Isometric rigidity of Wasserstein spaces: The graph metric case

Abstract

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