2025/09/12 by Rosendal, Christian
#Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Logic (math.LO)
paper · doi:10.48550/arxiv.2509.10686
Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose d is a compatible left-invariant metric on an amenable topological group G with no non-trivial homomorphisms to \mathbb R. Then, for every finite subset E⊆ G and ε>0, there is a finitely supported probability measure β on G such that maxg,h∈ E \sf W(βg, βh)lt;ε, where \sf W denotes the Wasserstein distance between probability measures on the metric space (G,d). When d is the word metric on a finitely generated group G, this strengthens a well known theorem of Reiter and, when d is bounded, recovers a result of Schneider and Thom. Furthermore, when G is locally compact, β may be replaced by an appropriate probability density f∈ L1(G). Also, when G\curvearrowright X is a continuous isometric action on a metric space, the space of Lipschitz functions on the quotient X/ /G is isometrically isomorphic to a 1-complemented subspace of the Lipschitz functions on X. And, when additionally G is skew-amenable, there is a G-invariant contraction \mathfrak Lip X \overset S\longrightarrow\mathfrakLip(X/ /G) so that (Sϕ)(Gx)=ϕ(x) whenever ϕ is constant on every orbit of G\curvearrowright X. This latter extends results of Cuth and Doucha from the setting of locally compact or balanced groups.