Amenability, Optimal Transport and Abstract Ergodic Theorems

Abstract: 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 $E\subseteq G$ and $\epsilon>0$, there is a finitely supported probability measure $\beta$ on $G$ such that $$ \max_{g,h\in E}\, {\sf W}(\beta g, \beta h)<\epsilon, $$ 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, $\beta$ may be replaced by an appropriate probability density $f\in L^1(G)$. 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 $$ \mathfrak {Lip}\, X \overset S\longrightarrow\mathfrak{Lip}(X/!!/G) $$ so that $(S\phi\big)\big(\overline{Gx}\big)=\phi(x)$ whenever $\phi$ 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.