某些有限群族的一致多对数直径界
Uniform poly-log diameter bounds for some families of finite groups
摘要：
用$p> m \geq 2$修复素数$p$和整数$m$。为$n=1,2,... $定义有限群族\[ G_n :=SL_m (\mathbb{Z}/p^{n}\mathbb{Z}) \]。我们将证明存在两个正常数$C$和$d$使得对于任何$n$和任何生成集$S\subseteq G_{n}$，当$diam (G,S)$是有限群$G$关于生成元集$S$的直径时，有\[ diam(G_n,S) \leq C \cdot log^d (|G_n|)\].它被定义为表示$g$的$S \cup S^{-1}$中最短字的长度在$g \in G$上的最大值。这一结果表明，这些有限群族的直径关于生成元的\emph{any}集有一个多对数界。这个结果的证明也提供了一个有效的算法，找到这样的多对数表示的任何元素。此外，它表明，在$log$界的功率$d$可以是任意接近3的$m=2$和任意接近4的$m>2$。
英文摘要：
Fix a prime $p$ and an integer $m$ with $p> m \geq 2$. Define the family of finite groups \[ G_n :=SL_m (\mathbb{Z}/p^{n}\mathbb{Z}) \] for $n=1,2,... $. We will prove that there exist two positive constants $C$ and $d$ such that for any $n$ and any generating set $S\subseteq G_{n}$, \[ diam(G_n,S) \leq C \cdot log^d (|G_n|)\] when $diam (G,S)$ is the diameter of the finite group $G$ with respect to the set of generators $S$. It is defined as the maximum over $g \in G$ of the length of the shortest word in $S \cup S^{-1}$ representing $g$. This result shows that these families of finite groups have a poly-logarithmic bound on the diameter with respect to \emph{any} set of generators. The proof of this result also provides a efficient algorithm for finding such a poly-logarithmic representation of any element. In addition it shows that the power $d$ in the $log$ bound can be arbitrary close to 3 for $m=2$ and arbitrary close to 4 for $m>2$.
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