不可约快速区间同胚集生成汤普森群\(F_n\)的副本
Irreducible fast sets of bump homeomorphisms generate copies of Thompson's groups $F_n$
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中文总结 AI 辅助
研究由不可约几何快速正凸起集生成的群，证明对于\(n\geq2\)，由\(n\)个此类集合生成的群同构于\(n\)元汤普森群\(F_n\)，回答了布林和扎雷姆斯基提出问题的强版本。
中文摘要 AI 辅助
区间的同胚映射若其支撑集为单个开区间且在此区间上每个点都向右移动，则称为正凸起。为正凸起\(b\)在其支撑集上的作用选择一个基本域\([m,b(m))\)，会将支撑集的其余部分分成两个区间，称为\(b\)的脚。若能选择基本域使所有得到的脚两两不相交，则有限正凸起集在几何上是快速的。这样一个集合的交叉图以凸起为顶点，当两个凸起的支撑集重叠但不嵌套时它们相邻，若交叉图连通则该集合是不可约的。我们证明，对于每个\(n\geq2\)，由\(n\)个不可约几何快速正凸起集生成的每个群都同构于\(n\)元汤普森群\(F_n\)。这回答了布林和扎雷姆斯基提出的一个问题（奥伯沃尔法赫报告15（2018））的强版本。
英文摘要
A homeomorphism of an interval is a positive bump if its support is a single open interval on which it moves every point to the right. Choosing a fundamental domain $[m,b(m))$ for the action of a positive bump $b$ on its support splits the remainder of the support into two intervals, called the feet of $b$. A finite set of positive bumps is geometrically fast if fundamental domains can be chosen so that all the resulting feet are pairwise disjoint. The crossing graph of such a set has the bumps as its vertices, two bumps being adjacent whenever their supports overlap but are not nested, and the set is irreducible if its crossing graph is connected. We prove that for every $n\geq 2$, every group generated by an irreducible geometrically fast set of $n$ positive bumps is isomorphic to the $n$-ary Thompson group $F_n$. This answers the strong version of a problem posed by Brin and Zaremsky (Oberwolfach Rep. 15 (2018)).
