Markoff Triples and Strong Approximation

Jean Bourgain, Alex Gamburd, Peter Sarnak

Published 2015-05-24Version 1

We study the connectedness of the set of solutions ($\mod p$) of the Markoff equation $x_1^2+x_2^2+x_3^2 = 3x_1 x_2x_3$ under the action of the group of nonlinear morphisms generated by coordinate permutations and Vieta involutions. In particular it is shown that for almost all $p$, the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.