TY - JOUR AU - Ghosh, Amit AU - Meiri, Chen AU - Sarnak, Peter PY - 2022/01/18 Y2 - 2024/03/30 TI - Commutators in $SL_2$ and Markoff surfaces I JF - New Zealand Journal of Mathematics JA - NZ J Math VL - 52 IS - SE - Vaughan Jones Memorial Special Issue DO - 10.53733/198 UR - https://nzjmath.org/index.php/NZJMATH/article/view/198 SP - 773-819 AB - <p>We show that the commutator equation over $SL_2\Z$ satisfies a profinite local to global principle, while it can fail with infinitely many<span class="Apple-converted-space">  </span>exceptions for $SL_2(\Z[\frac{1}{p}])$. The source of the failure is a reciprocity obstruction<span class="Apple-converted-space"> </span>to the Hasse Principle for cubic Markoff surfaces.</p> ER -