Determining solubility for finitely generated groups of PL homeomorphisms

TL;DR

This paper classifies finitely generated subgroups of $PL_+(I)$ as either soluble or containing a non-soluble subgroup, providing algorithms to determine solubility, derived length, and membership within these groups.

Contribution

It proves a dichotomy for finitely generated subgroups of $PL_+(I)$, introduces a computable subgroup class, and develops algorithms for solubility and membership problems.

Findings

Every finitely generated subgroup is either soluble or contains an embedded copy of Brin's group B.

An algorithm determines solubility and derived length in finite time for certain subgroups.

The membership problem is solved for a family of soluble subgroups.

Abstract

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 breakpoints of any finite set of generators. 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 $\langle X\rangle$. Finally, we give a solution of the membership problem for a family of finitely generated soluble subgroups of any computable subgroup of $PL_+(I)$.