Isometric rigidity of Wasserstein spaces: the graph metric case

TL;DR

This paper proves that Wasserstein spaces over countable graph metric spaces are isometrically rigid for all p ≥ 1, and constructs such spaces with prescribed isometry groups, revealing deep geometric and group-theoretic properties.

Contribution

It establishes isometric rigidity of Wasserstein spaces over countable graph metrics and constructs spaces with arbitrary countable groups as their isometry groups.

Findings

Wasserstein spaces over countable graph metrics are isometrically rigid for all p ≥ 1.

Existence of Wasserstein spaces with isometry groups isomorphic to any given countable group.

The results connect geometric properties of Wasserstein spaces with algebraic group structures.

Abstract

The aim of this paper is to prove that the $p$-Wasserstein space $\mathcal{W}_p(X)$ is isometrically rigid for all $p\geq 1$ whenever $X$ is a countable graph metric space. As a consequence, we obtain that for every countable group $H$ and any $p\geq 1$ there exists a $p$-Wasserstein space whose isometry group is isomorphic to $H$.