TY - JOUR

T1 - The markoff group of transformations in prime and composite Moduli

AU - Meiri, Chen

AU - Puder, Doron

N1 - Publisher Copyright:
© 2018.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85055264436&partnerID=8YFLogxK

U2 - 10.1215/00127094-2018-0024

DO - 10.1215/00127094-2018-0024

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85055264436

SN - 0012-7094

VL - 167

SP - 2679

EP - 2720

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 14

ER -