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.
Categories: math.DS