1 October 2018 The Markoff group of transformations in prime and composite moduli
Chen Meiri, Doron Puder
Duke Math. J. 167(14): 2679-2720 (1 October 2018). DOI: 10.1215/00127094-2018-0024


The Markoff group of transformations is a group Γ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation x2+y2+z2=xyz. The fundamental strong approximation conjecture for the Markoff equation states that for every prime p, the group Γ acts transitively on the set X(p) of nonzero solutions to the same equation over Z/pZ. Recently, Bourgain, Gamburd, and Sarnak proved this conjecture for all primes outside a small exceptional set. Here, we study a group of permutations obtained by the action of Γ on X(p), and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that Γ 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 Gilman and Evans regarding “Tr-systems” of PSL(2,p).


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Chen Meiri. Doron Puder. "The Markoff group of transformations in prime and composite moduli." Duke Math. J. 167 (14) 2679 - 2720, 1 October 2018. https://doi.org/10.1215/00127094-2018-0024


Received: 16 October 2017; Revised: 12 February 2018; Published: 1 October 2018
First available in Project Euclid: 28 September 2018

zbMATH: 06982204
MathSciNet: MR3859362
Digital Object Identifier: 10.1215/00127094-2018-0024

Primary: 11D25
Secondary: 20B15 , 20B25 , 20E05

Keywords: Markoff equation , Markoff triples , T-systems

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 14 • 1 October 2018
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