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 $PL_+(I)$.

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中文翻译：

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确定有限生成的 PL 同胚群的溶解度

摘要：单位区间的保向分段线性同胚群$PL_+(I)$的有限生成子群的集合包括许多重要的群，最著名的是R. Thompson群$F$。在这里，我们证明每个有限生成的子群 $G<PL_+(I)$ 要么是可解的，要么包含有限生成的、不可解的 Brin-Navas 群 $B$ 的嵌入副本，证实了第一作者的猜想从 2009 年开始。在 $G$ 可解的情况下，我们表明 $G$ 的导出长度受任何有限生成器集的断点数量的限制。我们指定了一组 $PL_+(I)$（包括 R. Thompson 群 $F$）的“可计算”子群，并给出了一个算法来确定这样一个可计算群的给定有限子集 $X$ 是否生成可溶性基团。当基团溶解时，该算法还确定了 $\langle X\rangle$ 的导出长度。最后，我们给出了$PL_+(I)$ 的任何可计算子群的有限生成的可溶子群的特定族的成员资格问题的解决方案。

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