## A class of spaces that admit no sensitive commutative group actions

### Volume 217 / 2012

#### Abstract

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 $\varepsilon>0$ there exists a cover of $X$ which consists of finitely many connected sets with diameter less than $\varepsilon$; (2) $X$ contains a free $n$-network, that is, there exists a nonempty open set $W$ in $X$ having no isolated point and $n\in\mathbb N$ such that, for any nonempty open set $U\subset W$, there is a nonempty connected open set $V\subset U$ such that the boundary $\partial_X(V)$ 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.