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

Abstract

Let $X$ be a compact metric space and $G$ a finitely generated group. Suppose $\varphi :G\to \mathrm{Homeo}(X)$ is a continuous action. We show that if $\varphi$ 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.

References in corpus (3)

• Distal higher rank lattice actions on surfaces
• Continua having distal minimal actions by amenable groups
• The nonexistence of expansive actions on Suslinian continua by groups of subexponential growth