Families Index for Manifolds with Hyperbolic Cusp Singularities

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of $\mathbb Q$-rank 1. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weakening 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).