The nonexistence of expansive commutative group actions on Peano continua having free dendrites

A dendrite D in a metric space X is said to be free if there exists a connected open set U in X such that U ¯ = D . In this paper, we prove that there is no expansive commutative group action on any Peano continuum having a free dendrite. In particular, no 1-dimensional compact ANR admits an expansive commutative group action.