Asymptotic behavior of the Riemannian Heisenberg group and its horoboundary

The paper is devoted to the large-scale geometry of the Heisenberg group H equipped with left-invariant Riemannian metrics. We prove that two such metrics 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 metric d on H there is a unique subRiemannian 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.