自相似群与图的Z形积和置换积
Self-similar groups and the zig-zag and replacement products of graphs
摘要：
每一个自相似群都自然地生成一个无穷序列的有限$d$ -正则图$Γ_n$。我们构造了自相似群，其图$Γ_n$可以表示为迭代之字形积和图幂：$Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$ z $}}}Γ$（$k\geq 1$）。我们还构造了自相似群，其图$Γ_n$可以表示为迭代置换积和图幂：$Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$ r $}}}Γ$（$k\geq 1$）.这给出了自相似群的简单明确的例子，其图形$Γ_n$形成一个扩展的家庭，以及自动机群的例子，其图形$Γ_n$具有线性直径${\rm diam}(Γ_n)=O(n)$和有界围长。
英文摘要：
Every finitely generated self-similar group naturally produces an infinite sequence of finite $d$-regular graphs $Γ_n$. We construct self-similar groups, whose graphs $Γ_n$ can be represented as an iterated zig-zag product and graph powering: $Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$z$}}}Γ$ ($k\geq 1$). Also we construct self-similar groups, whose graphs $Γ_n$ can be represented as an iterated replacement product and graph powering: $Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$r$}}}Γ$ ($k\geq 1$). This gives simple explicit examples of self-similar groups, whose graphs $Γ_n$ form an expanding family, and examples of automaton groups, whose graphs $Γ_n$ have linear diameters ${\rm diam}(Γ_n)=O(n)$ and bounded girth.
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