Lipschitz-Free Spaces: A Topometric Approach and Group Actions (2408.15208v6)

Abstract: We investigate actions of topological groups $G$ on Lipschitz-free spaces $\mathcal{F}(M)$, induced by isometric actions on pointed metric spaces $M$. In particular, we study the associated dynamical $G$-systems under the weak-star topology, focusing on the dual action on $\mathrm{Lip}_0(M) = \mathcal{F}(M)^*$ and the bidual $\mathcal{F}(M)^{**}$. Two natural compactification constructions are considered: the metric compactification of isometric $G$-spaces over pointed metric spaces, and the Gromov $G$-compactification of bounded metric $G$-spaces. We show that for every bounded stable metric $G$-space $(M,d,\mathbf{0})$, the corresponding metric $G$-compactification defines a weakly almost periodic $G$-flow. We examine some metric versions of amenability. We introduce also a topometric version of Lipschitz-free spaces and study its universal property.