Relative $L^p$-cohomology and Heintze groups

We introduce the notion of \textit{relative $L^p$-cohomology} as a quasi-isometry invariant defined for Gromov-hyperbolic spaces, and apply it 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 of the form $\mathbb{R}^{n-1}\rtimes_\alpha\mathbb{R}$, which gives a way to prove that the eigenvalues of $\alpha$, up to a scalar multiple, are invariant by quasi-isometries. In the case of degree $1$ we show a relation between the relative and the classical $L^p$-cohomology.