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Abstract
The set of finitely generated subgroups of the group PL_{+}(I) of orientationpreserving piecewiselinear 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, nonsoluble BrinNavas 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).
Original language  English 

Pages (fromto)  68156837 
Number of pages  23 
Journal  Transactions of the American Mathematical Society 
Volume  374 
Issue number  10 
DOIs  
Publication status  Published  19 Jul 2021 
Keywords
 Piecewise linear homeomorphism
 Thompson's group
 Soluble
 Membership problem
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Dive into the research topics of 'Determining solubility for finitely generated groups of PL homeomorphisms'. Together they form a unique fingerprint.Projects
 1 Finished

Automata Languages Decidability: Automata, Languages, Decidability in Algebra
1/03/10 → 31/05/14
Project: Standard