# Finite subquotients of R. Thompson's group $F$

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.]

• There are very exotic subgroups of $F$. I do not know their finite quotients. – Mark Sapir Aug 16 at 19:34
• @MarkSapir do you have a reference for "exotic" subgroups of $F$? – YCor Aug 17 at 5:34
• 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. – grok Aug 17 at 7:40
• @YCor: See papers by Brin (alone and with others) and papers by Golan (alone and with me). – Mark Sapir Aug 17 at 10:17
• @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. – Mark Sapir Aug 17 at 22:34