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

De Rham $L^p$-Cohomology For Higher Rank Spaces And Groups: Critical Exponents And Rigidity