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arXiv · 2110.11030

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

Amit Ghosh·Chen Meiri·Peter Sarnak
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Abstract

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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Amit Ghosh, Chen Meiri, Peter Sarnak. 2021-10-21. Commutators in $\mathrm{SL}_2$ and Markoff surfaces I. https://doi.org/10.53733/198

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