Determining solubility for finitely generated groups of PL homeomorphisms

Collin Bleak, Tara Brough, Susan Hermiller

Published 2015-07-24Version 1

The set of finitely generated subgroups of the group $PL_+(I)$ of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson's group $F$. In this paper we show that every finitely generated subgroup $G<PL_+(I)$ is either soluble, or contains an embedded copy of Brin's group $B$, a finitely generated, non-soluble group, which verifies a conjecture of the first author from 2009. In the case that $G$ is soluble, we show that the derived length of $G$ is bounded above by the number of $G$-orbit classes of the breakpoints of any finite set of generators. Finally, we specify a set of `computable' subgroups of $PL_+(I)$ (which includes R. Thompson's group $F$) and we give an algorithm which determines in finite time whether or not any given finite subset $X$ of such a computable group generates a soluble group. When the group is soluble, the algorithm also determines the derived length of $\langle X\rangle$.