Burnside's Problem, spanning trees, and tilings
Burnside's Problem, spanning trees, and tilings
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伯恩赛德问题，跨越树和瓷砖
DOI：
10.2140/gt.2014.18.179
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Brandon Seward
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文献类型：
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作者：
Brandon Seward
本文研究了伯恩赛德问题和冯诺依曼猜想的几何形式。这是通过考虑一个类似预防措施的行动的概念来完成的。类平移作用是由凯文·怀特（Kevin Whyte）引入的，作为子群包容的几何模拟。怀特证明了冯·诺依曼猜想的一个几何版本，证明了一个非阿贝尔生成群是不服从的，当且仅当它允许任何（等价于每一个）非阿贝尔自由群的类服从作用。我们通过证明当作用自由群是可传递群时，这个类传递作用可以被选择为传递的，从而加强了Whyte的结果。我们进一步证明了伯恩赛德问题的几何形式成立。也就是说，每一个非线性生成的无限群都允许$\Z$的类仿射作用。这回答了怀特提出的一个问题。在追求这些结果中，我们发现了凯莱图的一个有趣的性质：每一个无限生成的群G$有一些凯莱图有一个正则的生成树。当且仅当$G$有2个端点时，这个正则生成树可以被选择为具有2度（因此是一个双无限哈密顿路径），并且当且仅当$G$是不顺从的时，它可以被选择为具有大于2度的任何度数。我们使用这个最后的结果，然后研究平铺组。我们定义了一个一般的概念的polytilings和推广的概念MT群和ccc群的设置polytilings。证明了每个可数群是poly-MT，每个可数生成群是poly-ccc。
In this paper we study geometric versions of Burnside's Problem and the von Neumann Conjecture. This is done by considering the notion of a translation-like action. Translation-like actions were introduced by Kevin Whyte as a geometric analogue of subgroup containment. Whyte proved a geometric version of the von Neumann Conjecture by showing that a finitely generated group is non-amenable if and only if it admits a translation-like action by any (equivalently every) non-abelian free group. We strengthen Whyte's result by proving that this translation-like action can be chosen to be transitive when the acting free group is finitely generated. We furthermore prove that the geometric version of Burnside's Problem holds true. That is, every finitely generated infinite group admits a translation-like action by $\Z$. This answers a question posed by Whyte. In pursuit of these results we discover an interesting property of Cayley graphs: every finitely generated infinite group $G$ has some Cayley graph having a regular spanning tree. This regular spanning tree can be chosen to have degree 2 (and hence be a bi-infinite Hamiltonian path) if and only if $G$ has finitely many ends, and it can be chosen to have any degree greater than 2 if and only if $G$ is non-amenable. We use this last result to then study tilings of groups. We define a general notion of polytilings and extend the notion of MT groups and ccc groups to the setting of polytilings. We prove that every countable group is poly-MT and every finitely generated group is poly-ccc.