Isometric Actions and Finite Approximations

TL;DR

This paper demonstrates that all isometric actions on Cantor sets can be represented as inverse limits of finite set actions, and that actions by finitely generated amenable groups are residually finite.

Contribution

It establishes a new connection between isometric actions on Cantor sets and inverse limits, and proves residual finiteness for actions of finitely generated amenable groups.

Findings

Isometric actions on Cantor sets are conjugate to inverse limits of finite actions.

Finitely generated amenable group actions are residually finite.

Provides a structural understanding of isometric actions on Cantor sets.

Abstract

We show that every isometric action on a Cantor set is conjugate to an inverse limit of actions on finite sets; and that every isometric action by a finitely generated amenable group is residually finite.