Arithmetic Groups

We discuss a local to global profinite principle for being a commutator in some arithmetic groups. Specifically we show that $SL_2(Z)$ satisfies such a principle, while it can fail with infinitely many exceptions for $SL_2(Z[1/p])$. The source of the failure is a reciprocity obstruction to the integral Hasse Principle for cubic Markoff surfaces. Joint work with Amit Ghosh and Peter Sarnak.