The Markoff group of transformations in prime and composite moduli. With an appendix by Dan Carmon. (English) Zbl 1447.11049

Summary: 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 \(X^\ast(p)\) of nonzero solutions to the same equation over \(\mathbb{Z}/p\mathbb{Z}\). Recently, Bourgain, Gamburd, and Sarnak [J. Bourgain et al., C. R., Math., Acad. Sci. Paris 354, No. 2, 131–135 (2016; Zbl 1378.11043)] proved this conjecture for all primes outside a small exceptional set.
Here, we study a group of permutations obtained by the action of \(\Gamma\) on \(X^\ast(p)\), 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 nonzero solutions in a big class of composite moduli.
Finally, our result also translates to a parallel in the case \(r=2\) of a well-known theorem of M. Evans [Math. Proc. Camb. Philos. Soc. 113, 9–22 (1993; Zbl 0781.20021)] and R. Gilman [Can. J. Math. 29, 541–551 (1977; Zbl 0332.20010)] regarding “\(T_r\)-systems” of \(\mathrm{PSL}(2,p)\).


11D25 Cubic and quartic Diophantine equations
20B15 Primitive groups
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20E05 Free nonabelian groups
Full Text: DOI arXiv Euclid


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