Relative $L^p$-cohomology and Heintze groups

Abstract

We define the L p -cohomology of a Gromov hyperbolic Riemannian manifold relative to a point on its boundary at infinity and prove that it is, as in the classical case, a quasi-isometry invariant. We obtain an application to the problem of quasi-isometry classification of Heintze groups. More precisely, we explicitly construct non-zero relative L p -cohomology classes on a Heintze group R n ⋊ α R, which allows us to prove that the eigenvalues of α, up to a scalar multiple, are invariants under quasi-isometries.