Commensurating actions and self-similar groups

Commensurating actions govern how a group can act on CAT(0) cube complexes. We classify them for every finitely generated contracting self-similar branch group. As a main application, we establish Property FW for the iterated monodromy group of the subdivision rule generating the classical square Sierpi\'nski carpet. This is the first example of an infinite finitely generated amenable group with Property FW, answering a question of Cornulier. As another application, we show that a contracting self-similar regular branch group does not have Property PW. In particular, the first Grigorchuk group does not have Property PW, answering another question in the literature.