Determining solubility for finitely generated groups of PL homeomorphisms
MetadataShow full item record
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).
Bleak , C , Brough , T & Hermiller , S 2021 , ' Determining solubility for finitely generated groups of PL homeomorphisms ' , Transactions of the American Mathematical Society , vol. 374 , no. 10 , pp. 6815-6837 . https://doi.org/10.1090/tran/8421
Transactions of the American Mathematical Society
Copyright © 2021 American Mathematical Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at http://www.ams.org/journals/tran/
DescriptionFunding: The first and second authors were partially supported by EPSRC grant EP/H011978/1. The third author was partially supported by grants from the Simons Foundation (#245625) and the National Science Foundation (DMS-1313559)
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.