋮ Joining measures for horocycle flows on abelian covers ⋮ Topological proof of Benoist-Quint's orbit closure theorem for \(\mathrm{SO}(d, 1) \) ⋮ Asymptotic Counting in Conformal Dynamical Systems ⋮ Spectral theory of the frame flow on hyperbolic 3-manifolds ⋮ Ergodicity of unipotent flows and Kleinian groups ⋮ Invariant Radon measures for Unipotent flows and products of Kleinian groups ⋮ Matrix coefficients, counting and primes for orbits of geometrically finite groups

Cites Work

• Classifying the isometric extensions of a Bernoulli shift
• Horocycle flows, joinings and rigidity of products
• Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
• The density at infinity of a discrete group of hyperbolic motions
• The limit set of a Fuchsian group
• Rigidity of horocycle flows
• Factors of horocycle flows
• An extension of Ratner's rigidity theorem to n-dimensional hyperbolic space
• Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature
• Zero-Entropy Affine Maps on Homogeneous Spaces
• Strong Rigidity of Locally Symmetric Spaces. (AM-78)
• Rigidity of some translations on homogeneous spaces
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