Commensurating actions and self-similar groups

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 Sierpiński 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.

Disclosure

“the mathematics in this paper was carried out by humans; no AI was involved in the proofs or in the search for ideas or references. Once the first draft was complete, we made limited use of AI to revise and improve the presentation: • Mistral AI (CNRS subscription) helped drafting parts of the introduction; • Claude Opus 4.8 (Pro plan) was asked to produce a full referee report, which helped us find some typos and minor inaccuracies (none affecting the core validity o”

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2026-08-23T18:53:55.839250Z

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