On finite local approximations of isometric actions of residually finite groups

TL;DR

This paper demonstrates that isometric actions of residually finite groups can be approximated locally by finite models, enabling their embedding into ultraproducts of finite actions, thus bridging infinite and finite group actions.

Contribution

It introduces a method to approximate isometric actions of residually finite groups locally by finite models and embeds these actions into ultraproducts of finite isometric actions.

Findings

Any isometric action of a residually finite group admits approximate local finite models.

Such actions can be embedded into metric ultraproducts of finite isometric actions.

The results connect infinite group actions with finite approximations via ultraproducts.

Abstract

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.