The generation problem in Thompson group $F$
The generation problem in Thompson group $F$
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汤普森群 $F$ 的生成问题
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Gili Golan
中科院分区：
文献类型：
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作者：
Gili Golan
我们证明了Thompson群$F$中的生成问题是可判定的，即，有一种算法可以判断F的元素的有限集合是否生成整个F。该算法利用$F$的子群的Stallings $2$-核，它可以以类似于有限生成自由群的子群的Stallings核的方式定义。进一步研究$F$子群的Stallings $2$-核，为$F$中的另一个算法问题提供了解决方案。也就是说，给定$F$的一个n-生成子群$H$，如果$H$传递地作用在有限并元分式集$mathcal D$上，则它是可判定的。研究的其他应用包括构造新的极大子群的$F$的无限指数，其中，一个极大子群的无限指数的传递作用在集合$mathcal D$和建设的初等顺从子群的$F$这是极大的正规子群的$F $。
We show that the generation problem in Thompson group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of $F$, which can be defined in an analogue way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings $2$-core of subgroups of $F$ provides a solution to another algorithmic problem in $F$. Namely, given a finitely generated subgroup $H$ of $F$, it is decidable if $H$ acts transitively on the set of finite dyadic fractions $mathcal D$. Other applications of the study include the construction of new maximal subgroups of $F$ of infinite index, among which, a maximal subgroup of infinite index which acts transitively on the set $mathcal D$ and the construction of an elementary amenable subgroup of $F$ which is maximal in a normal subgroup of $F$.