Question: Do there exist amenable Thompson-like groups?

I realise that my question is vague, but defining and studying groups which look like Thompson's groups $F$, $T$ and $V$ seems to be an independent field of research in the litterature, so my question should make sense.

Among the examples of such groups I know, typically either they contain a non-abelian free group or they contain Thompson's group $F$. In the former case, of course the group is not amenable; and in the latter case, the amenability of the entire group would imply the amenability of $F$, so a proof of the amenability should not be available.

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    It's indeed very vague, especially given the number of inequivalent ways to define Thompson's groups. At least some full-topological groups including ones occurring as subgroups of IET have been proved (by Juschenko-Monod + generalizations) to be amenable (while they're not elementary amenable). – YCor Jul 8 at 13:41
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    Also some piecewise projective f.g. groups of self-homeomorphisms of the segments, very similar to Thompson's $F$ have been proved by Monod to be non-amenable. – YCor Jul 8 at 13:47

There are two types of groups which are sometimes considered similar to the R. Thompson group $F$.

  1. Some "fractal groups" acting on rooted trees. One of the "closest" to $F$ groups in this class is probably the Basilica group discovered by Grigorchuk and Zuk whose amenability was proved by Bartholdi and Virag. You can try finding lectures by Grigorchuk where he advertised the amenability problem for the Basilica group and its similarity to $F$. The reason for similarity is that both $F$ and the Basilica group act nicely on the Cantor set.

  2. Topological full groups of minimal subshifts. Their amenability was proved by Juschenko and Monod. The reason for similarity is that $F$ can also be viewed as topological full group of some amenable (but not cyclic) group (an affine group of homeomorphisms of $\mathbb{Z}[1/2]$) acting on a (non-compact) metric space.

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