Asymptotic behavior of the Riemannian Heisenberg group and its horoboundary

TL;DR

This paper investigates the large-scale geometry of the Riemannian Heisenberg group, establishing conditions for asymptotic distances, and explicitly describing its horoboundary, revealing its relation to the asymptotic cone.

Contribution

It characterizes when two left-invariant Riemannian distances are asymptotic and describes the horoboundary of the Riemannian Heisenberg group explicitly.

Findings

Distances with bounded difference are asymptotic.

Existence of a unique subRiemannian metric close at infinity.

The Riemannian Heisenberg group is at bounded distance from its asymptotic cone.

Abstract

The paper is devoted to the large scale geometry of the Heisenberg group $\mathbb H$ equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian distance $d$ on $\mathbb H$ there is a unique subRiemanniann metric $d'$ for which $d-d'$ goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its asymptotic cone. The second consequence, which was our aim, is the explicit description of the horoboundary of the Riemannian Heisenberg group.