  Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds
with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's
treatment of Dirac-type operators associated to these metrics by weaking the
hypotheses on the boundary families through the use of Fredholm perturbations
as in the family index theorem of Melrose and Piazza and by treating the index
of families of such operators. We also extend the index theorem of Moroianu and
Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators
associated to fibered cusp metrics (sometimes known as fibered boundary
metrics).
