关于Cayley图中球面连通性的一个注记
A remark on the connectedness of spheres in Cayley graphs
摘要：
这篇小笔记的目的是要证明一个基本的但有用的性质的群。设G是一个有一端的有限生成群.固定一个（有限）生成集，设$B_n$为围绕$e$的半径为$n$的球。令$B_n^{c,\infty}$是$B_n$的补的无限连通分支。如果存在一个$r >0$使得所有$n \geq 0$的$B_{n+r} \cap B_n^{c,\infty}$连通，则G有连通球面.这注记表明，如果G是双存在的，那么它有连通球面。
英文摘要：
The aim of this small note is to prove an elementary yet useful properties of finitely presented groups. Let G be a finitely generated group with one end. Fix a (finite) generating set and let $B_n$ be the ball of radius $n$ around $e$. Let $B_n^{c,\infty}$ be the infinite connected component of the complement of $B_n$. Then G has connected spheres if there exists a $r >0$ such that $B_{n+r} \cap B_n^{c,\infty}$ is connected for all $n \geq 0$. This note shows that if G is finitely presented then it has connected spheres.
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