*Submitted Paper*

**Inserted:** 18 sep 2015

**Last Updated:** 18 sep 2015

**Year:** 2015

**Abstract:**

The paper is devoted to the large scale geometry of the Heisenberg group *H* equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are symptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian distance *d* on *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.

**Keywords:**
Riemannian Geometry, Heisenberg group, sub-Riemannian geometry, horoboundary, asymptotic cone

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