Concentration of maps and group action

TL;DR

This paper investigates how compact and Lévy group actions influence the concentration properties of various metric spaces, including R-trees, doubling spaces, metric graphs, and Hadamard manifolds.

Contribution

It introduces a new perspective using concentration theory to analyze group actions on diverse metric spaces, expanding understanding of their geometric and probabilistic properties.

Findings

Group actions induce concentration phenomena in the studied spaces.

Lévy groups exhibit strong concentration effects on these metric spaces.

Results unify and extend previous work on group actions and metric space geometry.

Abstract

In this paper, from the viewpoint of the concentration theory of maps, we study a compact group and a L\'{e}vy group action to a large class of metric spaces, such as R-trees, doubling spaces, metric graphs, and Hadamard manifolds.