We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-\'Emery inequality for the corresponding sub-Laplacian implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generali...
Roughly speaking, a map between metric spaces is asymptotically Möbius if it induces quasi-Möbius maps on asymptotic cones. We show that under such maps, some large-scale notions of dimension increases: asymptotic dimension for finitely generated nilpotent groups, telescopic dimension for CAT (0) spaces.
We study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. ...
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In this work, we approach the interesting problem of representing and studying the position, velocity, acceleration and arc-length of trajectories of points defined in a two-dimensional geometrical space, which has been chosen for being simpler than three-dimensional spaces while retaining much of their structure, and richer than the one-dimensiona...
Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.
Using Patterson-Sullivan measures we investigate growth problems for groups acting on a metric space with a strongly contracting element.
The median of a set of vertices $P$ of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all vertices of $P$. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medi...
Any Lipschitz map $f\colon M \to N$ between metric spaces can be ``linearised'' in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$. The purpose of this note is to explore the connections between the injectivity of $f$ and the injectivity of $...
