素数模与复合模变换的Markoff群
The Markoff Group of Transformations in Prime and Composite Moduli
摘要：
马尔可夫变换群是仿射积分态射的群$Γ$，已知其传递作用于方程$x^{2}+y^{2}+z^{2}=xyz$的所有正整数解的集合。马尔可夫方程的基本强逼近猜想指出，对于每个素数$p$，群$Γ$传递地作用在$\mathbb{Z}/p\mathbb{Z}$上同一方程的非零解的集合$X^{*}\left(p\right)$上。最近，Bourgain，Gamburd和Sarnak证明了这个猜想的所有素数以外的一个小的例外集。在本文中，我们研究了$Γ$作用于$X^{*}\left(p\right)$而获得的一组排列，并表明对于大多数素数，它是完全对称或交替群。我们利用这个结果推导出$Γ$也传递作用于一个大类的复合模的非零解的集合。我们的结果也与Gilman的一个著名定理有关，即对于任何有限的非交换单群$G$和$r\ge3$，群$\mathrm{Aut}\left(F_{r}\right)$至少作用于$G$的一个$T_{r}$ -系统作为交替群或对称群。在这种语言中，我们的主要结果转化为，对于大多数素数$p$，群$\mathrm{Aut}\left(F_{2}\right)$作用于$\mathrm{PSL}\left(2,p\right)$的特定$T_{2}$ -系统作为交替或对称群。
英文摘要：
The Markoff group of transformations is a group $Γ$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$. The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$, the group $Γ$ acts transitively on the set $X^{*}\left(p\right)$ of non-zero solutions to the same equation over $\mathbb{Z}/p\mathbb{Z}$. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set. In the current paper, we study a group of permutations obtained by the action of $Γ$ on $X^{*}\left(p\right)$, and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $Γ$ acts transitively also on the set of non-zero solutions in a big class of composite moduli. Our result is also related to a well-known theorem of Gilman, stating that for any finite non-abelian simple group $G$ and $r\ge3$, the group $\mathrm{Aut}\left(F_{r}\right)$ acts on at least one $T_{r}$-system of $G$ as the alternating or symmetric group. In this language, our main result translates to that for most primes $p$, the group $\mathrm{Aut}\left(F_{2}\right)$ acts on a particular $T_{2}$-system of $\mathrm{PSL}\left(2,p\right)$ as the alternating or symmetric group.
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