# The Markoff Group of Transformations in Prime and Composite Moduli

## TL;DR

This paper investigates the action of the Markoff group on solutions to the Markoff equation modulo primes and composites, showing it often acts as the full symmetric or alternating group, with implications for group actions on finite simple groups.

## Contribution

It proves that for most primes, the Markoff group acts as the full symmetric or alternating group on solutions, and extends this transitivity to many composite moduli, connecting to automorphism group actions.

## Key findings

- The group acts as the full symmetric or alternating group for most primes.
- Transitivity extends to solutions modulo many composite numbers.
- Connections established with automorphism groups of free groups and simple groups.

## 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.