Commutators in $SL_2$ and Markoff surfaces I

@article{Ghosh2022CommutatorsI,
  title={Commutators in \$SL\_2\$ and Markoff surfaces I},
  author={Amit Ghosh and Chen Meiri and Peter Sarnak},
  journal={New Zealand Journal of Mathematics},
  year={2022}
}
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. 

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