Relative L^p-cohomology and application to Heintze groups

Abstract

We introduce the notion of relative $L^p$-cohomology as a quasi-isometry invariant defined for a Gromov-hyperbolic space and a point on its boundary at infinity and reproduce some basic properties of $L^p$-cohomology in this context. In the case of degree 1 we show a relation between the relative and the classical $L^p$-cohomology. As an application, we explicitly construct non-zero relative $L^p$-cohomology classes for a purely real Heintze group of the form ${\mathbb{R}}^{n-1}{⋊}_{\alpha }\mathbb{R}$, which gives a way to prove that the eigenvalues of $\alpha$, up to a scalar multiple, are invariant under quasi-isometries.