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- W4291250892 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 $F$. In this paper we show that every finitely generated subgroup $G<PL_+(I)$ is either soluble, or contains an embedded copy of Brin's group $B$, a finitely generated, non-soluble group, which verifies a conjecture of the first author from 2009. In the case that $G$ is soluble, we show that 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 we give an algorithm which determines in finite time whether or not any 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 $langle Xrangle$. Finally, we give a solution of the membership problem for a family of finitely generated soluble subgroups of any computable subgroup of $PL_+(I)$." @default.
- W4291250892 created "2022-08-13" @default.
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- W4291250892 date "2015-07-24" @default.
- W4291250892 modified "2023-10-16" @default.
- W4291250892 title "Determining solubility for finitely generated groups of PL homeomorphisms" @default.
- W4291250892 doi "https://doi.org/10.48550/arxiv.1507.06908" @default.
- W4291250892 hasPublicationYear "2015" @default.
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