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For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zeroPythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbitO = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with criticalexponent exceeding 1/2.  We find infinitely many Pythagorean triples in O whose hypotenuse, area, andproduct of side lengths have few prime factors, where "few" is explicitlyquantified. We also compute the asymptotic of the number of such Pythagoreantriples of norm at most T, up to bounded constants.

Almost prime Pythagorean triples in thin orbits