Highly transitive actions of groups acting on trees

TL;DR

This paper demonstrates that groups acting on trees with certain stabilizer properties can have faithful, highly transitive actions on countable sets, extending known results to broader classes of groups.

Contribution

It introduces the concept of highly core-free subgroups and shows that groups with specific stabilizer conditions admit highly transitive actions, generalizing previous results.

Findings

Groups with finite edge stabilizers and ICC vertex stabilizers have faithful, highly transitive actions.

The notion of highly core-free subgroups is introduced and exemplified.

The results recover and extend known cases such as surface groups.

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.