Sub-Finsler Horofunction Boundaries of the Heisenberg Group

: We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics, that is, those that arise as asymptotic cones of word metrics, on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.


Introduction . Describing the horofunction boundary
The study of boundaries of metric spaces has a rich history and has been fundamental in building bridges between the elds of algebra, topology, geometry, and dynamical systems. Understanding the boundary was essential in the proof of Mostow's rigidity theorem for closed hyperbolic manifolds, and boundaries have also been used to classify isometries of metric spaces, to understand algebraic splittings of groups, and to study the asymptotic behavior of random walks.
The simplest and most classical setting for horofunctions is in the study of isometries of the hyperbolic plane. There, the isometry group splits and induces a geodesic ow and a horocycle ow on the tangent bundle; horocycles, or orbits of the horocycle ow, are level sets of horofunctions. The notion has since been abstracted by Busemann, generalized by Gromov, and used by Rie el, Karlsson-Ledrappier, and many others to derive results in various elds. The horofunction boundary is obtained by embedding a metric space X into the space of continuous real-valued functions on X via the metric, as we will de ne below.
In this paper, we develop tools to study the horofunction boundary of homogeneous groups, in particular the real Heisenberg group H. The horofunction boundary of the Heisenberg group has been the subject of study in several publications. Klein and Nicas described the boundary of H for the Korányi and sub-Riemannian metrics [13,14], while several others have studied the boundaries of discrete word metrics in the integer Heisenberg group [1,25]. In this paper, we aim to understand the horofunction boundary of the real Heisenberg group H for a family of polygonal sub-Finsler metrics which arise as the asymptotic cones of the integer Heisenberg group for di erent word metrics [22].
While horofunction boundaries are not (yet) used as widely as visual boundaries or Poisson boundaries, they admit a theory which is useful across several elds including geometry, analysis, and dynamical systems. Whether it is classifying Busemann functions, giving explicit formulas for the horofunctions, describing the Nate Fisher: Department of Mathematics, Tufts University, Medford, USA, E-mail: nathan. sher@tufts.edu *Corresponding Author: Sebastiano Nicolussi Golo: Department of Mathematics and Statistics, University of Jyväskylä, Finland, E-mail: sebastiano2.72@gmail.com topology of the boundary, or studying the action of isometries on the boundary, the understanding and the description of a horofunction boundary varies signi cantly between works.
In this paper, as is done in for the ∞ metric on R n in [6], we hope to combine these analytic, topological, and dynamical descriptions while also introducing a more geometric approach. In particular, we want to associate a "direction" to every horofunction as well as a geometric condition for a sequence of points to induce a horofunction. In some settings, the horofunction boundary is made up entirely of limit points induced by geodesic rays, or, in other words, every horofunction is a Busemann function. It is known that in CAT(0) spaces [2] as well as in polyhedral normed vector spaces [12], the horofunction boundary is composed only of Busemann functions. This connection between horofunctions and geodesic rays provides a natural notion of directionality to the horofunction boundary, which is not present in settings of mixed curvature, as described in [17]. For the model we develop in homogeneous metrics, sequences converging to a horofunction can often be dilated back to a well-de ned point on the unit sphere, which we can then regard as a direction. In these sub-Finsler metrics, there are many directions with no in nite geodesics at all, so this provides one of the motivating senses in which the horofunction boundary is a better choice to capture the geometry and dynamics in nilpotent groups.

. Outline of paper
For any homogeneous group, we convert the problem of describing the horofunction boundary to a study of generalized directional derivatives, called Pansu derivatives, of the distance function. It su ces to understand Pansu derivatives on the unit sphere, i.e., the set of points at distance one from the neutral element of the group. Therefore, in any homogeneous group where the unit sphere is understood, our method allows a description of the horofunction boundary.
Pansu-di erentiable points on the sphere, i.e., points p at which distance to the origin has a well de ned Pansu derivative, can be thought of as directions of horofunctions. Not all horofunctions are directional; the rest are blow-ups of non-di erentiable points. Background on homogeneous groups, Pansu derivatives, and horofunctions is provided in §2. We use Kuratowski limits, a notion of set convergence in a metric space, to de ne the blow-up of a function in §3.
In the remainder of the paper, we focus on the Heisenberg group H. For sub-Riemannian metrics on H, Klein-Nicas showed that the horofunction boundary is a topological disk [14]. In Theorem 4.1 of §4 we show that an analogous disk belongs to the boundary for the larger class of sub-Finsler metrics, but is a proper subset in many cases.
Our main theorem (Theorem 5.4 in §5) describes the horoboundary of polygonal sub-Finsler metrics on H in terms of blow-ups. From this, we are able to give explicit expressions for the horofunctions, to describe the topology of the boundary, and to identify Busemann points. This description is extremely explicit and allows us to visualize the horofunction boundary and to understand it geometrically. We get a correspondence between "directions" on the sphere and functions in the boundary, as indicated in Figure 1. This description allows us to realize the horofunction boundary as a kind of dual to the unit sphere, generalizing previous observations for normed vector spaces and for the sub-Riemannian metric on H [6,10,12,14,24].
Finally, using our description of the boundary, we also study the group action on the boundary in §6, generalizing results of Walsh and Bader-Finkelshtein [1,25].

Preliminaries on homogeneous groups and horofunctions
We begin with a brief introduction to graded Lie groups, homogeneous metrics, Pansu derivatives, and horofunctions. For a survey on graded Lie groups and homogeneous metrics, we refer the interested reader to [15].
Sphere Sphere Sphere Sphere Boundary Boundary Boundary Boundary Figure 1: The duality between unit spheres and horofunction boundaries for various metric spaces, where colors indicate a correspondence between directions on the spheres and points in the boundary. Note that in both the round and hexagonal cases, the 2D spheres and boundaries embed in the Heisenberg spheres (along the equators) and boundaries.

. Graded Lie groups
Let V be a real vector space with nite dimension and [·, ·] : V × V → V be the Lie bracket of a Lie algebra g = (V , [·, ·]). We say that g is graded if subspaces V , . . . , Vs are xed so that V = V ⊕ · · · ⊕ Vs and [V i , The sum in the formula above is nite because g is nilpotent. The resulting Lie group, which we denote by G, is nilpotent and simply connected; we will call it graded group or strati ed group, depending on the type of grading of the Lie algebra. The identi cation G = V = g corresponds to the identi cation between Lie algebra and Lie group via the exponential map exp : g → G. Notice that p − = −p for every p ∈ G and that is the neutral element of G.
If g is another graded Lie algebra with underlying vector space V and Lie group G , then, with the same identi cations as above, a map V → V is a Lie algebra morphism if and only if it is a Lie group morphism, and all such maps are linear. In particular, we denote by Hom h (G; G ) the space of all homogeneous morphisms from G to G , that is, all linear maps V → V that are Lie algebra morphisms (equivalently, Lie group morphisms) and that map V j to V j . If g is strati ed, then homogeneous morphisms are uniquely determined by their restriction to V .
For λ > , de ne the dilations as the maps δ λ : V → V such that δ λ v = λ j v for v ∈ V j . Notice that δ λ δµ = δ λµ and that δ λ ∈ Hom h (G; G), for all λ, µ > . Notice also that a Lie group morphism F : G → G is homogeneous if and only if F • δ λ = δ λ • F for all λ > , where δ λ denotes the dilations in G . We say that a subset M of V is homogeneous if δ λ (M) = M for all λ > .
A homogeneous distance on G is a distance function d that is left-invariant and 1-homogeneous with respect to dilations, i.e., When a strati ed group G is endowed with a homogeneous distance d, we call the metric Lie group (G, d) a Carnot group. Homogeneous distances induce the topology of G, see [18,Proposition 2.26], and are biLipschitz equivalent to each other. Every homogeneous distance de nes a homogeneous norm de( , where e is the neutral element of G. We denote by | · | the Euclidean norm in R .

. Pansu derivatives
Let G and G be two Carnot groups with homogeneous metrics d and d , respectively, and let The map L is called Pansu derivative of f at p and it is denoted by P Df (p) or P Df |p. .

Sub-Finsler metrics
Let G be a strati ed group and · a norm on the rst layer V ⊂ TeG of the strati cation. Using lefttranslations, we extend the norm · to the sub-bundle ∆ ⊂ TG of left-translates of V . We call a curve in G admissible if it is tangent to ∆ almost everywhere, and using the norm · we can measure the length of any admissible curve. A classical result tells us that in a strati ed group, where V bracket-generates the whole Lie algebra, any two points in G are connected by an admissible curve. We then de ne a sub-Finsler length metric, also called Carnot-Carathéodory metric, by where the in mum is taken over all admissible γ connecting p to q.
and we conclude that P Dde|p = .

. The Heisenberg group
The Heisenberg group H is the simply connected Lie group whose Lie algebra h is generated by three vectors X, Y, and Z, with the only nontrivial Lie bracket [X, Y] = Z. The strati cation is given by V = span{X, Y} and V = span{Z}. Via the exponential map and the above basis for h, the Heisenberg group can be coordinatized as R with the following group multiplication: Under this group operation, the generating vectors in the Lie algebra correspond to the left-invariant vector elds X = ∂x − y∂z , Y = ∂y + x∂z, Z = ∂z .
It will sometimes be convenient to coordinatize H as R × R, in which case the group operation can be written where ω is the standard symplectic form on the plane, ω((x, y), (x , y )) = xy − x y. Denote by ∆ the horizontal distribution, the sub-bundle (or plane eld) generated by the vector elds X and Y. A curve is admissible if its derivative belongs to ∆.
Let π : H → R be the projection of a point to its horizontal components, π(x, y, z) = (x, y), which is a group morphism.
Given a path γ : [ , T] → R and an initial height z , there exists a unique lift to an admissible pathγ in H such thatγ has height z at time zero and π(γ) = γ. Using Green's theorem and applying an elementary observation, we have that the third component ofγ, is given by the sum of z and the balayage area of γ, i.e., the signed area enclosed by γ.
Let d be the sub-Finsler metric on H induced by a norm · on R with unit disk Q. The length in (H, d) of an admissible curveγ is equal to the length in (R , · ) of the projected curve π(γ). A well-known result is that geodesics in sub-Finsler metrics are lifts of solutions to the Dido problem with respect to · ; that is, geodesics are lifts of arcs which trace the perimeter of the isoperimetrix I for the given norm.

. Horoboundary of a metric space
Let (X, d) be a metric space and C (X) the space of continuous functions X → R endowed with the topology of the uniform convergence on compact sets. The map ι : X → C (X), (ι(x))(y) := d(x, y), is an embedding, i.e., a homeomorphism onto its image.
Let C (X)/R be the topological quotient of C (X) with kernel the constant functions, i.e., for every f , g ∈ C (X) we set the equivalence relation Then the restriction C (X)o → C (X)/R of the quotient map is an isomorphism of topological vector spaces, for each o ∈ X. Indeed, one easily checks that it is both injective and surjective, and that its inverse map is is constant for all z ∈ X, then taking z = x and then z = x in turn tells us that c = d(x, x ) = −d(x , x). Hence c = and Moreover,ι is continuous, but it does not need to be an embedding, as we learned from [5,Proposition 4.5]. In the following lemma, which is a generalization of [5,Remark 4.3], we show thatι is an embedding under mild conditions on the distance function.

Lemma 2.2.
Let (X, d) be a proper metric space with the following property: Then the mapι : X → C (X)/R is an embedding. In particular, any proper metric space with path connected balls satisfy (2.2) with r = and s = . And so do homogeneous distances on graded groups.
Proof. We need to show thatι maps closed sets to closed subsets ofι(X). Let A ⊂ X closed and p ∈ X \ A: we claim thatι(p) ∉ cl(ι(A)).
Notice that homogeneous distances on graded groups satisfy the above connectedness condition.
De ne the horoboundary of (X, d) as ∂ h X := cl(ι(X)) \ι(X) ⊂ C (X)/R, where cl(ι(X)) is the topological closure. Another description of the horoboundary is possible, as we identify ∂ h X with a subset of C (X)o for some o ∈ X. More explicitly: f ∈ C (X)o belongs to ∂ h X if and only if there is a sequence pn ∈ X such that pn → ∞ (i.e., for every compact K ⊂ X there is N ∈ N such that pn ∉ K for all n > N) and the sequence of functions fn ∈ C (X)o, converge uniformly on compact sets to f . If γ : [ , ∞) → X is a geodesic ray, one can check that lim t→∞ι (γ(t)) exists, and the geodesic ray converges to a horofunction. Indeed, one can check that for each x in a compact set K, {d(γ(t), x) − d(γ(t), γ( ))} is non-increasing and bounded below. These horofunctions which are the limits of geodesic rays, Busemann functions, have been widely studied and inspired the de nition of general horofunctions.

. Horofunctions and the Pansu derivative
On homogeneous groups, we observe a fundamental connection between horofunctions and Pansu derivatives of the function de : x → d(e, x).
Let d be a homogeneous metric on G with unit ball B and unit sphere ∂B. Again, we denote by e the neutral element of G and by de the function x → d(e, x).
The horofunction f is limit of the sequence of points Moreover, if de is strictly Pansu di erentiable at p, then f = P Dde|p; if pn ≡ p and de is Pansu di erentiable at p, then f = P Dde|p.
Then (2.5) holds and, up to passing to a subsequence, pn converges to a point p ∈ ∂B.
The opposite direction is also clear.

. Horofunctions on vertical bers
From the basic ingredients above, we can deduce that all horofunctions are constant on vertical bers, i.e., cosets of [G, G], when a Lipschitz property holds for de : x → d(e, x). Notice that, by [16, Proposition 3.3 and Theorem A.1], the Lipschitz property 2.7 is satis ed for all homogeneous distances on G, whenever g is strongly bracket generating, that is, the strati cation The Heisenberg group H is an example of such groups.

Proposition 2.4 (Vertical invariance of horofunctions). Suppose that G is a Carnot group and d a homogeneous distance satisfying
for some Riemannian distance ρ on G. Then, horofunctions of (G, d) are constant along the cosets of Proof. Let ρ be a left-invariant Riemannian metric on G. Recall that, by the Ball-Box Theorem [9,[19][20][21] , and let pn ∈ ∂B and ϵn → as in Lemma 2.3. Then, for every ζ ∈ [G, G] and x ∈ G,

Blow-ups of sets and functions in homogeneous groups
As we observed in Lemma 2.3, in homogeneous groups there is a connection between horofunctions in the boundary and directional derivatives along the unit sphere. Wherever the unit sphere is smooth, this directional derivative is the Pansu derivative. While the unit sphere is Pansu di erentiable almost everywhere, the nonsmooth points must be studied using a di erent strategy. In this section, we overview the Kuratowski convergence of closed sets, sometimes credited to Kuratowski-Painlevé, and we use it de ne the blow-up of functions.

. Kuratowski limits in metric spaces
Let (X, d) be a locally compact metric space and let CL(X) be the family of all closed subsets of X. If x ∈ X and C ⊂ X, we set d(x, C) := inf{d(x, y) : y ∈ C}. The Kuratowski limit inferior of a sequence {Cn} n∈N ⊂ CL(X) is de ned to be Li n→∞ Cn := q ∈ X : lim sup n→∞ d(q, Cn) = = q ∈ X : ∀n ∈ N ∃xn ∈ Cn s.t. lim n→∞ xn = q , while the Kuratowski limit superior is de ned to be It is clear that Lin Cn ⊆ Lsn Cn and that they are both closed.
If Li Cn = Ls Cn = C, then we say that the C is the Kuratowski limit of {Cn}n and we write If, for all n ∈ N, Ωn ⊂ X are closed sets and fn : Ωn → R continuous functions, then we say that, for some Ω ⊂ X closed and f : Ω → R continuous, if Ω = K-limn Ωn and if, for every x ∈ Ω and every sequence {xn} n∈N with xn ∈ Ωn and xn → x, we have f (x) = limn fn(xn). Notice that this is equivalent to say that If C n , . . . , C J n are sequences of closed sets, then one easily checks that Therefore, if the limit K-limn→∞ C j n exists for each j, then we have It is a classical result of Zarankiewicz that under mild conditions, CL(X) is sequentially compact with respect to Kuratowski convergence. Theorem 3.1 (Zarankiewicz [26]). If (X, d) is a separable metric space, then the family of closed sets is sequentially compact with respect to the Kuratowski convergence, that is, if {Cn} n∈N is a sequence of closed sets, then there is N ⊂ N in nite and C ⊂ X closed such that K-lim N n→∞ Cn = C.
For ϵ ≥ and C ⊂ X, let Lemma 3.2. Assume X to be locally compact. Let fn : X → R be a sequence of continuous functions locally uniformly converging to f∞ : X → R. Then Proof. For the rst inclusion in (3.2), let p ∈ X with f∞(p) < −ϵ < for some ϵ ≤ . Then there is r > such thatB(p, r) is compact and f∞(x) < −ϵ for all x ∈B(p, r). By the uniform convergence on compact sets, there exist N ∈ N such that fn(p) < −ϵ/ < for all n > N. Therefore, p ∈ Lin→∞{fn ≤ }. For the third inclusion in (3.2), consider a sequence {pn} n∈N ⊂ X with pn → p and fn(pn) ≤ . Then, by the uniform convergence on compact sets, we have limn fn(pn) = f (p) and thus f (p) ≤ . The last statement is a direct consequence of the fact that Kuratowski superior and inferior limits are both closed.
Then pn = γ(− /n) is a sequence of points converging to p with max F (pn) < . We conclude that p ∈ {max F < }.

Lemma 3.4.
Assume that X is locally compact. For each j integer between and J ∈ N, let {f j n } n∈N be a sequence of continuous functions f j n : X → R converging uniformly on compact sets to f j ∞ : X → R. Then the sequence of continuous functions gn := max{f j n } j converges uniformly on compact sets to g∞ Proof. We give a proof only for J = : the general case can then be proved by induction.
So, we assume J = . Let K X, ϵ > , and let To prove the claim we need to check four cases, which by symmetry reduce to the following two: In the rst case, Therefore, This proves the claim and the rst part of the lemma.

. Blow-ups of sets in homogeneous groups
Let G be a homogeneous group with a homogeneous distance d.
, we can assume, up to passing to a subsequence, that the limit = +∞, then t j = +∞; 3. otherwise, there are q j n ∈ {F j = } such that, up to a subsequence, limn δ /ϵn ((q j n ) − pn) =: v j , and we set t j = − P DF j |p(v j ). and note that near δ /ϵn (p − n p), the locus j {f j n ≤ } is a local description of the translated and dilated set δ /ϵn (p − n Ω) for all n > . We then observe that

Proof. Let
By the strict Pansu di erentiability of F j at p, the functions f j n converge uniformly on compact sets to f j ∞ (x) := P DF j |p(x) + P DF j |p(v j ).
Since zero does not belong to the convex hull of { P DF j |p} j∈J , there is w ∈ V such that P DF j |p(w) > for all j. De ne γ(t) = p exp(tw). Then d dt Proof. De ne the function Since J is nite and F j are all smooth, we have Now, we de ne pn := p(δ ϵ / n w )(δϵ n w ). As in the proof of Proposition 3.6, we de ne f j n (x) = F j (pn δϵ n x) ϵn and recall that j {f j n ≤ } gives a local description of δ /ϵn (p − n Ω). In the limit, our choice of pn will allow us to express BU(Ω, {pn}n , {ϵn}n) = j {f j ∞ ≤ } as in equation (3.5). Indeed, by our choice of pn, for any x ∈ G, it follows that Finally, using the same strategy as in the second part of the proof of Proposition 3.6, we conclude that (3.5) holds.

. Blow-ups of functions in homogeneous groups
For a continuous function f : Ω → R, we de ne If Q is a closed set, we say that a function f : Q → R is smooth if there exists a smooth extension of f in a neighborhood of Q. In particular, the derivative of f at points p ∈ ∂Q is well de ned. Theorem 3.9. Let Ω ⊂ G be a closed set such that there is a family Q of regular closed sets with disjoint interiors such that Ω = Q∈Q Q. For each Q ∈ Q, let f Q : G → R smooth such that the function f : Ω → R de ned by  Notice that the constants c Q can be determined by the continuity of g and g(e) = . If there are more than one choice of such constants, the resulting function is still the same: indeed, if g and g are two functions as in (3.6) with di erent constants, then g − g is a piecewise constant and continuous function that is in e, and thus g = g . Moreover, we remark that we don't need the limit sets R Q to have disjoint interiors. Proof. The fact that G = Q∈Q R Q follows from p ∈ Ω is nonempty and for every N ⊂ N in nite there is N ∈ N with N ⊂ N. For every N ∈ N, de ne g N := lim N n→∞ gn. We aim to prove that g N = g for all N ∈ N. Let x ∈ R Q for some Q ∈ Q. Then there exist yn ∈ Q such that xn := δ /ϵn (p − n yn) → x. Therefore, Finally, g N is continuous and g N (e) = . So, for any pair N, N ∈ N, the di erence g N − g N is a piecewise constant and continuous function that takes the value 0 at e. Hence, g N − g N ≡ , for all N ∈ N.
This theorem will allow us to nish our description of the horofunction boundary. At non-smooth points, horofunctions do not necessarily correspond to Pansu derivatives, but instead are piecewise de ned by Pansu derivatives in each blow-up region. Theorem 3.9 can also be used to recover results about the horofunction boundaries of normed spaces as in [10,24].

Vertical sequences in the Heisenberg group H
In this section, we focus on the Heisenberg group, see Section 2.4. We extend to sub-Finsler distances a result that Klein-Nicas proved for the sub-Riemannian and the Korany distances in [13,14]. In particular, we show that, for any sub-Finsler metric in the Heisenberg group H, vertical sequences induce a topological disk in the horoboundary. The result is not true for all homogeneous distances in H, see Remark 4.4. There is, therefore, a topological disk {p → w − π(p) − w : w ∈ R } ⊂ C(H) in the horofunction boundary.
We need a couple of lemmas before the proof of the theorem. We start with a technical lemma concerning convex geometry. Fix b > and an open bounded convex set Q ⊂ R . Dilate Q by λ ≥ , and take two points p, q ∈ ∂(λQ) so that |p − q| ≤ b. The line passing through p and q cuts λQ into two parts with areas s and t respectively, say s ≤ t. Then the lemma says there is M such that that s < M for all λ ≥ . Proof. Since L (Q − λ ) ≤ λ L (Q), we must show that L (Q − λ ) remains bounded for λ large. Taking λ large enough, we can assumex(λ) < +∞. De ne V λ (x) = L {y : (x, y) ∈ λQ}, and note that V λ (x) = λV (x/λ). Up to translating Q, we can assume V (x) = for all x ≤ and V (x) > for small x > . Moreover, since Q is convex, V is a concave function. If lim x→ + V (x) > , thenx(λ) = and thus L (Q − λ ) = for λ large. Now assuming that lim x→ + V (x) = , we have thatx(λ) > for all λ. By concavity, there are ϵ, m > such that V (x) ≥ mx for all x ∈ ( , ϵ]. By the de nition ofx, if for λ large enough. Moreover, the convergence is uniform in v on compact sets.
Proof. By the triangle inequality, we have for all t. Let Q * ⊂ R be the convex set dual to the unit ball Q of the norm · on R . Let I be the rotation by de(( , t)). For t large enough, the projection γ : [ , ] → R of a geodesic from ( , ) to (v, t) is a portion of the boundary of λI, for some λ, with γ ( ) = ( , ) and γ ( ) = v. Notice that a is the length of γ , that b = v is the length of a chord of ∂(λI) and that t is the area one of the two parts of λI separated by the line passing through and v. Let s be the area of the other part and c the length of ∂(λI) \ γ . If A is the area of I and is the length of ∂I, we have a + c = λ and t + s = λ A. See Figure 2. The projection γ of a geodesic from ( , ) to ( , t) is the boundary of µC, for some µ so that t = L (µC) = µ A. Then h = de(( , t)) is the length of the boundary of µC. Therefore, For the uniform convergence, if we de ne f t (v) = de((v, t)) − de(( , t)) + de((v, )), then by the reverse triangle inequality, f t : and f t (e) = . Therefore, the pointwise convergence is uniform on compact sets. Proof of Theorem 4.1. It su ces to consider the case when sn → +∞. Notice that − de((wn , sn)) + de(( , sn)) − de((wn , )) where, ω is the standard symplectic form on R . Using Lemma 4.3 and the boundedness of wn, and lim n→∞ −de((wn , sn)) + de(( , sn)) − de((wn , )) = . Finally, For the last statement, x w ∈ R and set pn = (w, n) ∈ H. Then pn → f (v, t) = w − w − v in the horofunction boundary.   (( , M)). Therefore, ((v, )), for all n ∈ N. We conclude that (4.1) cannot hold for such d.

Horofunctions in polygonal sub-Finsler metrics on H
Before stating the main result of the section and the paper, we introduce the necessary notation for the description of sub-Finsler distances in H.

. Geometry of polygonal sub-Finsler metrics
On R , we denote by ·, · the standard scalar product, and by J the "multiplication by i", i.e., the anticlockwise rotation by π . Notice that ω(·, ·) = J·, · is the standard symplectic form. We will use the symplectic duality between R and (R ) * induced by ω via Let Q be a centrally-symmetric polygon in R with N vertices, and let · be the norm on R with unit metric disk Q. Enumerate the vertices {v k } k of Q with k ∈ Z modulo N, in an anticlockwise order. Notice that −v k = v k+N . De ne the k-th edge to be the vector e k := v k+ − v k . For each k, let α k ∈ (R ) * be the linear function such that α k (v k + te k ) = for all t ∈ R, that is, Let Q * ⊂ (R ) * be the unit disk of the norm dual to · , that is, the polar dual of Q. Note that Q * is the polygon with vertices {α k } k .
A result of Busemann (see the very rst paragraphs in [3]) tells us that the isoperimetric set I, or isoperimetrix, in (R , · ) is the image of Q * in R via the symplectic duality.¹ It follows that I is the polygon with vertices where ω(e k , v k ) = ω(v k+ , v k ) < , and edges σ k := α ω k − α ω k− . Figure 3 describes the situation for a hexagonal Q. We note that σ k is a scalar multiple of v k , For the case of polygonal sub-Finsler metrics on H, Duchin-Mooney [7] classify geodesics and describe the shape of the unit sphere. Here, we introduce some of their notation and summarize some key results.
Duchin-Mooney break geodesics into two categories: beelines and trace paths. Beeline geodesics are lifts of (R , · )-geodesics to admissible paths in H. Trace path geodesics, on the other hand, are lifts of paths in the plane which trace some portion of the boundary of rescaled versions of I.
As in Duchin-Mooney, we partition Q into quadrilateral regions which are reached by trace paths which trace the same edges of I. That is, for i < j < N + i, de ne Q ij ⊂ R to be the set of all endpoints of positively-oriented trace paths in the plane whose parametrizations start by tracing a portion of σ i , trace all of σ i+ , . . . , σ j− , and end by tracing a portion of σ j , rescaled so that the total length is 1: The unit sphere of a polygonal sub-Finsler distance is the set of all endpoints of unit-length geodesics and it can be described as a the region between the graphs of two functions Q → R, see Figure 4. Endpoints of beeline geodesics make up vertical wall panels on the edges of Q: we denote by Panel i,i+ the vertical wall panel which projects to edge Q i,i+ , through vertices v i and v i+ .
Endpoints of all unit-length, positively-oriented trace path geodesics make up the ceiling of the sphere: we denote by Panel + ij the ceiling panel above Q ij , that is the set of endpoints of lifts of all unit-length, positively-oriented trace paths whose endpoints lie in Q ij . It will be useful to have an explicit description of these panels. Fix a non-degenerate quadrilateral region  the geodesic lift of the trace path implicitly described by u(r, s); that is ϕ(r, s) is the balayage area spanned by the curve: ² 3) The  Figure 5. Therefore, if i < j, then (r, s)).
So, if p ∈ ∂B lies in the basement and π(p) = u ij (r, s), then Θ(p) ∈ ∂B Θ lies in the ceiling and π(Θ(p)) = i+N (s, r). See Figure 6. Finally, for all p, v ∈ H, we have .   Figure 6: Three trace paths with similar combinatorics whose lifts end at ceiling point p = (u, ϕ(u)), basement point p − = (−u, −ϕ(u)), and the image Θ(p) of p under the involution Θ, respectively. Note that the trace path of p − has the reverse parametrization as that of p.

. The theorem
Let d be a polygonal sub-Finsler metric on the Heisenberg group H. The fundamental lemma identifying horofunctions with Pansu derivatives (Lemma 2.3) applies in this case, but we need to take care in describing all possible blow-ups of the distance function at points on the sphere.
These blow-ups take two forms. As we will explain below, H is partitioned so that the function de is C ∞ in the interior of each region. So, on the one hand, we have the points of the unit sphere where de is smooth, and thus the only blow-up is the Pansu derivative of de. On the other hand, on the non-smooth part of the unit sphere, which we call the seam, the blow-up of de is de ned piecewise as in Theorem 3.9.
The unit sphere of d is made of smooth and non-smooth points. Smooth points are the interior points of the panels on ceiling, basement, and walls. Non-smooth points are on the seams between those panels, that is: north and south poles, star-like seams near the north and south poles, and seams between ceiling or basement and wall panels, vertices of Q. See Figure 7 for the seams along a hexagonal unit sphere: each type of seam point intersects di erent combinations of panel dilation cones and hence provides a di erent kind of blow-up function. We will study the blow-ups of the distance function de in each case separately. The results are summarized in the following theorem.

Smooth points:
(S1) If p is in the interior of Panel + ij such that the π(p) = u ij (r, s), then the Pansu derivative of de exists at p and

Non-smooth points:
(̸ S1) North and south poles (̸ S3) Star-like seams (a) Near the north pole, for C ∈ R ∪ {−∞, +∞} and s ∈ ( , ]: Near the south pole, for C ∈ R ∪ {−∞, +∞} and s ∈ ( , ]: (a) Between wall and ceiling panels, for C ∈ R ∪ {−∞, +∞} and s ∈ ( , ]: Between wall and basement panels, for C ∈ R ∪ {−∞, +∞} and s ∈ ( , ]: We remark that in each of the non-smooth cases, the constants c and c are uniquely determined by the value of C since by de nition f ( , ) = and f is continuous. The proof of this theorem is the content of the rest of the section. Before diving into it, we present several consequences, in particular the description of the horoboundary of (H, d).
Theorem 5.2 has corollaries concerning the regularity of de on the sphere. Indeed, since the Pansu derivative of de on the ceiling depends only on the endpoints of trace paths, it follows that P Dde is continuous on the ceiling and the basement of ∂B, except to the star-like seams near the poles, in red in Figure 8. We could draw a similar gure for the basement, where the families would spiral in anticlockwise, instead of clockwise.   of (H, d)). The horofunction boundary of (H, d) is the union of the image of the following embeddings in C(H): First, a disk given by K : For each i, the image of these four maps is two spheres glued together along a meridian. The second meridian of each of the two spheres is the segment between α i− and α i in ∂ h (R , · ) and −∂ h (R , · ), respectively. . This is analogous to results in the sub-Riemannian case; Klein-Nicas in [14] showed that the smooth points contribute a circle's worth of functions to the boundary, while the rest of the boundary comes from vertical sequences, analogous to our Theorem 4.1.
See Figures 10 and 9. In Figure 10, we introduce a sense of directionality to the horofunction boundary. Recall that to any sequence {qn} ⊂ H converging to a horofunction, we can associate sequences {pn}n ⊂ ∂B and {ϵn}n ⊂ R, where δϵ n qn = pn. For each horofunction f ∈ ∂ h H, there exist sequences {qn}n ↔ ({pn}n , {ϵn}n) such that qn → f and pn → p ∈ ∂B. This assigns directions to horofunctions in the boundary. This correspondence between the boundary and the unit sphere is far from bijective. There exist families of directions, such as each blue vertical wall panel, which collapse to single points in the boundary. On the other hand, there are directions, such as the purple north and south poles, which blow-up to 1-or 2-dimensional families in the boundary. In these cases, which boundary point you converge to will depend on how exactly qn goes o to in nity. The colors in the gures allow us to see which directions on the sphere converge to which families horofunctions.  .

Blow-ups of d e at smooth points
First we consider blow-ups of de at smooth points on ∂B, such as in the interior of each ceiling, basement, or wall panel making up the unit sphere. Since de is smooth in the interior of each of these panels, it is strictly Pansu di erentiable. We know from above that ceiling and basement points are reached by geodesics which are lifts of trace paths. It turns out that the Pansu derivative of de on the ceiling or basement depends only on where in the isoperimetrix I the trace path ends and is independent of the rest of the shape of the trace path. Similarly, if p ∈ ∂B is a basement point with π(p) = −u ij (r, s), i < j, then the Pansu derivative of de exists at p, and Proof. Given that p is in the interior of Panel + ij , de is smooth at p, and hence the Pansu derivative exists. Pansu derivatives are linear and invariant on vertical bers, so we are looking for a linear functional A ∈ (R ) * such that P Dde(p)(v, t) = A [v].
Let γ : [ , + ϵ] → H be the unit-speed trace path geodesic from the origin to γ ( ) = p. Since p is in the interior of Panel + ij , for su ciently small h we have γ ( + h) = pδ h v j , and so = lim Next, let γ : (−ϵ, ϵ) → H be a C path along the unit sphere ∂B which is horizontal at p, with γ ( ) = p and γ ( ) = w ∈ ∆p. Since γ is on the unit sphere, de(γ ) ≡ . A consequence of the horizontality of γ at p is the limit lim h→ δ /h (p − γ (h)) = γ ( ) = w, and so by the Pansu di erentiability of de at p, where again µ = r σ i + σ i+ + . . . + σ j− + s σ j . If p = (u, ϕ) has trace coordinates (r, s) in Panel + ij , then after rescaling and simplifying, we have (r, s) .
Meanwhile, the horizontal subspace at p is spanned by the left translations from the origin to p of (v i − u(r, s), ) and (v j − u(r, s), ).
This gives These two bases for Tp ∂B and ∆p allow us to nd w in the intersection as The vector w is the left-translation from the origin to p of the horizontal vector (ŵ, ), whereŵ : because Av j = . The proof of (5.6) is a long computation, of which we describe the main steps. The strategy is to write A[ŵ] in terms of the symplectic duals α ω i of the covectors α i . One can easily show that First of all, we have Secondly, one can check the following equalities: Finally, using these formulas to rewrite (5.7), one easily nds that a polynomial of order two in r and s. Each coe cient of this polynomial is easily shown to be zero, completing the proof of 5.6.
To show the result for basement points, we will use Remark 5.1 and the result just proved for ceiling points. Let p ∈ ∂B be in the basement with π(p) = u ij (r, s), i < j, so that Θ(p) lies in the ceiling of ∂B Θ and π(Θ(p)) = −u Θ −j,−i (s, r) = u Θ −j+N,−i+N (s, r). Then, for all v ∈ H, This completes the proof.

Proposition 5.7 (Wall Pansu derivatives). If p is in the interior of the wall panel
Proof. Let p = (u, t ) be in the interior of Panel i,i+ , and let q = (v, t) ∈ H. For su ciently small ϵ > , the point pδϵ q is inside the dilation cone of Panel i,i+ . In this dilation cone, de = α i • π. Thus, by de nition of the Pansu derivative and the linearity of α i , P Dde|p(q) = lim .

Blow-ups of d e at non-smooth points
We now consider blow-ups of the function de at points on the unit sphere which are not smooth, i.e., along the seams of the sphere.

. . Blow-ups near north and south poles
For each i, de ne the cones Recall from [7] that the non-degenerate Q ij containing ( , ) are Q i+ ,i for i = , . . . , N. For each i we also de ne the dilation cones U i := δ ( ,+∞) Panel + i+ ,i . Notice that Therefore, π(Panel + i+ ,i ) = Q i+ ,i ⊂ C − i , and thus U i ⊂ π − (C − i ).
Proposition 5.8 gives a second proof of Theorem 4.1 in the case of polygonal sub-Finsler distances. Proof. Suppose p is the north pole. A su ciently small neighborhood Ω of p is covered by the dilation cones U i . Moreover, up to shrinking Ω, we can suppose U i ∩ Ω = π − (C − i ) ∩ Ω. From Proposition 3.7, we conclude that all blow-ups of U i at p are H, ∅, left translations of π − (C − i ), and the half spaces π − ({v ω i ≤ }) and π − ({v ω i+ ≥ }). Next, we see from (5.2), (5.3) and (5.4) that (r, s) → (u i+ ,i (r, s), ϕ i+ ,i (r, s)) is well de ned in a neighborhood of ( , ) and the image is not tangent to [H, H] at ( , ). Notice that u i+ ,i ( , ) = ( , ). It follows that the map ψ i : (r, s, t) → δ t (u i+ ,i (r, s), ϕ i+ ,i (r, s)) is a di eomorphism near to ( , , ) and ψ i ( , , ) = p. Therefore, we can extend de| U i to a homogeneous smooth function f i de ned in a neighborhood of p by f i (q) = t(ψ − i (q)). Using Proposition 5.6 and the smoothness of f i , we deduce that We are now in the position to conclude the proof. On the one hand, if {pn} n∈N ⊂ H and {ϵn} n∈N ⊂ ( , +∞) are sequences with pn → p and ϵn → , then, up to passing to a subsequence, we can assume that BU(U i , {pn}n , {ϵn}n) exist for each i, by Theorem 3.1. Therefore, by Theorem 3.9, we obtain that For the south pole the proof is the same. Proposition 5.9 (Blow-ups at the north star seam). Let p = (u ii (r, ), ϕ ii (r, )), r ∈ ( , ), be a ceiling point above the star of Q in the degenerate panel Panel + ii . All the blow-ups of de at p are borhood Ω of p is covered by the two cones U i− and U i . Up to shrinking Ω, we have Thus, arguing like in the proof of Proposition 5.8, we can smoothly extend both de| U i and de| U i+ to Ω and show that all the blow-ups of de at p are those listed in the statement.
A similar analysis of points in star line segments in the basement of the unit sphere yields the following proposition. Checking the (r, s) coordinates of p in the three panels, one sees that the three pieces of the blow-up function are all equal to α i− . Thus the Pansu derivative of d at p exists.

. . Blow-ups along wall seams
For each i, de ne the cones is a convex combination of v i and v i+ . The boundary of W i is made up of a top and a bottom piece, each of which is smooth, which we denote by ∂W + i and ∂W − i , respectively. There exists a functionF : C + i → R whose graph is ∂W + i Indeed, ∂W i is parametrized by Using this parametrization, we solve for the height function, , which is smooth except in the {t = } plane. Notice that if s < .
(5.8) Proof. We consider three cases. First, if r = and s < , then π(p) = u i,i+ ( , s) = u i− ,i+ ( , s) ∈ Q i− ,i+ . Therefore, a neighborhood Ω of p is decomposed into two regions, Ω ∩ W i = Ω ∩ {F i ≤ } and In [25], Walsh proves that for any nitely generated nilpotent group, there is a one-to-one correspondence between nite orbits of Busemann functions under the action of the group and facets of a polyhedron de ned by the generators of the group. The following proposition generalizes this result to the real Heisenberg group for any polygonal sub-Finsler metric. Proposition 6.3. In the boundary of a polygonal sub-Finsler metric on H, there is a one-to-one correspondence between nite orbits of Busemann functions and edges of the metric-inducing polygon Q.
Proof. By Remark 6.2 and also by direct calculation, the action of the group on horofunctions of the form (v, t) → α i (v) is trivial. Since the α i are the blow-ups of de on vertical walls of the unit sphere, we get a correspondence between the facets of Q and nite orbits of the action.
It remains to show that no other Busemann functions are xed globally by the action of the group. For each vertex v i we have a family of blow-ups, in this case Busemann functions, A direct calculation shows that if g = (w, s), f ∈ F i , and ω(v i , w) ≠ , then g.f ∈ F i , but g.f ≠ f .

. Trivial action on reduced horofunction boundary
When de ning the horofunction boundary of a metric space, we de ned the maps ι : X → C (X) andι : X → C (X)/R. To de ne the reduced horofunction boundary we consider the image of ∂ h (X, d) in C (X)/C b (X), where C b (X) is the space of all continuous bounded functions. It is worth noting that the reduced horofunction boundary is not necessarily Hausdor , but as we show below, it has value in its strong relationship with the action of the group on ∂ h (X, d).
In [1], Bader-Finkelshtein show that the for any nitely generated abelian group and discrete Heisenberg group with any nite generating set, the action of the group on its reduced horofunction boundary is trivial. They further conjecture that this result should hold for any nitely generated nilpotent group. We are able to extend this result to the real Heisenberg group with a polygonal sub-Finsler metric. Proof. To prove this proposition, it will su ce to look at each of the families of functions described in the Theorem 5.2. We start by considering the three smooth families of horofunctions, which compose a circle in the boundary. These boundary points are all Pansu derivatives, and hence are linear. It is clear that two linear functions stay bounded distance from one another if and only if they are identical, and so each Pansu derivative remains distinct in the reduced horofunction boundary. By the de nition of action on the boundary, it is clear that if f is linear, then g.f = f for all g ∈ H, and so the action on these points in ∂ r h (H, d) is trivial. Next we consider the piecewise-linear horofunctions coming from the blow-ups of non-smooth points. Any (nontrivially) piecewise linear function cannot have bounded di erence from a linear function, and so they cannot be equivalent in the reduced horofunction boundary to the smooth families mentioned above. Our goal is to show that two horofunctions f and f di er by a bounded function if and only if f and f belong to the same orbit. Let f and f be distinct functions coming from the same family of functions in Theorem 5.2.