Groups acting on dendrons (Q291157)

A dendron is a compact, connected space in which every pair of points can be separated by deleting a third point. If \(G\) is a (discrete) group acting on a dendron \(X\) by homeomorphisms and if all orbits contain at least three points, then \(G\) contains a nonabelian free subgroup and the action is strongly proximal, hence \(G\) does not preserve any regular Borel probability measures on \(X\) and is not amenable. It follows from this result that every dendron \(X\) is a von Neumann space, i.e., every group acting on \(X\) by homeomorphisms either contains a nonabelian free subgroup or preserves some regular Borel probability measure.