Schreier图的端与自相似群极限空间的割点
Ends of Schreier graphs and cut-points of limit spaces of self-similar groups
摘要：
每个自相似群都作用于某个字母表$X$上的无限词空间$X^ω$上。研究了空间$X^ω$上有界自动机生成的自相似群作用的Schreier图$Γ_w$（$w\in X^ω$）.利用sofic子移位，我们确定了每个Schreier图$Γ_w$的端点数.几乎所有关于$X^ω$上一致测度的Schreier图$Γ_w$都有一个或两个端点，并且我们几乎必然地刻画了Schreier图有两个端点的有界自动机.建立了自相似群的极限空间与（局部）割点的联系。
英文摘要：
Every self-similar group acts on the space $X^ω$ of infinite words over some alphabet $X$. We study the Schreier graphs $Γ_w$ for $w\in X^ω$ of the action of self-similar groups generated by bounded automata on the space $X^ω$. Using sofic subshifts we determine the number of ends for every Schreier graph $Γ_w$. Almost all Schreier graphs $Γ_w$ with respect to the uniform measure on $X^ω$ have one or two ends, and we characterize bounded automata whose Schreier graphs have two ends almost surely. The connection with (local) cut-points of limit spaces of self-similar groups is established.
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