构造小直径有向Cayley图：一个有效的Solovay-Kitaev过程
Constructing Directed Cayley Graphs of Small Diameter: A Potent Solovay-Kitaev Procedure
摘要：
设$Γ$是群，$(Γ_n)_{n=1} ^{\infty}$是有限指数正规子群的降序序列.在对$Γ$中的幂和换位子字的行为的一些自然假设下，我们建立了$Γ/Γ_n$的有向Cayley图的直径的上界.我们得到的边界不依赖于生成集的选择。此外，在合理的条件下，我们的方法提供了一个快速的算法来构造直径满足我们的界限的有向凯莱图。这个证明非常类似于Solovay-Kitaev过程，它只使用交换子字，但也只构造小直径无向Cayley图。作为一个应用，我们给出了两个不同群：$SL_2 (\mathbb{F}_q [[t]])$（对于$q$偶群）和Fabrykowski和Gupta引入的三元有根树的自同构自同构群的有限直径界.
英文摘要：
Let $Γ$ be a group and $(Γ_n)_{n=1} ^{\infty}$ be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the $Γ/Γ_n$ , under some natural hypotheses on the behaviour of power and commutator words in $Γ$. The bounds we obtain do not depend on a choice of generating set. Moreover under reasonable conditions our method provides a fast algorithm for constructing directed Cayley graphs of diameter satisfying our bounds. The proof is closely analogous to the the Solovay-Kitaev procedure, which only uses commutator words, but also only constructs small-diameter undirected Cayley graphs. As an application we give directed diameter bounds on finite quotients of two very different groups: $SL_2 (\mathbb{F}_q [[t]])$ (for $q$ even) and a group of automorphisms of the ternary rooted tree introduced by Fabrykowski and Gupta.
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