  We initiate the investigation of critical exponents (in degree equal to the
rank) for the vanishing of L^p-cohomology of higher rank Lie groups and related
manifolds. We deal with the rank 2 case and exhibit such phenomena for
SL$_3$(R) and for a family of 5-dimensional solvable Lie groups. This leads us
to exhibit a continuum of quasi-isometry classes of rank 2 solvable Lie groups
of non-positive curvature. We provide a detailed description of the
$L^p$-cohomology of the real and complex hyperbolic spaces, to be combined with
a spectral sequence argument for our higher-rank results.

Group Theory;Differential Geometry