Determining solubility for finitely generated groups of PL homeomorphisms
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- by Collin Bleak, Tara Brough and Susan Hermiller PDF
- Trans. Amer. Math. Soc. 374 (2021), 6815-6837 Request permission
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 $F$. 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 $\langle X\rangle$. Finally, we give a solution of the membership problem for a particular family of finitely generated soluble subgroups of any computable subgroup of $PL_+(I)$.References
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Additional Information
- Collin Bleak
- Affiliation: School of Mathematics and Statistics, University of St. Andrews, North Haugh St Andrews, Fife KY16 9SS, Scotland
- MR Author ID: 831679
- ORCID: 0000-0001-5790-1940
- Email: cb211@st-andrews.ac.uk
- Tara Brough
- Affiliation: Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829–516 Caparica, Portugal
- MR Author ID: 1022829
- ORCID: 0000-0002-3576-0670
- Email: t.brough@fct.unl.pt
- Susan Hermiller
- Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
- MR Author ID: 311019
- Email: hermiller@unl.edu
- Received by editor(s): November 2, 2017
- Received by editor(s) in revised form: July 31, 2020
- Published electronically: July 19, 2021
- Additional Notes: The second author was supported by postdoctoral funding from the University of St Andrews, and by the FCT (Fundação para a Ciência e a Tecnologia / Portuguese Foundation for Science and Technology) fellowship SFRH/BPD/121469/2016 and the FCT projects UID/MAT/00297/2019 (Centro de Matemática e Aplicações) and PDTC/MAT-PUR/31174/2017.
The third author was partially supported by grants from the Simons Foundation (#245625) and the National Science Foundation (DMS-1313559). - © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 6815-6837
- MSC (2020): Primary 20F10; Secondary 20F16, 37C25
- DOI: https://doi.org/10.1090/tran/8421
- MathSciNet review: 4315590