群的圆性
Roundness properties of groups
摘要：
度量空间的圆性是由Per Enflo引入的，它是研究线性拓扑空间一致结构的工具。本文研究一般度量空间的圆性所反映的几何和拓扑性质。特别地，我们证明了圆度为2的测地空间是可收缩的，并且具有圆度$>1$的紧致黎曼流形必须是单连通的。然后，我们集中我们的调查的凯莱图的有限生成群。我们的主要结果之一是：$\geq 2$生成元上的自由阿尔贝群的每个Cayley图都有圆度$=1$ .证明了如果一个群没有圆度为$=1$的Cayley图，则它必是一个挠群，其每个元素的阶为$2,3,5$或7。
英文摘要：
Roundness of metric spaces was introduced by Per Enflo as a tool to study uniform structures of linear topological spaces. The present paper investigates geometric and topological properties detected by the roundness of general metric spaces. In particular, we show that geodesic spaces of roundness 2 are contractible, and that a compact Riemannian manifold with roundness $>1$ must be simply connected. We then focus our investigation on Cayley graphs of finitely generated groups. One of our main results is that every Cayley graph of a free abelian group on $\geq 2$ generators has roundness $=1$. We show that if a group has no Cayley graph of roundness $=1$, then it must be a torsion group with every element of order $2,3,5$, or 7.
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