Abstract
The paper is devoted to the largescale geometry of the Heisenberg group \({\mathbb {H}}\) equipped with leftinvariant 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 leftinvariant Riemannian metric d on \({\mathbb {H}}\) there is a unique subRiemannian metric \(d'\) for which \(dd'\) 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.
Introduction
In largescale geometry, various notions of space at infinity have received special interest for differently capturing the asymptotic geometric behavior. Two main examples of spaces at infinity are the asymptotic cone and the horoboundary. The description of the asymptotic cone for finitely generated groups is a crucial step in the algebraic characterization of groups of polynomial growth [4, 12, 21, 23, 30, 33]. The notion of horoboundary has been formulated by Gromov [11], inspired by the seminal work of Busemann on the theory of parallels on geodesic spaces [6]. The horoboundary has a fully satisfying visual description in the framework of CAT (0)spaces and of Gromovhyperbolic spaces [2, 3, 13]. It plays a major role in the study of dynamics and rigidity of negatively curved spaces [13, 15, 22, 24, 27, 29]. The visualboundary description breaks down for nonsimply connected manifolds [9] and when the curvature has variable sign, as we will make evident for the Riemannian Heisenberg group.
This paper contributes to the study of the asymptotic geometry of the simplest nonAbelian nilpotent group: the Heisenberg group. The asymptotic cone of the Heisenberg group equipped with a leftinvariant Riemannian metric \(d_R\) is the Heisenberg group equipped with a Carnot–Carathéodory metric \(d_{CC}\), see [23] and also [4]. Our contribution is a finer analysis of the asymptotic comparison of \(d_R\) and \(d_{CC}\). This leads to the explicit knowledge of the (Riemannian) horoboundary. We remark that the Heisenberg group is not hyperbolic; hence, one does not consider its visual boundary.
We recall the definition of horoboundary. Let (X, d) be a metric space. We consider the space of continuous real functions \(\mathscr {C}(X)\) endowed with the topology of uniform convergence on compact sets. We denote by \(\mathscr {C}(X)/{\mathbb {R}}\) the quotient with respect to the subspace of constant functions. The map \(x\mapsto d(x,\cdot )\) induces an embedding \(X\hookrightarrow \mathscr {C}(X)/{\mathbb {R}}\). The horoboundary of X is defined as \(\partial _hX := \bar{X}{\setminus } X\subset \mathscr {C}(X)/{\mathbb {R}}\). See Sect. 5 for a detailed exposition.
The horoboundary has been investigated for finitedimensional normed vector spaces [31], for Hilbert geometries [32], and for infinite graphs [34]. For nonsimply connected, negatively curved manifolds, it has been studied in [9]. Nicas and Klein computed the horoboundary of the Heisenberg group when endowed with the Korany metric in [17] and with the metric \(d_{CC}\) in [18].
We will show that the horoboundary of the Heisenberg group endowed with a leftinvariant Riemannian metric \(d_R\) coincides with the second case studied by Nicas and Klein, see Corollary 1.4. This will be an immediate consequence of our main result Theorem 1.3, which implies that the difference \(d_Rd_{CC}\) converges to zero when evaluated on points \((p,q_n)\) with \(q_n\) being a sequence that leaves every compact set.
Remark 1.1
Another term for Carnot–Carathéodory metric is subRiemannian metric. In this paper, we should discuss subRiemannian metrics that may actually be Riemannian. Therefore, we follow the convention that subRiemannian geometry includes as a particular case Riemannian geometry. This is in agreement with several established references in the field, see [1, 16, 26]. In the presence of a subRiemannian metric that is not Riemannian, we shall use the term strictly subRiemannian.
Detailed results
The Heisenberg group \({\mathbb {H}}\) is the simply connected Lie group whose Lie algebra \(\mathfrak {h}\) is generated by three vectors X, Y, Z with only nonzero relation \([X,Y]=Z\). A leftinvariant Riemannian metric d on \({\mathbb {H}}\) is determined by a scalar product g on \(\mathfrak {h}\); a leftinvariant strictly subRiemannian metric d is induced by a bracket generating plane \(V\subset \mathfrak {h}\) and a scalar product g on V (see Sect. 2 for detailed exposition). In both cases, we say that d is subRiemannian with horizontal space (V, g), where \(\dim V\) is either 2 or 3.
We are interested in the asymptotic comparison between these metrics. Given two leftinvariant subRiemannian metrics d and \(d'\) on \({\mathbb {H}}\), we deal with three asymptotic behaviors, in ascending order of strength, each of which defines an equivalence relation among subRiemannian metrics:

(i)
\(\lim _{d(p,q)\rightarrow \infty }\frac{d(p,q)}{d'(p,q)} = 1\);

(ii)
There exists \(c>0\) such that \(d(p,q)d'(p,q)< c\), for all p, q;

(iii)
\(\lim _{d(p,q)\rightarrow \infty }d(p,q)d'(p,q)=0\).
A first example of the implication \((i)\Rightarrow (ii)\) was proved by Burago in [5] for \({\mathbb {Z}}^n\)invariant metrics d on \({\mathbb {R}}^n\), by showing that d and the associated stable norm stay at bounded distance from each other. This result has been extended quantitatively for \({\mathbb {Z}}^n\)invariant metrics on geodesic metric spaces in [8]. Gromov and Burago asked for other interesting cases where the same implication holds. Another wellknown case where (i) is equivalent to (ii) is that of hyperbolic groups. Beyond Abelian and hyperbolic groups, Krat proved the equivalence for word metrics on the discrete Heisenberg group \({\mathbb {H}}({\mathbb {Z}})\) [19]. For general subFinsler metrics on Carnot groups, it has been proven in [4], following [28], that (i) is equivalent to the fact that the projections onto \({\mathbb {H}}/[{\mathbb {H}},{\mathbb {H}}]\) of the corresponding unit balls coincide, see (c) below. Our first result shows that this last condition is equivalent to each one of (i) and (ii) in the case of the Heisenberg group endowed with subRiemannian metrics.
Theorem 1.2
Let d and \(d'\) be two leftinvariant subRiemannian metrics on \({\mathbb {H}}\) whose horizontal spaces are (V, g) and \((V',g')\), respectively. Let \(\pi :\mathfrak {h}\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) be the quotient projection and \(\hat{\pi }:{\mathbb {H}}\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) the corresponding group morphism. Then, the following assertions are equivalent:

(a)
there exists \(c>0\) such that \(d(p,q)d'(p,q)< c\), for all p, q;

(b)
\(\frac{d(p,q)}{d'(p,q)} \rightarrow 1\) when \(d(p,q)\rightarrow \infty \);

(c)
\(\hat{\pi }\left( \{p\in {\mathbb {H}}:\ d(0,p)\le R\}\right) = \hat{\pi }\left( \{p\in {\mathbb {H}}:\ d'(0,p)\le R\}\right) \), for all \(R>0\), here 0 denotes the neutral element of \({\mathbb {H}}\);

(d)
\(\pi \left( \{v\in V:\ g(v,v)\le 1\}\right) = \pi \left( \{v'\in V':\ g'(v',v')\le 1\}\right) \);

(e)
there exists a scalar product \( \bar{g} \) on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) such that both
$$\begin{aligned} \pi _{V} : (V, g) \rightarrow (\mathfrak {h}/[\mathfrak {h},\mathfrak {h}], \bar{g}) \quad \text { and }\quad \pi _{V'} : (V', g') \rightarrow (\mathfrak {h}/[\mathfrak {h},\mathfrak {h}], \bar{g}) \end{aligned}$$are submetries, i.e., they map balls to balls.
Next, we prove that in every class of the equivalence relation (iii) there is exactly one strictly subRiemannian metric. To every leftinvariant subRiemannian metric d, we define the associated asymptotic metric \(d'\) as follows. If d is Riemannian defined by a scalar product g on \(\mathfrak {h}\), then \(d'\) is the strictly subRiemannian metric for which the horizontal space V is gorthogonal to \([\mathfrak {h},\mathfrak {h}]\) and the scalar product is \(g_V\). If d is strictly subRiemannian, then \(d'=d\).
Theorem 1.3
Let d and \(d'\) be two leftinvariant subRiemannian metrics on \({\mathbb {H}}\). Their associated asymptotic metrics are the same if and only if
Moreover, if (1) holds, then there is \(C>0\) such that
We remark that the estimate (2) in Theorem 1.3 is sharp, as we will show in Remark 4.2.
The above result can be interpreted in terms of asymptotic cones. Namely, if d is a leftinvariant Riemannian metric on \({\mathbb {H}}\) and \(d'\) is the associated asymptotic metric, then \(({\mathbb {H}},d')\) is the asymptotic cone of \(({\mathbb {H}},d)\). For the analogous result in arbitrary nilpotent groups, see [23]. By Theorem 1.3, more is true: \(({\mathbb {H}},d)\) is at bounded distance from \(({\mathbb {H}},d')\). Notice that this consequence cannot be deduced by the similar results for discrete subgroups of the Heisenberg group in [19] and [10], because the word metric is only quasiisometric to the Riemannian one. Moreover, we remark that there are examples of nilpotent groups of step two that are not at bounded distance from their asymptotic cone, see [4].
We now focus on the horoboundary. As a consequence of Theorem 1.3 and of the results of Klein–Nikas [18], we get:
Corollary 1.4
If \(d_R\) is a leftinvariant Riemannian metric on \({\mathbb {H}}\) with associated asymptotic metric \(d_{CC}\), then the horoboundary of \(({\mathbb {H}},d_R)\) coincides with the horoboundary of \(({\mathbb {H}},d_{CC})\); hence, it is homeomorphic to a twodimensional closed disk \(\bar{D}^2\).
More precisely, let g be the scalar product of \(d_R\) on \(\mathfrak {h}\) and \(W\subset \mathfrak {h}\) the orthogonal plane to \([\mathfrak {h},\mathfrak {h}]\). Define the norm \(\Vert w\Vert :=\sqrt{g(w,w)}\) on W. Fix a orthonormal basis (X, Y) for W and set \(Z:=[X,Y]\in [\mathfrak {h},\mathfrak {h}]\), so that (X, Y, Z) is a basis of \(\mathfrak {h}\). We identify \(\mathfrak {h}\simeq {\mathbb {H}}\) via the exponential map, which is a global diffeomorphism. So, we write \(p=w+zZ\) with \(w\in W\) and \(z\in {\mathbb {R}}\) for any point \(p\in {\mathbb {H}}\). We say that a sequence of points \(\{p_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {H}}\) diverges if it leaves every compact set. Moreover, we shall use the following terminology for a diverging sequence of the form \(p_n=w_n+z_nZ\):

(1)
vertical divergence, if there exists \(M<\infty \) such that \(\Vert w_n \Vert < M\) for all n;

(2)
nonvertical divergence with quadratic rate \(\nu \in [\infty , +\infty ]\), if \(w_n\) diverges and^{Footnote 1} \(\lim _{n \rightarrow \infty } \frac{z_n}{ 4\Vert w_n\Vert ^2} = \nu \).
Then, according to [18] (see Corollary 5.6, 5.9 and 5.13 therein), we deduce the following description of the Riemannian horofunctions:

(v): a vertically diverging sequence \(p_n = w_n+z_nZ\) converges to a horofunction h if and only if \(w_n \rightarrow w_\infty \), and in this case
$$\begin{aligned} h (w+zZ) = \Vert w_\infty \Vert  \Vert w_\infty  w \Vert ; \end{aligned}$$ 
(nv): a nonvertically diverging sequence \(p_n = w_n+z_nZ\) with quadratic rate \(\nu \) converges to a horofunction h if and only if \(\frac{w_n}{\Vert w_n\Vert } \rightarrow \hat{w}\), and then
$$\begin{aligned} h (w+zZ) = g (R_{\vartheta }(\hat{w}),w) \end{aligned}$$where \(R_{\vartheta }\) is the anticlockwise rotation in W of angle \(\vartheta =\mu ^{1} (\nu )\), and \(\mu : [\pi , \pi ] \rightarrow \overline{{\mathbb {R}}}\) is the extended Gaveau function
$$\begin{aligned} \mu ( \vartheta ) := \frac{\vartheta  \sin \vartheta \cos \vartheta }{\sin ^2(\vartheta )}. \end{aligned}$$
Moreover, all the horofunctions of \(({\mathbb {H}},d_R)\) are of type (v) or (nv), by Theorem 5.16 in [18]; it is also clear that neither is of both types.
In Sect. 5, we will also determine the Busemann points of \(\partial _h({\mathbb {H}},d_R)\), that is those horofunctions obtained by points diverging along quasigeodesics (see Definition 5.1). We obtain, as in the subRiemannian case:
Corollary 1.5
The Busemann points of \(({\mathbb {H}},d_R)\) are the horofunctions of type (nv) and can be identified to the boundary of the disk \(\bar{D}^2\).
The paper is organized as follows. In Sect. 2, we introduce the main objects and their basic properties. In Sect. 3, we estimate the difference between any two strictly subRiemannian leftinvariant metrics on \({\mathbb {H}}\). In Sect. 4, we compare any Riemannian leftinvariant metric on \({\mathbb {H}}\) and its associated asymptotic metric. At the end of the section, we shall prove Theorems 1.2 and 1.3. In Sect. 5, we concentrate on the horofunctions and we prove Corollaries 1.4 and 1.5. “Appendix” is devoted to the explicit description of subRiemannian geodesics.
Preliminaries
Definitions
The first Heisenberg group \({\mathbb {H}}\) is the connected, simply connected Lie group associated with the Heisenberg Lie algebra \(\mathfrak {h}\). The Heisenberg Lie algebra \(\mathfrak {h}\) is the only three dimensional nilpotent Lie algebra that is not commutative. It can be proven that, for any two linearly independent vectors \(X,Y\in \mathfrak {h}{\setminus }[\mathfrak {h},\mathfrak {h}]\), the triple (X, Y, [X, Y]) is a basis of \(\mathfrak {h}\) and \([X,[X,Y]]=[Y,[X,Y]]=0\).
We denote by \(\omega _{\mathbb {H}}:T{\mathbb {H}}\rightarrow \mathfrak {h}\) the leftinvariant Maurer–Cartan form. Namely, denoting by 0 the neutral element of \({\mathbb {H}}\) and identifying \(\mathfrak {h}\) with \(T_0{\mathbb {H}}\), we have \(\omega _{\mathbb {H}}(v) := \,\mathrm {d}L_{p}^{1}v\) for \(v\in T_p{\mathbb {H}}\), where \(L_p\) is the left translation by p.
Let \(\pi :\mathfrak {h}\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) be the quotient projection. Notice that \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is a commutative twodimensional Lie algebra. So the map \(\pi \) induces a Lie group epimorphism \(\hat{\pi }:{\mathbb {H}}\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\simeq {\mathbb {H}}/[{\mathbb {H}},{\mathbb {H}}]\).
SubRiemannian metrics in \({\mathbb {H}}\)
Let \(V\subset \mathfrak {h}\) be a bracket generating subspace. We have only two cases: either \(V=\mathfrak {h}\) or V is a plane and \(\mathfrak {h}=V\oplus [\mathfrak {h},\mathfrak {h}]\). In both cases, the restriction of the projection \(\pi _V:V\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is surjective. Let g be a scalar product on V and set the corresponding norm \(\Vert v\Vert :=\sqrt{g(v,v)}\) for \(v\in V\).
An absolutely continuous curve \(\gamma :[0,1]\rightarrow {\mathbb {H}}\) is said horizontal if \(\omega _{\mathbb {H}}(\gamma '(t))\in V\) for almost every t. For a horizontal curve, we have the length
A subRiemannian metric d is hence defined as
SubRiemannian metrics on \({\mathbb {H}}\) are complete, geodesic, and leftinvariant. They are either Riemannian, when \(V=\mathfrak {h}\), or strictly subRiemannian, when \(\dim V=2\). The pair (V, g) is called the horizontal space of d.
Since \(\pi _V:V\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is surjective, it induces a norm \(\Vert \cdot \Vert \) on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) such that \(\pi :(V,\Vert \cdot \Vert )\rightarrow (\mathfrak {h}/[\mathfrak {h},\mathfrak {h}],\Vert \cdot \Vert )\) is an submetry, i.e., for all \(w\in \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) it holds \(\Vert w\Vert = \inf \{\Vert v\Vert :\pi (v)=w\}\). Here we use the same notation for norms on V and on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\), because there will be no possibility of confusion. The norm on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is characterized by
for all \(R>0\).
Proposition 2.1
Let d be subRiemannian metric on \({\mathbb {H}}\) with horizontal space (V, g). Then, for all \(R>0\)
In particular, \(\hat{\pi }:({\mathbb {H}},d)\rightarrow (\mathfrak {h}/[\mathfrak {h},\mathfrak {h}],\Vert \cdot \cdot \Vert )\) is a submetry, i.e., for all \(v,w\in \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\)
Proof
Let \(v\in V\) with \(\Vert v\Vert \le R\). Set \(\gamma (t):=\exp (t v)\). Then, \(\gamma :[0,1]\rightarrow {\mathbb {H}}\) is a horizontal curve with \(d(0,\exp (v))\le \ell (\gamma ) =\Vert v\Vert \le R\). Since \(\hat{\pi }(\exp (v)) =\pi (v)\), then we have proven this inclusion.
Let \(p\in {\mathbb {H}}\) with \(d(0,p)\le R\) and let \(\gamma :[0,T]\rightarrow {\mathbb {H}}\) be a dlengthminimizing curve from 0 to p parametrized by arc length, so \(T = d(0,p)\). Then, \(\hat{\pi }\circ \gamma :[0,T]\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is a curve from 0 to \(\pi (p)\) and
In the first equality, we used the fact that \(\hat{\pi }\) is a morphism of Lie groups and its differential is \(\pi \), i.e., \(\omega _{{\mathbb {H}}/[{\mathbb {H}},{\mathbb {H}}]}\circ \,\mathrm {d}\hat{\pi }= \pi \circ \omega _{\mathbb {H}}\), where \(\omega _{{\mathbb {H}}/[{\mathbb {H}},{\mathbb {H}}]}\) is the Mauer–Cartan form of \({\mathbb {H}}/[{\mathbb {H}},{\mathbb {H}}]\). \(\square \)
Proposition 2.2
Let \(d,d'\) be two subRiemannian metrics on \({\mathbb {H}}\) such that
Then,
Proof
Let \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert '\) be the norms on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) induced by d and \(d'\), respectively. We will show that
which easily implies \(\Vert \cdot \Vert =\Vert \cdot \Vert '\), because for any fixed \(v\in \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) one has \(1=\lim _{t\rightarrow \infty }\frac{\Vert tv\Vert }{\Vert tv\Vert '} = \frac{\Vert v\Vert }{\Vert v\Vert '} \). Moreover, the equality (4) follows from Proposition 2.1 combined with (3) and (5).
Since both maps \(\hat{\pi }:({\mathbb {H}},d)\rightarrow (\mathfrak {h}/[\mathfrak {h},\mathfrak {h}],\Vert \cdot \Vert )\) and \(\hat{\pi }:({\mathbb {H}},d')\rightarrow (\mathfrak {h}/[\mathfrak {h},\mathfrak {h}],\Vert \cdot \Vert ')\) are submetries, for every \(v\in \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) there are \(p_v,p_v'\in {\mathbb {H}}\) such that \(\hat{\pi }(p_v)=\hat{\pi }(p'_v) = v\), \(\Vert v\Vert =d(0,p_v)\) and \(\Vert v\Vert '=d'(0,p_v')\).
Moreover, it holds \(\Vert v\Vert ' \le d'(0,p_v)\) and \(\Vert v\Vert \le d(0,p_v')\), again because \(\hat{\pi }\) is a submetry in both cases. Therefore,
Finally, if \(v\rightarrow \infty \), then both \(d(0,p_v)\) and \(d(0,p_v')\) go to infinity as well. The relation (5) is thus proven. \(\square \)
Balayage area and lifting of curves
Let \(V\subset \mathfrak {h}\) be a twodimensional subspace with \(V\cap [\mathfrak {h},\mathfrak {h}]=\{0\}\). Then, \([\mathfrak {h},\mathfrak {h}]=[V,V]\), i.e., V is bracket generating. Moreover, \(\pi _V:V\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is an isomorphism.
If \(\rho :[0,T]\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is a curve with \(\rho (0)=0\), then there is a unique \(\tilde{\rho }:[0,T]\rightarrow {\mathbb {H}}\) such that
Since \((\pi \circ \tilde{\rho })'=\rho '\), then \(\pi \circ \tilde{\rho }=\rho \). So, \(\tilde{\rho }\) is called the lift of \(\rho \).
The previous ODE system that defines \(\tilde{\rho }\) can be easily integrated. Let \(X,Y\in V\) be a basis, set \(Z:=[X,Y]\), so that (X, Y, Z) is a basis of \(\mathfrak {h}\). Let \((x,y,z)=\exp (xX+yY+zZ)\) be the exponential coordinates on \({\mathbb {H}}\) defined by (X, Y, Z). Using the Backer–Campbell–Hausdorff formula, one shows that X, Y, Z induce the following leftinvariant vector fields on \({\mathbb {H}}\):
Thanks to these vector fields, we can describe the Maurer–Cartan form as
The lift of \(\rho \) is hence given by the ODE
Take the coordinates (x, y) on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) given by the basis \((\pi (X),\pi (Y))\) and define the balayage area of a curve \(\rho :[0,T]\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) as
If \(\rho (0)=0\), then the balayage area of \(\rho \) corresponds to the signed area enclosed between the curve \(\rho \) and the line passing through 0 and \(\rho (T)\).
It follows that
In an implicit form, we can write
Notice that the lift \(\tilde{\rho }\) of a curve \(\rho \) depends on the choice of V. Moreover, both the area and the balayage area in \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) depend on the choice of the basis (X, Y). Nevertheless, once a plane \(V\subset \mathfrak {h}\) is fixed, the lift \(\tilde{\rho }\) does not depend on the choice of the basis X, Y.
If g is a scalar product on V and d is the corresponding strictly subRiemannian metric, the balayage area gives a characterization of dlengthminimizing curves. Let \(\bar{g}\) be the scalar product on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) induced by g. Then, the dlength of a curve \(\tilde{\rho }:[0,T]\rightarrow {\mathbb {H}}\) equals the length of \(\rho =\pi \circ \tilde{\rho }\).
Therefore, given \(p=(x,y,z)\in {\mathbb {H}}\), we have
This express the socalled Dido’s problem in the plane, and the solutions are arc of circles. It degenerates into a line if \(z=0\). We can summarize the last discussion in the following result.
Lemma 2.3
A curve \(\tilde{\rho }:[0,1]\rightarrow {\mathbb {H}}\) is dlengthminimizing from 0 to \(p=(x,y,z)\) if and only if \(\rho :=\hat{\pi }\circ \tilde{\rho }\) is an arc of a circle from 0 to \(\hat{\pi }(p)\) with \(\mathscr {A}(\rho )=z\).
Comparison between strictly subRiemannian metrics
The present section is devoted to comparing strictly subRiemannian metrics. For such metrics, Proposition 3.1 gives the only nontrivial implication in Theorem 1.2. The general case will follow from Proposition 4.1.
Proposition 3.1
Let d and \(d'\) be two strictly subRiemannian metrics on \({\mathbb {H}}\) with horizontal spaces (V, g) and \((V',g')\), respectively. Suppose that there exists a scalar product \( \bar{g} \) on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) such that both
are submetries.
Then,
Moreover, if \(d\ne d'\), then
In the proof, we will give the exact value of the supremum in (8). Indeed, by (11) and (12), we get \(\sup _{p\in {\mathbb {H}}}d(0,p)d'(0,p)=2h\), where h is defined below. For (8), we will first prove that two of such subRiemannian metrics are isometric via a conjugation \(x\mapsto gxg^{1}\) for some \(g\in {\mathbb {H}}\), and then, we apply Lemma 3.2. For (9), we will give a sequence \(p_n\rightarrow \infty \) and a constant \(c>0\) such that \(d(0,p_n)d'(0,p_n)>c\) for all \(n\in {\mathbb {N}}\).
Proof of (8)
Since \(\dim V=\dim V'=2\), then \(\pi _V\) and \(\pi _{V'}\) are isomorphisms. Therefore by the assumption, they are isometries onto \((\mathfrak {h}/[\mathfrak {h},\mathfrak {h}],\bar{g})\).
Let \(X\in V\cap V'\) be with \(g(X,X)=1\). Then, \(g'(X,X)=1\) as well.
Let \(Y\in V\) be orthogonal to X with \(g(Y,Y)=1\). Then, \(Z:=[X,Y]\ne 0\) and (X, Y, Z) is a basis of \(\mathfrak {h}\).
Let \(Y':=\pi _{V'}^{1}(\pi (Y))\in V'\). Then, \(g'(Y',Y')=1\) and \(g'(X,Y')=0\). Moreover, there is \(h\in {\mathbb {R}}\) such that \(Y'=Y+hZ\). In particular, \([X,Y']=Z\).
Using the formula \(\mathrm {Ad}_{\exp (hX)}(v) = e^\mathrm{{ad}_{hX}}v = v + h[X,v]\), we notice that
In particular, \(\mathrm {Ad}_{\exp (hX)}_V:(V,g)\rightarrow (V',g')\) is an isometry.
Therefore, the conjugation
is an isometry \(C_{\exp (hX)}:({\mathbb {H}},d)\rightarrow ({\mathbb {H}},d')\).
We can now use the following Lemma 3.2 and get
Lemma 3.2
Let G be a group with neutral element e and let \(d,d'\) be two leftinvariant distances on G.
If there is \(g\in G\) such that for all \(p\in G\)
then for all \(p\in G\)
Proof
Note that since d is left invariant, then for all \(a,b\in G\) we have \(d(e,ab) \le d(e,a) + d(e,b)\) and \(d(e,a) = d(e,a^{1})\). On the one side, we have \(d'(e,p) = d(e,gpg^{1}) \le d(e,p) + 2d(e,g)\). On the other side, we have \(d(e,p) = d(e, g^{1}gpg^{1}g) \le d(e,g^{1}) + d(e,gpg^{1}) + d(e,g) = 2 d(e,g) + d'(e,p)\). Hence, \(d(e,p)  d'(e,p) \le 2 d(e,g)\). By symmetry, we have also \(d(e,p)  d'(e,p) \le 2 d'(e,g)\). \(\square \)
Proof of (9)
We keep the same notation of the previous subsection. Up to switching V with \(V'\), we can assume \(h>0\).
Let (x, y, z) be the exponential coordinates on \({\mathbb {H}}\) induced by the basis (X, Y, Z) of \(\mathfrak {h}\), i.e., \((x,y,z) = \exp (xX+yY+zZ) \in {\mathbb {H}}\). Similarly, on \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) we have coordinates \((x,y)=x\pi (X)+y\pi (Y)\).
For \(R>0\), define
We will show that
Fix \(R>0\). Let \(\gamma :[0,T]\rightarrow {\mathbb {H}}\) be a \(d'\)minimizing curve from 0 to \(p_R\). Then, \(\hat{\pi }\circ \gamma :[0,T]\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is half circle of center (0, R) and radius R, see Fig. 1. The balayage area of \(\hat{\pi }\circ \gamma \) is
Let \(\eta :[0,T]\rightarrow {\mathbb {H}}\) be the dlengthminimizing curve from 0 to \(p_R\). Then, \(\hat{\pi }\circ \eta :[0,T]\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) is an arc of a circle of radius \(S_R\) whose balayage area is
It is clear that \(S_R>R\) and that the circle of \(\hat{\pi }\circ \eta \) has center \((\mu _R,R)\) for some \(\mu _R>0\). So we have
It is also clear from the picture that
Now, let’s look at the lengths. First of all, notice that \(\ell _{d'}(\gamma )=\ell (\hat{\pi }\circ \gamma )\) and \(\ell _d(\eta )=\ell (\hat{\pi }\circ \eta )\). For one curve, we have
and for the other, we have the estimate
which is clear from the picture. Hence,
We claim that
Let us start by checking that,
Indeed, from the first inequality of (14) together with (13) it follows
Since \(S_R>R\), then
i.e., the inequality (16).
From the second inequality of (14) together with (13), we get
Using the facts \(\mu _R\le h\) and \(S_R\le R+\mu _R\le R+ h\), from the above inequality one gets
Moreover, since \(h^2\ge \mu _R^2 = S_R^2  R^2 = (S_RR) (S_R+R)\), we also have
Finally, from (18) and (17) we obtain (15), as claimed. This completes the proof of (12) and of Proposition 3.1.
Comparison between Riemannian and strictly subRiemannian metrics
Let \(d_R\) be a Riemannian metric on \({\mathbb {H}}\) with horizontal space \((\mathfrak {h},g)\).
Let \(V\subset \mathfrak {h}\) be the plane orthogonal to \([\mathfrak {h},\mathfrak {h}]\) and let \(d_{CC}\) be the strictly subRiemannian metric on \({\mathbb {H}}\) with horizontal space \((V,g_V)\).
Fix a basis (X, Y, Z) for \(\mathfrak {h}\) such that (X, Y) is an orthonormal basis of \((V,g_V)\) and \(Z=[X,Y]\). The matrix representation of g with respect to (X, Y, Z) is
where \(\zeta >0\).
Let \(d_{CC}\) be the strictly subRiemannian metric on \({\mathbb {H}}\) with horizontal space \((V,g_V)\).
Our aim in this section is to prove the following proposition.
Proposition 4.1
If \(d_{CC}(0,p)\) is large enough, then:
In particular, it holds
For the proof of this statement, we need to know lengthminimizing curves for \(d_R\) and \(d_{CC}\), and a few properties of those, see the exposition in “Appendix.”
Proof
Let (x, y, z) be the exponential coordinates on \({\mathbb {H}}\) induced by the basis (X, Y, Z) of \(\mathfrak {h}\), i.e., \((x,y,z) = \exp (xX+yY+zZ) \in {\mathbb {H}}\). Fix \(p = (p_1,p_2,p_3)\in {\mathbb {H}}\).
Notice that both \(d_R\) and \(d_{CC}\) are generated as length metrics using the same length measure \(\ell \), with the difference that \(d_R\) minimizes the length among all the curves, while \(d_{CC}\) takes into account only the curves tangent to V. This implies that
therefore, we get the first inequality in (19). We need to prove the second inequality of (19).
If \(p\in \{z=0\}\), then \(d_{CC}(0,p)=d_R(0,p)\) by Corollary 5.9, and the claim is true.
Suppose \(p\notin \{z=0\}\) and let \(\gamma :[0,T]\rightarrow {\mathbb {H}}\) be a \(d_R\)lengthminimizing curve from \(0=\gamma (0)\) to \(p=\gamma (T)\). Since \(p\notin \{z=0\}\) and since we supposed that \(d_{CC}(0,p)\) is large enough, then by Corollary 5.7 we can parametrize \(\gamma \) in such a way that \(\gamma \) is exactly in the form expressed in Type II in Proposition 5.4 for some \(k>0\) and \(\theta \in {\mathbb {R}}\).
By Corollary 5.6, it holds
Moreover, by Corollary 5.10
Let \(\eta :[0,T]\rightarrow {\mathbb {H}}\) be the \(d_{CC}\)lengthminimizing curve corresponding to \(\gamma \) as shown in Corollary 5.10. Then, we know that \(d_{CC}(0,\eta (T)) = \ell (\eta )=T\), and
Hence, by Corollary 5.10 and (21)
i.e.,
Since \(\eta \) is a \(d_{CC}\)rectifiable curve, then \( \eta (T)_3 = \mathscr {A}(\hat{\pi }\circ \eta ) \), where \(\eta (T)_3\) is the third coordinate of the point in the exponential coordinates. Since \(\hat{\pi }\circ \gamma = \hat{\pi }\circ \eta \), then we have by (23)
Notice that \(\hat{\pi }\circ \gamma \) is an arc of a circle in \(\mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) of radius \(\frac{1}{k}\), see Proposition 5.4.
Now we want to define a horizontal curve \(\tilde{\rho }:[\epsilon ,T+\epsilon ]\rightarrow {\mathbb {H}}\), where \(\epsilon >0\) has to be chosen, such that \(\tilde{\rho }(\epsilon )=0\) and \(\tilde{\rho }(T+\epsilon )=p\). We first define a curve \(\rho :[\epsilon ,T+\epsilon ]\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) and then take its lift to \({\mathbb {H}}\).
For the definition of \(\rho \), we follow two different strategies for two different cases (see Fig. 2):
Case 1. Suppose that \(\hat{\pi }\circ \gamma \) doesn’t cover the half of the circle, i.e., \(T\le \frac{\pi }{k}\). Set \(\lambda = \hat{\pi }(p)\in \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\). Then, T is smaller than the circle of diameter \(\Vert \lambda \Vert \), i.e.,
Let \(\lambda ^\perp \in \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) be the unit vector perpendicular to \(\lambda \) and forming an angle smaller than \(\pi /2\) with the arc \(\hat{\pi }\circ \gamma \).
Let \(\epsilon >0\) such that
Now, define \(\rho :[\epsilon ,T+\epsilon ]\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) as
Notice that
and that \(\rho (T+\epsilon ) = \hat{\pi }\circ \gamma (T) = \hat{\pi }(p)\). Then, the horizontal lift \(\tilde{\rho }:[\epsilon ,T+\epsilon ]\rightarrow {\mathbb {H}}\) of \(\rho \) is a \(d_{CC}\)rectifiable curve from 0 to p.
Case 2. Suppose that \(\hat{\pi }\circ \gamma \) covers more than half of the circle. Let \(\lambda \in \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) be the diameter of the circle that contains 0. Since T is shorter than the whole circle, then
Let \(\lambda ^\perp \) be the unit vector perpendicular to \(\lambda \) and forming an angle smaller than \(\pi /2\) with the arc \(\hat{\pi }\circ \gamma \).
Let \(\epsilon >0\) be such that
Now, define \(\rho :[\epsilon ,T+\epsilon ]\rightarrow \mathfrak {h}/[\mathfrak {h},\mathfrak {h}]\) as
where we used the fact \(\lambda =\hat{\pi }\circ \gamma (\frac{\pi \Vert \lambda \Vert }{2}) \). Notice that
Then, the horizontal lift \(\tilde{\rho }:[\epsilon ,T+\epsilon ]\rightarrow {\mathbb {H}}\) of \(\rho \) is a \(d_{CC}\)rectifiable curve from 0 to p.
In both cases, \(\tilde{\rho }\) is a horizontal curve from 0 to p of length
Moreover, from (26) and (27) [respectively (28) and (29)] we get
Finally using in order (22), (30), (31), (24)
\(\square \)
Remark 4.2
The inequality (2) is sharp. Indeed, for \(z\rightarrow \infty \), we have the asymptotic equivalence
Proof of (32)
We claim that, for \(z>0\) large enough,
Let \(\gamma :[0,T]\rightarrow {\mathbb {H}}\) be a \(d_R\)lengthminimizing curve from 0 to (0, 0, z). Since z is large, we assume that \(\gamma \) is of (Type II), see Proposition 5.4, for some \(k>0\) and \(\theta =0\). Since the end point is on the Zaxis, we have
and \(z = \frac{T}{2k} + \frac{kT}{\zeta ^2}\), from which follows
We know also the length of \(\gamma \) (see Corollary 5.10) and so we get
Claim (33) is proved. From Corollary 5.8, we get \(d_{CC}(0,(0,0,z)) = 2\sqrt{\pi }\sqrt{z}\) and
\(\square \)
We are now ready to give the proof of the main theorems:
Proof of Theorem 1.2
The implication \((a)\Rightarrow (b)\) is trivial. The implication \((b)\Rightarrow (c)\) is proven in Proposition 2.2. The equivalence \((c)\Leftrightarrow (d)\) follows from Proposition 2.1. The assertion (e) is a restatement of (d). For \((d)\Rightarrow (a)\), one uses Proposition 4.1 in order to reduce to the case when both d and \(d'\) are strictly subRiemannian, and then, one applies Proposition 3.1. \(\square \)
Proof of Theorem 1.3
This is a consequence of Propositions 4.1 and of the sharpness result (9) of Proposition 3.1. \(\square \)
The horoboundary
Let (X, d) be a geodesic space and \(\mathscr {C}(X)\) the space of continuous functions \(X\rightarrow {\mathbb {R}}\) endowed with the topology of the uniform convergence on compact sets. The map \(\iota :X\hookrightarrow \mathscr {C}(X)\), \((\iota (x))(y):=d(x,y)\), is an embedding, i.e., a homeomorphism onto its image.
Let \(\mathscr {C}(X)/{\mathbb {R}}\) be the topological quotient of \(\mathscr {C}(X)\) with kernel the constant functions, i.e., for every \(f,g\in \mathscr {C}(X)\) we set the equivalence relation \(f\sim g\Leftrightarrow fg\) is constant.
Then, the map \(\hat{\iota }:X\hookrightarrow \mathscr {C}(X)/{\mathbb {R}}\) is still an embedding. Indeed, since the map \(\mathscr {C}(X)\rightarrow \mathscr {C}(X)/{\mathbb {R}}\) is continuous and open, we only need to show that \(\hat{\iota }\) is injective: if \(x,x'\in X\) are such that \(\iota (x)\iota (x')\) is constant, then one takes \(z\in Z\) such that \(d(x,z)=d(x',z)\), which exists because (X, d) is a geodesic space, and checks that
Define the horoboundary of (X, d) as
where \(cl(\hat{\iota }(X))\) is the topological closure.
Another description of the horoboundary is possible. Fix \(o\in X\) and set
Then, the restriction of the quotient projection \(\mathscr {C}(X)_o\rightarrow \mathscr {C}(X)/{\mathbb {R}}\) is an isomorphism of topological vector spaces. Indeed, one easily checks that it is both injective and surjective and that its inverse map is \([f]\mapsto ff(o)\), where \([f]\in \mathscr {C}(X)/{\mathbb {R}}\) is the class of equivalence of \(f\in \mathscr {C}(X)\).
Hence, we can identify \(\partial _hX\) with a subset of \(\mathscr {C}(X)_o\). More explicitly: \(f\in \mathscr {C}(X)_o\) belongs to \(\partial _hX\) if and only if there is a sequence \(p_n\in X\) such that \(p_n\rightarrow \infty \) (i.e., for every compact \(K\subset X\) there is \(N\in {\mathbb {N}}\) such that \(p_n\notin K\) for all \(n>N\)) and the sequence of functions \(f_n\in \mathscr {C}(X)_o\),
converge uniformly on compact sets to f.
Proof of Corollary 1.4
Let us first remark that if \(d,d'\) are two geodesic distances on X and
then
Indeed, first of all the space \(\mathscr {C}(X)_o\) depends only on the topology of X. Moreover, if \(f\in \partial _h(X,d)\), let \(p_n\in X\) be a sequence as in (36) and set \(f'_n(x):=d'(p_n,x)d'(p_n,o)\). Then,
and as a consequence of (37) we get \(f_n'\rightarrow f\) uniformly on compact sets. This shows \(\partial _h(X,d)\subset \partial _h(X,d')\). The other inclusion follows by the symmetry of (37) in d and \(d'\).
Now, if \(d_R\) and \(d_{CC}\) are metrics on \({\mathbb {H}}\) like in Corollary 1.4, then (37) is easily satisfied thanks to Theorem 1.3, and therefore, \(\partial _h({\mathbb {H}},d_R) = \partial _h({\mathbb {H}},d_{CC}) \) if the Riemannian metric \(d_R\) and the subRiemannian metric \(d_{CC}\) are compatible. The conclusion follows from [18]. \(\square \)
The Busemann points in the boundary \(\partial _h(X,d)\) are usually defined as the horofunctions associated with sequences of points \((p_n)\) diverging to infinity along rays or “almost geodesic rays”. However, in the literature there are different definitions of almost geodesic rays, according to the generality of the metric space (X, d) under consideration ([9, 14, 25]). A map \(\gamma : I=[0, +\infty ) \rightarrow (X,d)\) into a complete length space is called

a quasiray, if the length excess
$$\begin{aligned} \Delta _N (\gamma ) = \sup _{t,s\in [N,+\infty )} \ell (\gamma ; t,s) d(\gamma (t), \gamma (s)) \end{aligned}$$tends to zero for \(N \rightarrow +\infty \);

an almost geodesic ray, if
$$\begin{aligned} \Theta _N (\gamma )= \sup _{t,s\in [N,+\infty )} d (\gamma (t), \gamma (s)) + d (\gamma (s), \gamma (0)) t \end{aligned}$$tends to zero for \(N \rightarrow +\infty \).
(Notice that the second definition depends on the parametrization, while the first one is intrinsic.) We will use here a notion of Busemann points which is more general than both of them:
Definition 5.1
A sequence of points \((p_n)\) diverging to infinity in (X, d) is said to diverge almost straightly if for all \(\epsilon >0\), there exists L such that for every \(n \ge m \ge L\) we have
It is easy to verify that points diverging along a quasiray or along an almost geodesic ray diverge almost straightly. We then define a Busemann point as a horofunction f which is the limit of a sequence \(f_n(x)=d(p_n,x)d(p_n,o)\), for points \((p_n)\) diverging to infinity almost straightly.
To prove Corollary 1.5, we need the following lemma. We remind that a metric space is boundedly compact if closed balls are compact.
Lemma 5.2
Let (X, d) be a boundedly compact geodesic space, \(o\in X\) and \(\{p_n\}_{n\in {\mathbb {N}}}\subset X\) a sequence of points diverging almost straightly. Then:

(i)
the sequence \(f_n (x)=d(p_n,x)d(p_n,o)\) converges uniformly on compacts to a horofunction f;

(ii)
\(\lim _{n \rightarrow \infty } f(p_n) + d(o, p_n) =0\).
Proof
Since the 1Lipschitz functions \(f_n\) are uniformly bounded on compact sets and (X, d) is boundedly compact, then the family \(\{f_n\}_{n\in {\mathbb {N}}}\) is precompact with respect to the uniform convergence on compact sets. Hence, if we prove that there is a unique accumulation point, then we obtain that the whole sequence \(\{f_n\}_{n\in {\mathbb {N}}}\) converges.
So, let \(g,g'\in \mathscr {C}(X)\) and let \(\{f_{n_k}\}_{k\in {\mathbb {N}}}\) and \(\{f_{n'_k}\}_{k\in {\mathbb {N}}}\) be two subsequences of \(\{f_n\}_{n\in {\mathbb {N}}}\) such that \(f_{n_k}\rightarrow g\) and \(f_{n_k'}\rightarrow g'\) uniformly on compact sets. We claim
Let \(\epsilon >0\). Let \(L\in {\mathbb {N}}\) be such that (38) holds for every \(n \ge m \ge L\). Define for \(x\in X\)
Then, for \(n_i\ge n'_j\ge L\), we get for all \(x\in X\)
By taking the limit \(i\rightarrow \infty \) and \(j\rightarrow \infty \), we obtain for all \(x\in X\)
By the symmetry of the argument, also \(g'_L(x)g_L(x) \le \epsilon \) holds. Therefore for all \(x\in X\)
Setting \(R_\epsilon = g(p_L)  g'(p_L)\), we conclude the proof of claim (39).
It is now easy to conclude (i) from (39). Indeed, taking \(x=o\), we have \(R_\epsilon \le \epsilon \); therefore, for all \(\epsilon >0\) and for all \(x\in X\) \( g(x)g'(x) \le 2\epsilon , \) i.e., \(g=g'\). This completes the proof of (i).
To prove assertion (ii), fix \(\epsilon >0\) and let \(L\in {\mathbb {N}}\) be as above. Then, we have for all \(n\ge m\ge L\)
Taking first the limit \(n\rightarrow \infty \) and then \(m\rightarrow \infty \) in the above lines, we obtain the estimate
Since \(\epsilon >0\) is arbitrary, then (ii) holds true. \(\square \)
Then, the proof of Corollary 1.5 runs similarly to Theorem 6.5 of [18].
Proof of Corollary 1.5
The horofunctions of type (nv) clearly are Busemann points, as they are limits, in particular, of the Riemannian geodesic rays which are the horizontal half lines issued from the origin and which are always minimizing, see Proposition 5.4 and Corollary 5.5 in “Appendix.” On the other hand, consider a horofunction of type (v), \(h_u=(v,z) = u  u v\), for \(u \in W\). Assume that there exists an almost straightly diverging sequence of points \(p_n = v_n + z_nZ\) converging to \(h_u\). By Lemma 5.2 (ii), we deduce that
hence, \(\{v_n\}_{n\in {\mathbb {N}}}\) is necessarily an unbounded sequence. By Corollary 1.4 and the following description of horofunctions, it follows that \(h_u\) should be of type (nv), a contradiction. \(\square \)
Concluding remarks
The Riemannian Heisenberg group shows a number of counterintuitive features which is worth to stress:

(i)
in view of Corollary 1.4, all Riemannian metrics on \({\mathbb {H}}\) with the same associated asymptotic metric have the same Busemann functions, though they are not necessarily isometric (in contrast, notice that all strictly subRiemannian metrics on \({\mathbb {H}}\) are isometric). However, this is not surprising, because all leftinvariant Riemannian metrics on \({\mathbb {H}}\) are homothetic.

(ii)
there exist diverging sequences of points \(\{p_n\}_{n\in {\mathbb {N}}}\) that visually converge to a limit direction v (that is, the minimizing geodesics \(\gamma _n\) from o to \(p_n\) tend to a limit, minimizing geodesic \(\gamma _v\) with initial direction v), but whose associated limit point \(h_{\{p_n\}}\) is not given by the limit point \(\gamma _v (+\infty )\) of \(\gamma _v\). This happens for all vertically divergent sequences \(\{p_n\}_{n\in {\mathbb {N}}}\), as the limit geodesic \(\gamma _v\) is horizontal in this case (see Proposition 5.4 and Corollary 5.6 in the “Appendix”).

(iii)
there exist diverging trajectories \(\{p_n\}_{n\in {\mathbb {N}}}\), \(\{q_n\}_{n\in {\mathbb {N}}}\) staying at bounded distance from each other, but defining different limit points (e.g., vertically diverging sequences of points with different limit horofunctions).

(iv)
it is not true that, for a cocompact group of isometries G of \(({\mathbb {H}}, d_R)\), the limit set of G (which is the set of accumulation points of an orbit \(Gx_0\) in \(\partial ({\mathbb {H}}, d_R)\)) equals the whole Gromov boundary; for instance, the discrete Heisenberg group \(G= {\mathbb {H}}({\mathbb {Z}})\) has a limit set equal to the set of all Busemann points, plus a discrete subset of the interior of the disk boundary \(\bar{D}^2\). Also, the limit set may depend on the choice of the base point \(x_0 \in {\mathbb {H}}\).

(v)
The functions appearing in (nv) coincide with the Busemann functions of a Euclidean plane in the direction \(R_{\vartheta } (\hat{v}_\infty )\); that is, the horofunction h (v, z) associated with a diverging sequence \(P_n=(v_n,z_n)\) of \(({\mathbb {H}},d_R)\) is obtained just by dropping the vertical component z of the argument and then applying to v the usual Euclidean Busemann function in the direction which is opposite to the limit direction of the \(v_n\)’s, rotated by an angle \(\vartheta \) depending on the quadratic rate of divergence of the sequence (\(\vartheta \) is zero for points diverging subquadratically, and \(\vartheta =\pm \pi \) when the divergence is subquadratical).
These properties mark a remarkable difference to the theory of nonpositively curved, simply connected spaces.
Notes
 1.
From the paper [18], there is an extra 4 and a change of sign due to our different choices of coordinates.
References
 1.
Agrachëv, A.A., Barilari, D., Boscain, U.: Introduction to Riemannian and subRiemannian geometry (Lecture Notes) (2015). https://webusers.imjprg.fr/~davide.barilari/Notes.php
 2.
Ballmann, W.: Lectures on spaces of nonpositive curvature. In: Brin, M. (ed.) DMV Seminar with an Appendix, vol. 25, pp. viii+112, Birkhäuser Verlag, Basel (1995). ISBN: 37643 52426
 3.
Bridson, M.R., Haefliger, A.: Metric spaces of nonpositive curvature. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, pp. xxii+643, Springer, Berlin (1999). ISBN: 3540643249
 4.
Breuillard, E., Le Donne, E.: On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry. Proc. Natl. Acad. Sci. USA 110(48), 19220–19226 (2013). ISSN:10916490
 5.
Burago, D.: Periodic metrics. In: Seminar on Dynamical Systems (St. Petersburg, 1991), vol. 12, pp. 90–95. Progr. Nonlinear Differential Equations Appl. Birkhäuser, Basel (1994)
 6.
Busemann, H.: The Geometry of Geodesics, p. x+422. Academic Press Inc., NewYork, NY (1955)
 7.
Capogna, L. et al.: An introduction to the Heisenberg group and the subRiemannian isoperimetric problem. In: Progress in Mathematics, vol. 259, pp. xvi+223. Birkhäuser Verlag, Basel (2007). ISBN:9783764381325; 3764381329
 8.
Cerocchi, F., Sambusetti, A.: Quantitative bounded distance Theorem and Margulis’ Lemma for \(Z^n\) actions with applications to homology. Groups Geom. Dyn. (to appear)
 9.
Dal’bo, F., Peigné, M., Sambusetti, A.: On the horoboundary and the geometry of rays of negatively curved manifolds. Pac. J. Math. 259(1), 55–100 (2012). ISSN:00308730
 10.
Duchin, M., Shapiro, M.: Rational growth in the Heisenberg group. In: ArXiv eprints (2014). arXiv:1411.4201
 11.
Gromov, M.: Hyperbolic manifolds, groups and actions. In: Riemann Surfaces and Related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978). Annals of Mathematical Studies, vol. 97, pp. 183–213, Princeton Univ. Press, Princeton, NJ (1981)
 12.
Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981). ISSN:00738301
 13.
Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, vol. 8, pp. 75–263. Math. Sci. Res. Inst. Publ. Springer, New York (1987)
 14.
Haas, A.: Dirichlet points, Garnett points, and infinite ends of hyperbolic surfaces I. Ann. Acad. Sci. Fenn. Math. 21(1), 3–29 (1996). ISSN:00661953
 15.
Hadamard, J.: Les surfaces á courbures opposées et leurs lignes géodésiques. Jorn. de Math. 4(5), 27–74 (1898)
 16.
Jean, F.: Control of nonholonomic systems: from subRiemannian geometry to motion planning. In: Springer Briefs in Mathematics, pp. x+104. Springer, Cham (2014). ISBN:978 3319086897; 9783319086903
 17.
Klein, T., Nicas, A.: The horofunction boundary of the Heisenberg group. Pac. J. Math. 242(2), 299–310 (2009). ISSN:00308730
 18.
Klein, T., Nicas, A.: The horofunction boundary of the Heisenberg group: the Carnot–Carathéodory metric. Conform. Geom. Dyn. 14, 269–295 (2010). ISSN:10884173
 19.
Krat, S.A.: Asymptotic properties of the Heisenberg group. In: Zap. Nauchn. Sem. S.Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 261.Geom. i Topol. 4, vol. 268, pp. 125–154 (2014). ISSN:03732703x
 20.
Marenich, V.: Geodesics in Heisenberg groups. Geom. Dedic. 66(2), 175–185 (1997). ISSN:00465755
 21.
Milnor, J.: A note on curvature and fundamental group. J. Differ. Geom. 2, 1–7 (1968). ISSN:0022040X
 22.
Mostow, G.D.: Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies, vol. 78. pp. v+195. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1973)
 23.
Pansu, P.: Croissance des boules et des géodésiques fermées dans les nilvariétés. Ergod. Theory Dyn. Syst. 3(3), 415–445 (1983). ISSN:01433857
 24.
Patterson, S.J.: The limit set of a Fuchsian group. Acta Math. 136(3–4), 241–273 (1976). ISSN:00015962
 25.
Rieffel, M.A.: Group \(C^{*}\)algebras as compact quantum metric spaces. Doc. Math. (electronic) 7, 605–651 (2002). ISSN:14310635
 26.
Rifford, L.: SubRiemannian geometry and optimal transport. Springer Briefs in Mathematics, pp. viii+140. Springer, Cham (2014). ISBN:9783319048031; 9783319048048
 27.
Roblin, T.: Ergodicité et équidistribution en courbure négative . In: Mém. Soc. Math. Fr. (N.S.), vol. 95, pp. vi+96 (2003). ISSN:0249633X
 28.
Stoll, M.: On the asymptotics of the growth of 2step nilpotent groups. J. Lond. Math. Soc. (2) 58(1), 38–48 (1998). ISSN:00246107
 29.
Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50, 171–202 (1979). ISSN:00738301
 30.
Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972). ISSN:00218693
 31.
Walsh, C.: The horofunction boundary of finitedimensional normed spaces. Math. Proc. Camb. Philos. Soc. 142(3), 497–507 (2007). ISSN:03050041
 32.
Walsh, C.: The horofunction boundary of the Hilbert geometry. Adv. Geom. 8(4), 503–529 (2008). ISSN:1615715X
 33.
Wolf, J.A.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Differ. Geom. 2, 421–446 (1968). ISSN:0022040X
 34.
Webster, C., Winchester, A.: Busemann points of infinite graphs. Trans. Am. Math. Soc. (electronic) 358(9), 4209–4224 (2006). ISSN:00029947
Acknowledgments
The initial discussions for this work were done at the ‘2013 Workshop on Analytic and Geometric Group Theory‘ in Ventotene. We express our gratitude to the organizers: A. Iozzi, G. Kuhn, and M. Sageev. We also thank the anonymous referee for several improving suggestions.
Author information
Additional information
E.L.D. has been supported by the Academy of Finland Project no. 288501. S.N.G. has been supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/20072013/ under REA Grant agreement no. 607643.
Appendix: Lengthminimizing curves for \(d_{CC}\) and \(d_R\)
Appendix: Lengthminimizing curves for \(d_{CC}\) and \(d_R\)
In the Heisenberg group, locally lengthminimizing curves are smooth solutions of an Hamiltonian system both in the Riemannian and in the subRiemannian case. Locally, lengthminimizing curves are also called geodesics.
Let \(d_R\) be a Riemannian metric on \({\mathbb {H}}\) with horizontal space \((\mathfrak {h},g)\). Let \(V\subset \mathfrak {h}\) be the plane orthogonal to \([\mathfrak {h},\mathfrak {h}]\) and let \(d_{CC}\) be the strictly subRiemannian metric on \({\mathbb {H}}\) with horizontal space \((V,g_V)\).
Fix a basis (X, Y, Z) for \(\mathfrak {h}\) such that (X, Y) is an orthonormal basis of \((V,g_V)\) and \(Z=[X,Y]\). Set \(\zeta = \sqrt{g(Z,Z)}\).
The basis (X, Y, Z) induces the exponential coordinates (x, y, z) on \({\mathbb {H}}\), i.e., \((x,y,z)=\exp (xX+yY+zZ)\). We will work in this coordinate system.
The Riemannian and subRiemannian lengthminimizing curves are known, and we recall their parametrization in the following two propositions. SubRiemannian geodesics can be found with different notation in [7]. Riemannian geodesics are found in [20], in different coordinates and with parametrization by arc length.
Proposition 5.3
(subRiemannian geodesics) All the nonconstant locally lengthminimizing curves of \(d_{CC}\) starting from 0 and parametrized by arc length are the following: given \(k\in {\mathbb {R}}{\setminus }\{0\}\) and \(\theta \in {\mathbb {R}}\)

(Type I) The horizontal lines \(t\mapsto (t\cos \theta , t\sin \theta , 0)\);

(Type II) The curves \(t\mapsto (x(t),y(t),z(t))\) given by
$$\begin{aligned} \left\{ \begin{aligned} x(t)&= \frac{1}{k}\left( \cos \theta (\cos (kt)1)  \sin \theta \sin (kt) \right) \\ y(t)&= \frac{1}{k}\left( \sin \theta (\cos (kt)1) + \cos \theta \sin (kt) \right) \\ z(t)&= \frac{1}{2k} t \frac{1}{2k^2}\sin (kt){.} \end{aligned} \right. \end{aligned}$$Here the derivative at \(t=0\) is \((\sin \theta ,\cos \theta ,0)\).
Proposition 5.4
(Riemannian geodesics) All nonconstant locally lengthminimizing curves of \(d_R\) parametrized by a multiple of arc length and starting from 0 are the following: given \(k\in {\mathbb {R}}{\setminus }\{0\}\) and \(\theta \in {\mathbb {R}}\)

(Type 0) The vertical line \(t\mapsto (0,0,t)\);

(Type I) The horizontal lines \(t\mapsto (t\cos \theta , t\sin \theta , 0)\);

(Type II) The curves \(t\mapsto (x(t),y(t),z(t))\) given by
$$\begin{aligned} \left\{ \begin{aligned} x(t)&= \frac{1}{k}\left( \cos \theta (\cos (kt)1)  \sin \theta \sin (kt) \right) \\ y(t)&= \frac{1}{k}\left( \sin \theta (\cos (kt)1) + \cos \theta \sin (kt) \right) \\ z(t)&= \frac{1}{2k} t \frac{1}{2k^2}\sin (kt) + \frac{k}{\zeta ^2}t. \end{aligned} \right. \end{aligned}$$Here the derivative at \(t=0\) is \((\sin \theta ,\cos \theta ,\frac{k}{\zeta ^2})\), which has Riemannian length \(\sqrt{1+\frac{k^2}{\zeta ^2}}\).
The expression of geodesics helps us to prove the following facts.
Corollary 5.5
The horizontal lines of Type I are globally \(d_R\) and \(d_{CC}\)lengthminimizing curves.
Corollary 5.6
Both \(d_R\) and locally \(d_{CC}\)lengthminimizing curves \(\gamma \) of Type II are not minimizing from 0 to \(\gamma (t)\) if \(t > \frac{2\pi }{k}\).
Proof
This statement depends on the fact that in both cases, if we fix \(k\in {\mathbb {R}}{\setminus }\{0\}\), then for all \(\theta \) the corresponding lengthminimizing curves \(\gamma _{k,\theta }\) of Type II meet each other at the point \(\gamma _{k,\theta }(2\pi /k)\). \(\square \)
Corollary 5.7
The locally \(d_R\)lengthminimizing curve \(\gamma \) of Type 0, \(t\mapsto (0,0,t)\), is not minimizing from 0 to \(\gamma (t)\) for \(t>\frac{2\pi }{\zeta ^2}\).
Proof
For \(k>0\), let \(\gamma _k\) be the \(d_R\)lengthminimizing curve of Type II with this k and \(\theta =0\). Then, \((\gamma _k)_3(\frac{2\pi }{k}) = \frac{\pi }{k^2} + \frac{2\pi }{\zeta ^2}\). Letting \(k\rightarrow \infty \), we obtain \(\hat{z}:= \frac{2\pi }{\zeta ^2}\). This means that for every \(\epsilon >0\) there is \( z \le \hat{z}+\epsilon \) and \(k>0\) such that \(\gamma _k( \frac{2\pi }{k}) = (0,0,z)\). Therefore, \(t\mapsto (0,0,t)\) cannot be minimizing after z and therefore after \(\hat{z}\). \(\square \)
Corollary 5.8
If \(p=(x,y,p_3)\) and \(q=(x,y,q_3)\), then
Proof
First suppose \(p=0\): we have to prove that \(d_{CC}(0,(0,0,z))= 2\sqrt{\pi }\sqrt{z}\). This is done by looking at the lengthminimizing curves: they come from complete circle of perimeter \(2\pi R=d\) and area \(\pi R^2=z\), so that \(d_{CC}(0,(0,0,z))=2\pi \sqrt{\frac{z}{\pi }} = 2\sqrt{\pi }\sqrt{z}\).
The general case follows from the left invariance of \(d_{CC}\):
\(\square \)
Corollary 5.9
If \(p\in \{z=0\}\), then
Corollary 5.10
\(d_R\) and \(d_{CC}\)lengthminimizing curves of Type II are in bijection via the following rule: If \(\eta :[0,T]\rightarrow {\mathbb {H}}\) is a \(d_{CC}\)lengthminimizing curve of Type II, then
is \(d_R\)lengthminimizing of Type II, where \(k\in {\mathbb {R}}\) is given by \(\eta \). Moreover, it holds
and
Proof
All the statements come directly from the expression of the geodesics. Notice that a \(d_{CC}\)lengthminimizing curve \(\eta \) of Type II is parametrized by arc length, i.e., \(\Vert \omega _{\mathbb {H}}(\eta ')\Vert \equiv 1\).
On the other hand, the corresponding \(d_R\)lengthminimizing curve \(\gamma \) has derivative \(\omega _{\mathbb {H}}(\gamma ')=\omega _{\mathbb {H}}(\eta ') + \frac{k}{\zeta ^2} Z\), where \(\omega _{\mathbb {H}}(\eta ')\) is orthogonal to Z. Hence, \(\Vert \omega _{\mathbb {H}}(\gamma ')\Vert ^2 = 1 + \frac{k^2}{\zeta ^4}\) \(\square \)
Rights and permissions
About this article
Cite this article
Le Donne, E., NicolussiGolo, S. & Sambusetti, A. Asymptotic behavior of the Riemannian Heisenberg group and its horoboundary. Annali di Matematica 196, 1251–1272 (2017). https://doi.org/10.1007/s1023101606152
Received:
Accepted:
Published:
Issue Date:
Keywords
 Heisenberg group
 Horoboundary
 Asymptotic cone
 Riemannian geometry
 SubRiemannian geometry
Mathematics Subject Classification
 20F69
 53C23
 53C17