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

Let X be a compact metric space and G a finitely generated group. Suppose φ : G→ Homeo(X) is a continuous action. We show that if φ 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 AuslanderGlasner-Weiss which says that every distal action by a finitely generated group on a zerodimensional compactum is equicontinuous.