Concentration of maps and group actions

Abstract

In this paper we study actions of compact groups and of Lévy groups on a large class of metric spaces, such as R-trees, doubling spaces, metric graphs, and Hadamard manifolds, from the viewpoint of the theory of concentration of maps.

Concentration of 1-Lipschitz maps into an infinite dimensional $\ell^p$-ball with $\ell^q$-distance function

“…As applications of Theorem 1.1, by virtue of [3,Propositions 4.3 and 4.4], we obtain the following corollaries of a Lévy group action. A Lévy group was first introduced by Gromov and Milman in [10].…”

“…The Lévy families were first introduced and analyzed by Gromov and Milman in [10]. In our previous works [2], [3], [4], [5], the author proved that if a metric space Y is either an R-tree, a doubling space, a metric graph, or a Hadamard manifold, then lim n→∞ ObsDiam Y (X n ; −κ) = 0 holds for any κ > 0 and any Lévy family {X n } ∞ n=1 . To prove these results, we needed to assume the finiteness of the dimension of the target metric spaces.…”

Equivariant concentration in topological groups

“…The first estimates for orbit diameters, concerning Hölder actions on compact metric spaces, in terms of the isoperimetric behavior of the acting group and covering properties of the phase space belong to Milman [Mil87]. For generalizations of Milman's results as well as corresponding estimates for actions of Lévy groups on a certain class of non-compact metric spaces, we refer to Funano's work [Fun10].…”

Concentration of $1$-Lipschitz maps into an infinite dimensional $\ell ^p$-ball with the $\ell ^q$-distance function

“…The Lévy families were first introduced and analyzed by Gromov and Milman in [10]. In previous works [2], [3], [4], [5], the author proved that if a metric space Y is either an R-tree, a doubling space, a metric graph, or a Hadamard manifold, then lim n→∞ ObsDiam Y (X n ; −κ) = 0 holds for any κ > 0 and any Lévy family {X n } ∞ n=1 . To prove these results, we needed to assume the finiteness of the dimension of the target metric spaces.…”