Determining solubility for finitely generated groups of PL homeomorphisms

The set of finitely generated subgroups of the group PL₊(I) of orientation-preserving piecewise-linear homeomorphisms of the unitinterval includes many important groups, most notably R. Thompson’s group F. Here, we show that every finitely generated subgroup G < PL₊(I) is either soluble, or contains an embedded copy of the finitely generated, non-soluble Brin-Navas group B, affirming a conjecture of the first author from 2009. In the case that G is soluble, we show 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 give an algorithm which determines whether or not a 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 ⟨X⟩. Finally,we give a solution of the membership problem for a particular familyof finitely generated soluble subgroups of any computable subgroup of PL₊(I).