Commutators in $\mathrm{SL}_2$ and Markoff surfaces I

preprint2022arXivOpen access

We show that the commutator equation over $\mathrm{SL}_2(\mathbb{Z})$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $\mathrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$ . The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.

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Commutators in $\mathrm{SL}_2$ and Markoff surfaces I

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Commutators in SL2\mathrm{SL}_2SL2​ and Markoff surfaces I

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Commutators in $\mathrm{SL}_2$ and Markoff surfaces I