The Markoff Group of Transformations in Prime and Composite Moduli

Abstract

The Markoff group of transformations is a group $\Gamma$ 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 $\Gamma$ 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 $\Gamma$ 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 $\Gamma$ 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.