论文标题
关于无限组的传播
On the spread of infinite groups
论文作者
论文摘要
如果每个非平凡元素都是生成对的一部分，则组为$ \ frac32 $。 2019年，多诺芬（Donoven）和哈珀（Harper）表明，许多汤普森（Thompson）团体都是$ \ frac32 $生成的，并提出了五个问题。首先是是否存在一个2生成的组，其中每个适当的商循环不是$ \ frac32 $生成的。鉴于证明没有有限群体具有此财产的重要工作，这是一个自然的问题，但我们表明存在这样的无限群体。我们认为的组是霍顿组$ \ text {fsym}（\ mathbb {z}）\ rtimes \ mathbb {z} $的有限索引子组。然后，我们证明我们家庭中的前两个组是$ \ frac32 $生成的，并调查了这些组的相关概念。我们能够证明它们具有大于2的有限扩散。因此，这些是第一个显示出有限的正散布的无限群体，并且第一个被证明的散布大于2（$ \ Mathbb {z} $，而Tarski怪兽除外，这些怪物具有无限扩散）。
A group is $\frac32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups of the Houghton group $\text{FSym}(\mathbb{Z})\rtimes\mathbb{Z}$. We then show that the first two groups in our family are $\frac32$-generated, and investigate the related notion of spread for these groups. We are able to show that they have finite spread which is greater than 2. These are therefore the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread greater than 2 (other than $\mathbb{Z}$ and the Tarski monsters, which have infinite spread).
