Commutators in $\mathrm{SL}_2$ and Markoff surfaces I
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.
DOI
10.53733/198Citation
Ghosh, A., Meiri, C., & Sarnak, P. (2021). Commutators in $\mathrm{SL}_2$ and Markoff surfaces I. New Zealand Journal of Mathematics, Vaughan Jones Memorial Special Issue, 52, pp. 773-819. https://doi.org/10.53733/198