The Generation Problem in Thompson Group 𝐹

IF 2 4区数学 Q1 MATHEMATICS Memoirs of the American Mathematical Society Pub Date : 2023-12-01 DOI:10.1090/memo/1451

Gili Golan Polak

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引用次数: 0

Abstract

We show that the generation problem in Thompson’s group F F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F F generates the whole F F . The algorithm makes use of the Stallings 2 2 -core of subgroups of F F , which can be defined in an analogous way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings 2 2 -core of subgroups of F F provides a solution to another algorithmic problem in F F . Namely, given a finitely generated subgroup H H of F F , it is decidable if H H acts transitively on the set of finite dyadic fractions D \mathcal D . Other applications of the study include the construction of new maximal subgroups of F F of infinite index, among which, a maximal subgroup of infinite index which acts transitively on the set D \mathcal D and the construction of an elementary amenable subgroup of F F which is maximal in a normal subgroup of F F .

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汤普森群𝐹的生成问题

我们证明了Thompson群F F中的生成问题是可判定的，即存在一种算法来判定F F的有限元素集是否生成整个F F。该算法利用了F - F的子群的Stallings 22核，它可以用类似于有限生成自由群的子群的Stallings核的方式来定义。对F - F子群的Stallings - 22核的进一步研究，为F - F中的另一个算法问题提供了解决方案。即，给定F F的有限生成子群H H，当H H传递作用于有限并进分数集D \数学D时，H H是可判定的。本研究的其他应用还包括构造无穷指标F的新的极大子群，其中一个传递作用于集合D \数学D的无穷指标F的极大子群，以及F F的一个初等可服从子群的构造，该子群在F F的一个正则子群中是极大的。

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来源期刊

Memoirs of the American Mathematical Society 数学-数学

CiteScore

3.50

自引率

5.30%

发文量

审稿时长

>12 weeks

期刊介绍： Memoirs of the American Mathematical Society is devoted to the publication of research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers or groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the AMS. To be accepted by the editorial board, manuscripts must be correct, new, and significant. Further, they must be well written and of interest to a substantial number of mathematicians.

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