De Rham's theorem for Orlicz cohomology in the case of Lie groups

TL;DR

This paper establishes the equivalence and invariance of Orlicz cohomology theories for Lie groups and Gromov-hyperbolic spaces, extending classical de Rham theorems to these contexts.

Contribution

It proves the equivalence between simplicial and de Rham Orlicz cohomology for Lie groups and shows invariance under quasi-isometries, extending to Gromov-hyperbolic spaces.

Findings

Equivalence of simplicial and Orlicz-de Rham cohomology for Lie groups

Invariance of Orlicz cohomology under quasi-isometries for Lie groups

Extension of results to Gromov-hyperbolic spaces relative to boundary points

Abstract

We prove the equivalence between the simplicial Orlicz cohomology and the Orlicz-de Rham cohomology in the case of Lie groups. Since the first one is a quasi-isometry invariant for uniformly contractible simplicial complexes with bounded geometry, we obtain the invariance of the second one in the case of contractible Lie groups. We also define the Orlicz cohomology of a Gromov-hyperbolic space relative to a point on its boundary at infinity, for which the same results are true.