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

Abstract

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.