A Minimal Non-solvable Group of Homeomorphisms

Abstract

Let $P{L}_0(I)$ 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 $P{L}_0(I)$. We find mild conditions under which other non-solvable subgroups ($B$, $(\wr \mathbb{Z}\wr {)}^{\infty }$, $(\mathbb{Z}\wr {)}^{\infty }$, and ${}^{\infty }(\wr \mathbb{Z})$) embed in subgroups of $P{L}_0(I)$. We show that all solvable subgroups of $P{L}_0(I)$ embed in all non-solvable subgroups of $P{L}_0(I)$. These results continue to apply if we replace $P{L}_0(I)$ by any generalized R. Thompson group $F_n$.

29 pages, 1 figure

References in corpus (2)

• Automorphisms of generalized Thompson groups
• Solvability in Groups of Piecewise-linear Homeomorphisms of the Unit Interval