  We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi
invariant. Any translation surface whose Veech group is commensurable to any of
a large class of triangle groups is shown to have an affine pseudo-Anosov
homeomorphism of this type. We also apply a reduction to finite triangle groups
and thereby show the existence of non-parabolic elements in the periodic field
of certain translation surfaces.
