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Recall R. Thompson's group $F$ acting on the interval $[0,1]$: it consists of piecewise linear oriented maps with slopes a power of $2$ and dyadic breakpoints.

Is every finite subquotient (= quotient of a subgroup) soluble?

[my money is on "yes". Finite quotients of $F$ itself are abelian. I know subgroups of $F$ that are made of copies of $F$ and of soluble groups, but nothing more exotic.]

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  • $\begingroup$ There are very exotic subgroups of $F$. I do not know their finite quotients. $\endgroup$
    – user6976
    Commented Aug 16, 2019 at 19:34
  • $\begingroup$ @MarkSapir do you have a reference for "exotic" subgroups of $F$? $\endgroup$
    – YCor
    Commented Aug 17, 2019 at 5:34
  • $\begingroup$ I'm aware of a rich collection of f.g. subgroups, see pi.math.cornell.edu/~justin/Ftp/complexity_subgrp_F.pdf but they're all elementary amenable. $\endgroup$
    – grok
    Commented Aug 17, 2019 at 7:40
  • $\begingroup$ @YCor: See papers by Brin (alone and with others) and papers by Golan (alone and with me). $\endgroup$
    – user6976
    Commented Aug 17, 2019 at 10:17
  • $\begingroup$ @grok: Yes, all known subgroups of $F$ are either elementary amenable or contains a copy of $F$. But all finite groups are elementary amenable, so it does not answer your question. $\endgroup$
    – user6976
    Commented Aug 17, 2019 at 22:34

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