On copies of the absolute Galois group in $\mathrm{Out}\hat{F}_2$
  In this article we consider outer Galois actions on a free profinite group of rank two, induced by the \'etale fundamental group of a projective line minus three points or of a pointed elliptic curve over a number field. Under mild technical assumptions their respective images uniquely determine the curves and the number fields. 
Robert A. Kucharczyk
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