On differentiable compactifications of the hyperbolic space

The group of direct isometries of the real n-dimensional hyperbolic space is G=SOo(n,1). This isometric action admits many differentiable compactifications into an action on the closed ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the classifications of real analytic and smooth compactifications.Comment: 11

On Lipschitz compactifications of trees

We study the Lipschitz structures on the geodesic compactification of a regular tree, that are preserved by the automorphism group. They are shown to be similar to the compactifications introduced by William Floyd, and a complete description is given.Comment: 6 page

Approximation by finitely supported measures

Given a compactly supported probability measure on a Riemannian manifold, we study the asymptotic speed at which it can be approximated (in Wasserstein distance of any exponent p) by finitely supported measure. This question has been studied under the names of ``quantization of distributions'' and, when p=1, ``location problem''. When p=2, it is linked with Centroidal Voronoi Tessellations.Comment: v2: the main result is extended to measures defined on a manifold. v3: references added. 25 pp. To appear in ESAIM:COC

On differentiable compactifications of the hyperbolic plane and algebraic actions of SL(2;R) on surfaces

18 p.International audienceIt is known that the hyperbolic plane admits a countable infinity of compactifications into a closed disk such that the isometric action of SL(2;R) acts analytically on the compactified space. We prove that among those compactifications, only the two most classical ones (namely the closures of Poincaré's disk and Klein's disk) are algebraic, that is to say obtained as a union of orbits of a projectivized linear representation of SL(2;R). More generally, we classify all algebraic actions of SL(2;R) on surfaces