(\mathcal{Z}\)-stability of crossed products by topological full groups

We introduce the property of having good subgroups for actions of countable discrete groups on compact metrizable spaces. We show that it implies comparison for actions of amenable groups and amenable actions of groups containing the free group on two generators. As a result, free actions on finite-dimensional spaces of topological full groups of many étale groupoids are almost finite whenever the group is amenable, or have comparison if instead it contains a free group on two generators and the action is minimal and amenable. In particular, free minimal actions give rise to classifiable crossed products in both of the above cases.