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arXiv · 2607.13776

Commensurating actions and self-similar groups

Nicolás Matte Bon·Volodymyr Nekrashevych·Tianyi Zheng
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Abstract

Commensurating actions govern how a group can act on non-positively curved cube complexes. We obtain a complete picture of them for a class of finitely generated groups acting on rooted trees: contracting self-similar branch groups. The main application is a proof of 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 Grigorchuk group does not have Property PW, answering another question in the literature.

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Nicolás Matte Bon, Volodymyr Nekrashevych, Tianyi Zheng. 2026-07-15. Commensurating actions and self-similar groups. https://arxiv.org/abs/2607.13776

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