论文标题
汤普森的$ f $几乎是$ \ frac {3} {2} $ - 生成
Thompson's group $F$ is almost $\frac{3}{2}$-generated
论文作者
论文摘要
回想一下，如果$ g $的每个非平凡元素都属于$ g $的生成对，则据说$ g $是$ \ frac {3} {2} $ - 生成的。事实证明，汤普森（Thompson）的$ v $是$ \ frac {3} {2} $ - 由Donoven and Harper于2019年生成。这是无限呈现的无限呈现的非循环$ \ frac {3} {3} {2} {2} $生成的组的第一个例子。最近，Break，Harper和Skipper证明了Thompson的Group $ T $也是$ \ frac {3} {2} $生成的。在本文中，我们证明，汤普森的$ f $是“几乎” $ \ frac {3} {2} $，从某种意义上说，$ f $的每个元素的图像构成了$ f $的每个元素构成一对生成对的$ \ mathbb {z}^2 $的一部分。我们还证明，对于f $中的每个非平凡元素$ f \ f \ in f $ in f $中都有一个元素$ g \，因此子组$ \ langle f，g \ rangle $包含$ f $的派生子组。此外，如果$ f $不属于$ f $的派生子组，则f $中有一个元素$ g \，因此$ \ langle f，g \ rangle $具有$ f $的有限索引。
Recall that a group $G$ is said to be $\frac{3}{2}$-generated if every non-trivial element of $G$ belongs to a generating pair of $G$. Thompson's group $V$ was proved to be $\frac{3}{2}$-generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic $\frac{3}{2}$-generated group. Recently, Bleak, Harper and Skipper proved that Thompson's group $T$ is also $\frac{3}{2}$-generated. In this paper, we prove that Thompson's group $F$ is "almost" $\frac{3}{2}$-generated in the sense that every element of $F$ whose image in the abelianization forms part of a generating pair of $\mathbb{Z}^2$ is part of a generating pair of $F$. We also prove that for every non-trivial element $f\in F$ there is an element $g\in F$ such that the subgroup $\langle f,g\rangle$ contains the derived subgroup of $F$. Moreover, if $f$ does not belong to the derived subgroup of $F$, then there is an element $g\in F$ such that $\langle f,g\rangle$ has finite index in $F$.
