The Markoff Group of Transformations in Prime and Composite Moduli

• Published in: arXiv.org 2017-10
• Main Authors: Chen Meiri, Puder, Doron, Carmon, Dan
• Format: Article
• Language: English
• Subjects: Group theory; Permutations; Transformations; Traveling salesman problem
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1702.08358

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.