The horofunction boundary of the infinite dimensional hyperbolic space

In this paper we give a complete description of the horofunction boundary of the infinite dimensional real hyperbolic space, and characterise its Busemann points.


Introduction
The geometry of the infinite dimensional real hyperbolic space H ∞ has been the object of study since it was first suggested by Gromov in [9]. Already much work has been done on the subject and for a detailed study of H ∞ ; see [5,6,19,22].
The main goal of this paper is to determine the horofunction boundary of H ∞ and its Busemann points. The horofunction boundary is a natural way to embed a, possibly nonproper, metric space into a compact topological space. In general the horofunction boundary is much larger than other commonly used boundaries, such as the visual boundary, also known as the bordification [3,21], and the Satake compatification [25]. By now the horofunction boundary has been explicitly determined in a variety of metric spaces, including special cases of, normed spaces [10][11][12]26], Hilbert's metric spaces [16,29] and Teichmüller spaces [14,30], see [15] for an overview.
Although the horofunction boundary tends to be large, it has proven to be very useful in a variety of applications, see [15] and the references therein. In particular, it has been used in the study of isometry groups of metric spaces, see [20,28,29], the analysis of Denjoy-Wolff type problems in metric spaces, see [1,7,13,16,17], and establishing multiplicative ergodic theorems [8].
B Floris Claassens f.c.claassens@kent.ac.uk 1 School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NX, UK An important subset of the horofunction boundary consists of the Busemann points, which were introduced by Rieffel in [24]. They are horofunctions that are the limits of almost geodesics. It is of interest to understand for which spaces all horofunctions are Busemann points, see for instance [27,31].
Before stating the main result we briefly recall the definition of the real hyperbolic space. Let (H , ·, · ) be an infinite dimensional Hilbert space and let V = R ⊕ H . Let Q : V → R be the quadratic form, for u, v ∈ V + , which is a metric between pairs of rays in V + . If we restrict d h to a hyperboloid we obtain H ∞ = (H, d h ) the hyperboloid model of the infinite dimensional real hyperbolic space. Please note that we do not make any assumptions on the separability or the cardinality of the orthonormal basis of the Hilbert space H , as it does not play a role here. In this paper we will work with Klein's model of the infinite dimensional real hyperbolic space H ∞ = (D, d h ), which is defined on the disc On D the metric d h coincides with Hilbert's cross-ratio metric which is defined as follows. Let u and v be different elements of D and let l u,v be the line through u and v. Let u and v be the intersection of l x,y and the boundary of D such that u is between u and v and v is between u and v . The Hilbert distance between u and v is given by In the study of Hilbert's metric the factor 1 2 is usually ignored as it plays no role, except for fixing the curvature of the space.
We can describe the horofunction boundary and Busemann points of H ∞ as follows.
where 0 < r ≤ 1 andû ∈ D such that 0 ≤ 1 − r 2 < Q(û) or r = 1 andû ∈ ∂D. Furthermore, ξ is a Busemann point if and only if r = 1 andû ∈ ∂D, in which case The proof of Theorem 1 uses the order structure, which is connected to the hyperbolic distance as noted by Birkhoff in [2].

Preliminaries
Hilbert's metric Let (H , ·, · ) be a Hilbert space and let The closure of the cone defines a partial order structure on V by u ≤ v if and only if v − u ∈ V + . The order structure can be used to give an alternative formula for d h as follows. Define the gauge function Note that for λ, μ > 0 we have that δ(λu, μv) = δ(u, v). Birkhoff's version of Hilbert's metric was popularised by Bushell in [4] and is well known to coincide with Hilbert's cross ratio metric [18]. Using Birkhoff's version of Hilbert's metric one can generalise Hilbert's metric to the interior of any cone of a normed vector space. It is well known that on D the topology defined by Hilbert's metric and the norm topology coincide; see [17, Corollary 2.5.6].
Horofunctions and Busemann points For finite dimensional real hyperbolic spaces it is well known that ∂H n coincides with the horofunction boundary. In infinite dimensions this is no longer the case as the space is no longer proper, i.e., closed balls are not compact. Let us briefly recall the construction of the horofunction boundary. Let (X , d) a metric space and let b ∈ X be a base point. Consider the natural embedding i : X → C(X ) given by where C(X ) is equipped with the topology of compact convergence; see [23, §46]. By the triangle inequality we have so i(X ) is equicontinuous. By the same methods we also find for all x, y ∈ X that is called the horofunction boundary of (X , d) and its elements are called horofunctions. The horofunction compactification is metrizable if (X , d) is proper, which means it is sequentially compact. If X is not proper i(X ) need not be sequentially compact any more, so for the rest of the article we will use nets. It is known that the topology of compact convergence on i(X ) coincides with the topology of pointwise convergence, see the proof of Ascoli's Theorem in [23,Page 291].
We like to point out that the bordification, as discussed in Bridson and Heafliger [3], is obtained in a similar way by equipping C(X ) with the topology of uniform convergence on bounded sets. It is known that the bordification and the horofunction compactification coincide on proper geodesic metric spaces. There is the following useful example by Bader; see [15], illustrating the difference between the bordification and the horocompactification.
Example 1 Consider the intervals [0, n] for n ∈ N which are glued together to a point x 0 at the point 0. We can equip this space with a metric d for which d(x, y) = |x − y| if x and y are in the same interval and d(x, y) = x + y if x and y are in different intervals. This space is known as an R-tree, see [3]. Using x 0 as the base point, one can easily verify that there are no horofunctions in the bordification, but every unbounded sequence converges to ξ(y) = |y| in the horocompactification, which is in fact i(x 0 ).
We call a horofunction ξ ∈ i(X )\i(X ) a Busemann point if there exists an almost geodesic net (x α ) in X such that ξ = lim α i(x α ).
Note that if H = (D, d h ) has finite dimension a net (x α ) in D will only give rise to a horofunction if (x α ) is unbounded with respect to the metric, as D is norm compact. For H ∞ however, D is not proper and in Theorem 2 we will show there are horofunctions which are limits of bounded nets. Busemann points however, always are limits of unbounded nets as observed in [31]. For convenience of the reader we have included the proof.

Proposition 1
Let (x α ) be a net in a complete metric space X . If (x α ) is almost geodesic and bounded, then (x α ) converges to some x ∈ X.
Proof Let b be a base point. The first step is to prove that d(x α , b) converges to some r ∈ R. To see this we define for an index A the supremum r A = sup α≥A d(x α , b) which exists as the net is bounded. Let ε > 0 and let A be an index such that for all α ≥ α ≥ A we have Hence (d(x α , b)) is a Cauchy net from which it follows that lim α d(x α , b) = r for some r ∈ R. Now let ε > 0 and let A be an index such that for all α ≥ α ≥ A we have |r −d(x α , b)| < ε and It follows that Hence (x α ) is a Cauchy net. The proposition follows by completeness.

Classification of the horofunction boundary of H ∞
To prove Theorem 1, we will first calculate the gauge functions.
For all u, v ∈ V + we can rewrite this result using the quadratic and bilinear forms as Note that if Q(u) = Q(v) = 1, then using Proposition 2 we find which shows that indeed on V + the hyperbolic metric d h coincides with Birkhoff's version of the Hilbert metric. We also need the following basic result from functional analysis.

Lemma 1
Let (x α ) be a net in a Hilbert space H such that x α converges in the weak topology to some x ∈ H and ( x α ) converges to some r ≥ 0. Then r ≥ x . Moreover, if r = x , then (x α ) converges to x in the norm topology.
Proof Note that Using this we can now classify the horofunctions of H ∞ .

Theorem 2
Let (H , ·, · ) be an infinite dimensional Hilbert space and let V = R ⊕ H. The horofunction of the Hilbert geometry are the functions of the following form: where either x < 1 and x < r ≤ 1 or x = r = 1.
Proof Let ((1, x α )) be a net in V + such that δ(·, (1, x α )) − δ ((1, 0), (1, x α )) converges to a horofunction. By taking a subnet we may assume that (x α ) weakly converges to somê x ∈ H as the unit ball is weakly compact and ( x α ) converges to some r ≤ 1. Note that by Lemma 1, r ≥ x . Let (γ , y) ∈ V + . Using Proposition 2 we find Hence i ((1, x α ))((γ , y)) = Taking the limit gives us Note that if r = x < 1, then ξ = i (1,x). So r > x , if x < 1. Now suppose that a function is of the form as described above. Note that all we need to do is find a net ((1, x α )) in V + such that (x α ) converges weakly tox and ( x α ) converges to r . Then it will give rise to the desired horofunction by the above. If x = 1, consider the sequence ((1, (1 − 1 n )x)), clearly this sequence converges strongly to (1,x) and gives rise to a horofunction by the above. If x < 1 then let (e n ) be an orthonormal sequence in H , which exists as dim(H ) = ∞, and consider the sequence ((1,x + r 2 − x 2 e n )). Note that this converges weakly tox since (e n ) converges weakly to 0. Also note that Note that the proof of Theorem 2 also shows that ξ is a Busemann point if x = 1. We can show that these are the only horofunctions that are Busemann points. Proof In Theorem 2 we already proved that if x = r = 1, then ξ is a Busemann point. Now suppose that ξ is a Busemann point and let ((1, x α )) be an almost geodesic net such that i ((1, x α )) converges to ξ . Combining Proposition 1 and Theorem 2 gives us that δ ((1, 0), (1, x α )) is not bounded, so lim α x α = r = 1. Note that we can rewrite the horofunction as ξ ((1, y) Now suppose x < 1. Let > 0 and let A be such that for all α ≥ α ≥ A we have ε + δ ((1, 0), (1, x α )) ≥ δ ((1, 0), (1, x α )) + δ ((1, x α ), (1, x α

))
As in the proof of Theorem 2, using Proposition 2 we find Taking the exponential we find As this holds for all α ≥ α, we can take the limit with respect to α to get Finally, as this holds for all α ≥ A, we can take the limit with respect to α to find which is a contradiction.
To conclude this paper, we note that recently Gutiérrez has found the following description of the horofunction compactification of infinite dimensional Hilbert spaces; see [11]. where z ∈ H , with z < 1 and 0 < c < ∞ or ξ(y) = − y, z (y ∈ H ) (3.2) where z ∈ H and z ≤ 1.
As with H ∞ , a horofunction of the norm geometry ξ = lim α i(x α ) is uniquely determined by the weak limit z = lim α x α x α and the limit of the norms c = lim α x α . In particular if (x α ) is unbounded, which in the case of H ∞ corresponds with r = 1 and in the case of Hilbert spaces corresponds with c = ∞, we see that the general form (3.1) becomes a simplified form (3.2). Using similar arguments as in the proof of Theorem 3 one can easily check that the Busemann points are exactly the horofunctions of the form (3.2) where z = 1.
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