Extensions of amenable groups by recurrent groupoids

We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This applies to a wide class of groups, amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk’s group, Basilica group, groups associated with Fibonacci and Penrose tilings, the topological full groups of Cantor minimal systems, groups acting on rooted trees by bounded automorphisms, groups generated by finite automata of linear activity growth, and groups naturally appearing in holomorphic dynamics.