  We give a lower bound for the bottom of the $L^2$ differential form spectrum
on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan
and Corlette in the function case. Our method is based on the study of the
resolvent associated with the Hodge-de Rham Laplacian and leads to applications
for the (co)homology and topology of certain classes of hyperbolic manifolds.
