A minimal non-solvable group of homeomorphisms

Abstract

Let ${\mathrm{PL}}_o(I)$ represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group $W$ and show that $W$ embeds in every non-solvable subgroup of ${\mathrm{PL}}_o(I)$. We find mild conditions under which other non-solvable subgroups $B$, $(\wr Z\wr {)}^{\infty }$, $(Z\wr {)}^{\infty }$, and ${}^{\infty }(\wr Z)$ embed in subgroups of Let ${\mathrm{PL}}_o(I)$. We show that all solvable subgroups of ${\mathrm{PL}}_o(I)$ embed in all non-solvable subgroups of PLo(I). These results continue to apply if we replace ${\mathrm{PL}}_o(I)$ by any generalized Thompson group $F_n$.