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