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 $X^{\ast }\left(p\right)$ 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 $X^{\ast }\left(p\right)$, 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 $r\ge 3$, 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.

31 pages, by Chen Meiri and Doron Puder, with an appendix by Dan Carmon. Better exposition than in last version, and some non-accurate statements fixed

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