The asymptotic geometry of the Riemannian Heisenberg group and its horoboundary.




Two metrics d and d' on a space X are "asymptotic" if their ratio d'(p,q)/d(p,q) tends to 1 when the distance d(p,q) goes to infinity. A celebrated result of D.Burago implies that any two periodic metrics on R^n are asymptotic if and only if they stay at bounded distance from each other, i.e. there exists some constant C such that d'(p,q)-d(p,q) less than C. A similar phenomenon occurs for cocompact, Gromov-hyperbolic spaces and for word metrics on the discrete Heisenberg group.
We show that the same is true for left-invariant Riemannian (or sub-Riemannian) metrics on the real Heisenberg group H^3: actually, two such metrics d,d' on H^3 are asymptotic if and only if their difference goes to zero when d(p,q) diverge. It follows that, unsurprisingly, the Riemannian Heisenberg group is at bounded distance from its asymptotic cone; moreover, we are able to describe the Riemannian horoboundary of the Heisenberg group, using Klein and Nikas' work on its boundary with respect to the natural Carnot-Caratheodory metric.
(This is a joint work with E. Le Donne and S. Nicolussi Golo.)