Boundary at infinity of symmetric rank one spaces

TL;DR

This paper demonstrates that certain canonical metrics on the boundary at infinity of rank one symmetric spaces are bilipschitz equivalent to visual metrics derived from Gromov products, establishing a universal geometric relationship.

Contribution

It establishes the bilipschitz equivalence of Carnot-Caratheodory metrics with visual metrics on the boundary at infinity of rank one symmetric spaces.

Findings

Canonical metrics are bilipschitz equivalent to visual metrics.

Universal bilipschitz constants apply to these equivalences.

Results apply to all rank one symmetric spaces of non-compact type.

Abstract

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in the space and on the boundary at infinity respectively.