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

Meiri, Chen; Puder, Doron

Duke Math. J. 167, No. 14, 2679-2720 (2018).

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 \(\mathrm{PSL}(2,p)\).

MSC:

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

Keywords:

Markoff equation; Markoff triples; \(T\)-systems

Citations:

Zbl 1378.11043; Zbl 0781.20021; Zbl 0332.20010

References:

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