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.

non-amenable. $\endgroup$ – YCor Jul 8 '18 at 13:47