Abstract
We show that the commutator equation over $SL_2\Z$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $SL_2(\Z[\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.
Highlights
A necessary condition for Z ∈ ΓD to be a commutator is that it be so in every finite quotient of ΓD. If this condition is sufficient we say that ΓD satisfies a profinite local to global principle, or a profinite Hasse principle
SL2(Z) satisfies the profinite Hasse principle for commutators
Every finitely generated subgroup of PSL2(Z) contains only finitely many conjugacy classes of elements of trace t. For each such conjugacy class we show that it either consists of commutators or that its elements are not commutators in some finite quotient of Gm,n
Summary
Gk} contains a representative of every PSL2(Z)-conjugacy class of trace |t|. Item (3) of Lemma 2.42 implies that every Gm,n-conjugacy class of elements of trace t contains an element in {gi,j | 1 i k, 1 j ri}. Lemma 2.42 implies that ∪1 i kYi contains at least one element of every conjugacy class of Gm,n of element of trace t. Under Notation 2.26, in the following table, for the given values of (m, n) and t, the set Rt non-trivially intersects every Gm,n-conjugacy class consisting of elements of trace t:. Lemmas 2.37 and 2.42 imply that every g ∈ G2,∞ with trace 2 is conjugate in G2,∞ to br or b−r or (ab)r or (ab−1)r for some r 1. In all the other cases, it follows from Lemma 2.45 that d is a commutator
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