Families Index for Manifolds with Hyperbolic Cusp Singularities

Manifolds with fibered hyperbolic cusp metrics include hyperbolicmanifolds with cusps and locally symmetric spaces of -rank 1. We extend Vaillant's treatment of Dirac-typeoperators associated to these metrics by weakening the hypotheseson the boundary families through the use of Fredholm perturbationsas in the family index theorem of Melrose and Piazza, and bytreating the index of families of such operators. We also extendthe index theorem of Moroianu and Leichtnam–Mazzeo–Piazzato families of perturbed Dirac-type operators associated tofibered cusp metrics (sometimes known as fibered boundary metrics).