The non-coexistence of distality and expansivity for group actions on infinite compacta

TL;DR

This paper proves that a continuous group action on an infinite compact metric space cannot be both distal and expansive unless the space is finite, highlighting the incompatibility of these properties under certain conditions.

Contribution

It establishes the non-coexistence of distality and expansivity for group actions on infinite compacta, and demonstrates the necessity of finite generation of the group with a counterexample.

Findings

Distal and expansive actions on infinite compacta are incompatible.

Finite generation of the acting group is necessary for the coexistence of distality and expansivity.

Provides a counterexample illustrating the necessity of finite generation.

Abstract

Let $X$ be a compact metric space and $G$ a finitely generated group. Suppose $\phi:G\rightarrow {\rm Homeo}(X)$ is a continuous action. We show that if $\phi$ is both distal and expansive, then $X$ must be finite. A counterexample is constructed to show the necessity of finite generation condition on $G$. This is also a supplement to a result due to Auslander-Glasner-Weiss which says that every distal action by a finitely generated group on a zero-dimensional compactum is equicontinuous.