Orlicz spaces and the large scale geometry of Heintze groups

TL;DR

This paper develops an Orlicz space-based cohomology theory for metric spaces, proving quasi-isometry invariance and applying it to analyze the large-scale geometry of negatively curved homogeneous spaces, especially Heintze groups.

Contribution

It introduces a new Orlicz space cohomology framework and applies it to characterize the boundary behavior and quasi-isometries of Heintze groups with negative curvature.

Findings

Quasi-isometry invariance of Orlicz cohomology for general Young functions.

Identification of degree one cohomology with Orlicz-Besov spaces on the boundary.

Self quasi-isometries fix boundary points and preserve foliations in non-Carnot Heintze groups.

Abstract

We consider an Orlicz space based cohomology for metric (measured) spaces with bounded geometry. We prove the quasi-isometry invariance for a general Young function. In the hyperbolic case, we prove that the degree one cohomology can be identified with an Orlicz-Besov function space on the boundary at infinity. We give some applications to the large scale geometry of homogeneous spaces with negative curvature (Heintze groups). As our main result, we prove that if the Heintze group is not of Carnot type, any self quasi-isometry fixes a distinguished point on the boundary and preserves a certain foliation on the complement of that point.