A finitely generated group embeds in Thompson's group V if and only if it is the transition group of a context-free graph, which rules out intermediate-growth groups and the Basilica and Hanoi Towers groups.
Commensurating actions and self-similar groups
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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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A graph-theoretical characterisation of subgroups of Thompson's group $V$
A finitely generated group embeds in Thompson's group V if and only if it is the transition group of a context-free graph, which rules out intermediate-growth groups and the Basilica and Hanoi Towers groups.