Relative L^p-cohomology and application to Heintze groups

Abstract

We introduce the notion ofrelativeLp-cohomologyas a quasi-isometry invariantdefined for a Gromov-hyperbolic space and a point on its boundary at infinity and reproduce somebasic properties ofLp-cohomology in this context. In the case of degree1we show a relation betweenthe relative and the classicalLp-cohomology. As an application, we explicitly construct non-zerorelativeLp-cohomology classes for a purely real Heintze group of the formRn−1⋊αR, which gives away to prove that the eigenvalues ofα, up to a scalar multiple, are invariant under quasi-isometries.