Complexity among the finitely generated subgroups of Thompson's group
Complexity among the finitely generated subgroups of Thompson's group
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DOI：
10.4171/jca/49
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发表时间：
2017-11
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通讯作者：
C. Bleak;Matthew G. Brin;J. Moore
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文献类型：
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作者：
C. Bleak;Matthew G. Brin;J. Moore
本文证明了RichardThompson群F$的一个由嵌入关系严格良序的群族的存在性。除了这个族的最大元素（它是$F$本身）之外，所有的元素都是初等顺从群。事实上，我们还得到，对于每个$\alpha <\epsilon_0 $，一个生成的初等服从子群的$F$，其EA类是$\alpha + 2$。这些群都有简单、明确的描述，可以看作是以$\mathbf{Z} + \mathbf {Z}$、$\mathbf{Z} \wr \mathbf {Z}$和Brin-Navas群$B$开始的级数的自然延续。我们还给出了一个例子，一对双生成的基本顺从子群的$F$的财产，既不是嵌入到其他。
We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $\epsilon_0 +1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In fact we also obtain, for each $\alpha < \epsilon_0$, a finitely generated elementary amenable subgroup of $F$ whose EA-class is $\alpha + 2$. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with $\mathbf{Z} + \mathbf{Z}$, $\mathbf{Z} \wr \mathbf{Z}$, and the Brin-Navas group $B$. We also give an example of a pair of finitely generated elementary amenable subgroups of $F$ with the property that neither is embeddable into the other.