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Dimension conforme et sphère à l’infini des variétés à courbure négative. (Conformal dimension and the ideal boundary of manifolds with negative curvature). (French) Zbl 0722.53028

It is shown that the ideal boundary of a Hadamard manifold M carries a natural quasiconformal structure if the curvature of M is negatively pinched or M has a cocompact group of isometries. For any topological space X carrying a quasiconformal structure \(\beta\), a quasiconformal invariant that generalizes the notion of modulus of a curve family is introduced and used to define the conformal dimension of (X,\(\beta\)). This dimension increases under quasiconformal imbedding; for example, a quasiisometric imbedding between Hadamard manifolds of pinched negative curvature extends to a quasiconformal imbedding between their ideal boundaries. Calculations on the conformal dimension at infinity yield a lower bound for the pinching of negatively curved Riemannian metrics carried by compact quotients of rank one symmetric spaces, and a sharp lower bound for the Hausdorff dimension of the limit set of certain quasiconformal groups.

MSC:

53C20 Global Riemannian geometry, including pinching
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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