group theory

Commensurating actions and self-similar groups

arXiv:2607.13776

summary

The authors classify commensurating actions of finitely generated contracting self‑similar branch groups on CAT(0) cube complexes and use this classification to prove that the iterated monodromy group of the square Sierpiński carpet has Property FW (the first amenable example) and that such groups do not have Property PW.

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.

34 pages, 3 figures. v2: minor revision

Topics & keywords

#self-similar groups#branch groups#property fw#property pw#cube complexescommensurating actionscontracting self-similarregular branch groupamenable groupiterated monodromy group
Commensurating actions and self-similar groups · wovepaper