Groups Acting on Dendrons

Abstract

A dendron is defined as a continuum (a nonempty, connected, compact Hausdorff space) in which every two distinct points have a separation point. It is proved that if a group G acts on a dendron D by homeomorphisms, then either D contains a G-invariant subset consisting of one or two points or G contains a free noncommutative subgroup and, furthermore, the action is strongly proximal.