A class of spaces that admit no sensitive commutative group actions

We show that a metric space X admits no sensitive commutative group action if it satisfies the following two conditions: (1) X has property S, that is, for each epsilon > 0 there exists a cover of X which consists of finitely many connected sets with diameter less than epsilon; (2) X contains a free n-network, that is, there exists a nonempty open set W in X having no isolated point and n is an element of N such that, for any nonempty open set U subset of W, there is a nonempty connected open set V subset of U such that the boundary partial derivative(x)(V) contains at most n points. As a corollary, we show that no Peano continuum containing a free dendrite admits a sensitive commutative group action. This generalizes some previous results in the literature.