HIGHLY TRANSITIVE ACTIONS OF GROUPS ACTING ON TREES

Abstract. We show that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set. This result is actually true for infinite vertex stabilizers and some more general, finite of infinite, edge stabilizers that we call highly core-free. We study the notion of highly core-free subgroups and give some examples. In the case of amalgamated free products over highly core-free subgroups and HNN extensions with highly core-free base groups we obtain a genericity result for faithful and highly transitive actions. In particular, we recover the result of D. Kitroser stating that the fundamental group of a closed, orientable surface of genus g> 1 admits a faithful and highly transitive action. An action of a countable group Γ on an infinite countable set X is called highly transitive if it is n-transitive for all n ≥ 1. It is easy to see that an action Γ y X is highly transitive if and only if the image of Γ in the Polish group S(X) of bijections of X is dense. An obvious example of a highly transitive and faithful action is given by the action of the countable group of finitely supported permutations of X. This group is not finitely generated and far from being free (it is amenable)