Determining solubility for finitely generated groups of PL homeomorphisms

Abstract

The set of finitely generated subgroups of the group P L + ( I ) PL_+(I) of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson’s group F F . Here, we show that every finitely generated subgroup G > P L + ( I ) G>PL_+(I) is either soluble, or contains an embedded copy of the finitely generated, non-soluble Brin-Navas group B B , affirming a conjecture of the first author from 2009. In the case that G G is soluble, we show the derived length of G G is bounded above by the number of breakpoints of any finite set of generators. We specify a set of ‘computable’ subgroups of P L + ( I ) PL_+(I) (which includes R. Thompson’s group F F ) and give an algorithm which determines whether or not a given finite subset X X of such a computable group generates a soluble group. When the group is soluble, the algorithm also determines the derived length of ⟨ X ⟩ \langle X\rangle . Finally, we give a solution of the membership problem for a particular family of finitely generated soluble subgroups of any computable subgroup of P L + ( I ) PL_+(I) .