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A new characterization of Gromov hyperbolicity for negatively curved surfaces.

José M. Rodríguez, Eva Tourís (2006)

Publicacions Matemàtiques

In this paper we show that to check Gromov hyperbolicity of any surface of constant negative curvature, or Riemann surface, we only need to verify the Rips condition on a very small class of triangles, namely, those obtained by marking three points in a simple closed geodesic. This result is, in fact, a new characterization of Gromov hyperbolicity for Riemann surfaces.

A Non-Probabilistic Proof of the Assouad Embedding Theorem with Bounds on the Dimension

Guy David, Marie Snipes (2013)

Analysis and Geometry in Metric Spaces

We give a non-probabilistic proof of a theorem of Naor and Neiman that asserts that if (E, d) is a doubling metric space, there is an integer N > 0, depending only on the metric doubling constant, such that for each exponent α ∈ (1/2; 1), one can find a bilipschitz mapping F = (E; dα ) ⃗ ℝ RN.

A rough curvature-dimension condition for metric measure spaces

Anca-Iuliana Bonciocat (2014)

Open Mathematics

We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as...

Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian

Kei Funano (2016)

Analysis and Geometry in Metric Spaces

We apply Gromov’s ham sandwich method to get: (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in Euclidean space.

Asymptotic isoperimetry of balls in metric measure spaces.

Romain Tessera (2006)

Publicacions Matemàtiques

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of a metric measure space...

Atoroïdalité complète et annulation de l’invariant λ ¯ de Perelman

Pablo Suárez-Serrato (2007/2008)

Séminaire de théorie spectrale et géométrie

On résume les proprietés de l’invariant λ ¯ de Perelman, et en combinaison avec l’invariant de Yamabe on exprime certaines proprietés géométriques des variétés de dimension 3 en fonction de λ ¯ . On décrit des exemples d’annulation de λ ¯ en dimension 4 , où on trouve des liens entre l’effondrement et l’existence de métriques à courbure scalaire positive. On montre qu’une version d’atoroïdalité qu’on appelle atoroïdalité complète est détectée par λ ¯ sur les variétés de courbure négative ou nulle de dimension...

Au bord de certains polyèdres hyperboliques

Marc Bourdon (1995)

Annales de l'institut Fourier

Le cadre de cet article est celui des groupes et des espaces hyperboliques de M.  Gromov. Il est motivé par la question suivante : comment différencier deux groupes hyperboliques à quasi-isométrie près ? On illustre ce problème en détaillant un exemple de M. Gromov issu de Asymptotic invariants for infinite groups. On décrit une famille infinie de groupes hyperboliques, deux à deux non quasi-isométriques, de bord la courbe de Menger. La méthode consiste à étudier leur structure quasi-conforme au...

Coarse topology, enlargeability, and essentialness

Bernhard Hanke, Dieter Kotschick, John Roe, Thomas Schick (2008)

Annales scientifiques de l'École Normale Supérieure

Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K -theory of the corresponding reduced C * -algebras. Our proofs do not depend on the Baum–Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.

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