  We give a construction of hyperbolic 3-manifolds with rank two fundamental
groups and report an experimental search to find such manifolds. Our manifolds
are all surface bundles over the circle with genus two surface fiber. For the
manifolds so obtained, we then examine whether they are of Heegaard genus two
or not. As a byproduct, we give an infinite family of fibered knots of genus
two in the 3-sphere whose knot groups are of rank two.
