On finite local approximations of isometric actions of residually finite groups

We show that any isometric action of a residually finite group admits approximate local finite models. As a consequence, if $G$ is residually finite, every isometric $G$-action embeds isometrically into a metric ultraproduct of finite isometric $G$-actions.

💡 Research Summary

The paper investigates finite local approximations of isometric actions of residually finite groups. The authors prove that for any residually finite group (G) and any isometric action (\phi\colon G\curvearrowright (X,d)), one can approximate the action on any prescribed finite subset (X_{0}\subset X) and any finite set of group elements (A\subset G) up to an arbitrary error (\varepsilon>0) by a genuine isometric action on a finite metric space. Concretely, there exist a finite metric space ((Y,\eta)), an isometric action (\psi\colon G\curvearrowright Y) and a map (f\colon X_{0}\to Y) such that for all (g,h\in A) and (x,y\in X_{0}),

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