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 \(\mathbb{R}^{n-1}\rtimes_\alpha\mathbb{R}\), which gives a way to prove that the eigenvalues of \(\alpha\), up to a scalar multiple, are invariant under quasi-isometries.