Commutators in $SL_2$ and Markoff surfaces I

Abstract

We show that the commutator equation over $SL_2\Z$ satisfies a profinite local to global principle, while it can fail with infinitely many  exceptions for $SL_2(\Z[\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.