The markoff group of transformations in prime and composite Moduli

Chen Meiri, Doron Puder

Research output: Contribution to journalArticlepeer-review


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).

Original languageEnglish
Pages (from-to)2679-2720
Number of pages42
JournalDuke Mathematical Journal
Issue number14
StatePublished - 1 Oct 2018


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