Boundaries of Hyperbolic Metric Spaces
Abstract.
We investigate the relationship between the metric boundary and the Gromov boundary of a hyperbolic metric space. We show that the Gromov boundary is a quotient topological space of the metric boundary, and that therefore a wordhyperbolic group has an amenable action on the metric boundary of its Cayley graph. This result has significance for the study of Lipnorms on group C*algebras.
Key words and phrases:
Hyperbolic space, hyperbolic group, Gromov boundary, Cayley graph, group C*algebra, quantum metric space1991 Mathematics Subject Classification:
Primary 20F65; Secondary 46L87, 53C231. Introduction
The Gromov boundary of a hyperbolic metric space has been extensively studied, but the Gromov boundary is not guaranteed to exist for nonhyperbolic metric spaces. Gromov [4] introduced another boundary which makes sense for any metric space, but this was little studied until Rieffel [8] showed that this second boundary, called the metric boundary in his papers, is important in the study of metrics on the state spaces of group C*algebras.
If is a countable discrete group equipped with a length function , and is its reduced C*algebra, then one has a seminorm defined on a dense *subalgebra of , where is multiplication by and operates by convolution on . This in turn gives a metric on the state space of by
and a natural question to ask is whether the topology generated by this metric coincides with the weak* topology on the state space, ie. the seminorm is a Lipnorm [6, 7, 9]. Rieffel proves that this is in fact the case for with certain length functions, and a critical requirement in his proof is that the action of on its metric boundary is always amenable.
There is some interest, then, in knowing when the action of a group is amenable on its metric boundary. In the case of wordhyperbolic groups with the standard wordlength metric, it is known that the action of a wordhyperbolic group on its Gromov boundary is amenable [2, 3], and as Rieffel points out in [8], if there is an equivariant, continuous surjection from the metric boundary onto the Gromov boundary, then the action of the group on the metric boundary must be amenable.
We show that this is in fact the case, and more: the Gromov boundary is a quotient topological space of the metric boundary in a completely natural way, and that the quotient map is therefore such an equivariant, continuous surjection from the metric boundary to the Gromov boundary.
We note here that Ozawa and Rieffel [5] have shown that, for hyperbolic groups, is in fact a Lipnorm using techniques which do not use the notion of the metric boundary. However these methods do not work for , and we hope that our result may lead to a unified way of showing that the seminorms for these groups are in fact Lipnorms.
This paper is part of an undergraduate research project between the authors. The authors would like to thank Michelle Schultz for organizing the undergraduate research seminar at UNLV, and Marc Rieffel for encouraging this line of research.
2. The Gromov Boundary
There are many different but equivalent definitions for a hyperbolic metric space, but for our purposes we are only interested in a couple. We follow Alonso, Smith, et. al. [1], in our presentation, and a more complete discussion of hyperbolic spaces can be found there.
Definition 2.1.
A metric space is geodesic if given any two points , , there is an isometry from the interval into .
If is a metric space, with a basepoint , we define an inner product by
Where the base point is implicit, we will just write .
The metric space is hyperbolic if it is geodesic and there is some such that
(1) 
for all .
One can show that although the constant may be different for different basepoints, whether or not the space is hyperbolic does not depend on the choice of basepoint.
We have a particular interest in groups whose Cayley graphs are hyperbolic, and there is an equivalent definition based on properties of generators and relations alone. We note that if is a group with a finite presentation , then given a reduced word in the generators, , with in , we can write as a product
where is a word in , and . For a given , let be the smallest possible number of terms in such a product, and let be the length of .
Definition 2.2.
Let be a group with a finite presentation . We say that is wordhyperbolic if it satisfies a linear isoperimetric inequality: there is some such that
for all reduced words with in .
One can show that the choice of generators and relations does not affect whether or not the group is wordhyperbolic and, moreover, a group is wordhyperbolic if and only if its Cayley graph (regarded as a 1complex with the graph metric) is hyperbolic.
Perhaps the simplest way to consider the Gromov boundary is as the limit points of geodesic rays, where two geodesic rays are considered equivalent if they are a finite distance apart. This definition highlights similarities between the Gromov boundary and the metric boundary discussed in the next section. However, the most useful definition of the Gromov boundary for our purposes is in terms of the inner product.
Definition 2.3.
Let be a metric space. We say that a sequence converges to infinity (in the Gromov sense) if
Given two sequences and which both converge to infinity, we define a relation by
If is a hyperbolic metric space, then is in fact an equivalence relation on sequences which converge to infinity. It is worthwhile noting that if is hyperbolic then
If is not hyperbolic, the relation will not be an equivalence relation, in general:
Example 2.1.
Consider the Cayley graph of with the standard generators and relations. Let , and . All three sequences converge to infinity, but although and , .
We define the Gromov boundary of a hyperbolic metric space to be the set of equivalence classes of sequences which converge to infinity. We will say that a sequence in converges to an equivalence class in if it is an element of the equivalence class.
We can topologise the boundary by extending the inner product to .
Definition 2.4.
Let be a hyperbolic metric space, and let , . Then we define
One can show that if this inner product is restricted to , it is the same as the original inner product on . Indeed, if , and , we have
It is also the case that if is hyperbolic, with
for all , and , then the same identity holds for this extended inner product. We have
for all , and .
We then can say that a sequence converges to if and only if
With this definition, it can be shown that is a compactification of .
3. The Metric Boundary
We now consider the metric compactification and the metric boundary. The most succinct definition is that the metric compactification of a metric space corresponds to the pure states of the commutative, unital, C*algebra generated by the functions which vanish at infinity on , the constant functions, and the functions of the form
where is some fixed basepoint (which does not affect the resulting algebra). The metric boundary is simply .
More concretely, we can understand the metric boundary as a limit of rays in much the same way as the simple definition of the Gromov boundary.
Definition 3.1.
Let be a metric space, and an unbounded subset of containing , and let . We say that

is a geodesic ray if
for all , .

is an almostgeodesic ray if for every , there is an integer such that
for all , with .

is a weaklygeodesic ray if for every and every , there is an integer such that
and
for all , with , .
It is immediate that every geodesic ray is an almostgeodesic ray. Rieffel showed that every almostgeodesic ray is a weaklygeodesic ray. The significance of weakly geodesic rays is that they give the points on the metric boundary in reasonable metric spaces.
Theorem 3.1 (Rieffel).
Let be a complete, locally compact metric space, and let be a weakly geodesic ray in . Then
exists for every , and defines an element of . Conversely, if is proper and if has a countable base, then every point of is determined as above by a weaklygeodesic ray.
This is similar in character to the definition of the Gromov boundary, although the reliance on weaklygeodesic rays is necessary in general. Rieffel defined any point which is the limit of an almostgeodesic ray to be a Busemann point, and it was shown in [10] that even for simple hyperbolic spaces the metric boundary may have nonBusemann points. It is an open question as to whether this phenomenon can occur with wordhyperbolic groups.
Unlike the Gromov boundary, the metric boundary is, in general, dependent upon the choice of metric. For example, different generating sets for an infinite discrete group generally give distinct metric boundaries for the corresponding wordlength metrics.
From a practical viewpoint, the initial definition of the metric boundary means that a sequence converges to a point on the metric boundary iff is eventually outside any compact subset of , and converges for all . Two sequences converge to the same point on the metric boundary iff
for every . We can extend the functions to the boundary by letting
for any sequence . Then a sequence converges to iff for all , and this is sufficient to determine the topology of the metric compactification.
4. The Gromov Boundary as a Quotient
We observe that the functions and the inner product are closely related, since
and furthermore, they play similar roles in the definitions of Gromov and metric boundaries. It is natural, therefore, to ask what relationship there may be between the two different boundaries.
The key observation is that the triangle inequality implies that for any ,
with equality iff lies on a geodesic path . We will want to show that that gets large for elements from various sequences, and this implies that all we need do is find a so that is large.
The following lemma tells us that as we get close to a metric boundary point, we can find such that is large.
Lemma 4.1.
Let be a proper geodesic metric space with a distinguished basepoint 0. Then for any in the metric boundary of , and any , there is a point such that .
Proof.
Let be any sequence which converges to .
Let and consider a collection of minimal paths for large enough that . Because is a geodesic metric space, there must be a unique point in each of these paths in the sphere of radius , centred at 0. Since is proper the sphere is compact, and so given any we must be able to find at least one point such that an infinite number of the lie within of . Let be the subsequence of corresponding to this infinite subset. Then we have, for and sufficiently large,
or, equivalently,
Taking limits, we conclude that
Hence, given any , we can choose and such that , and obtain a point such that
∎
This lemma has two immediate corollaries:
Corollary 4.2.
Let be a proper geodesic metric space with a distinguished basepoint , and let . Then converges to infinity in the Gromov sense.
Proof.
We know that for all , eventually gets close to . Hence by the previous lemma, for any can find a such that for all sufficiently large.
However, we than have that if and are large enough that both and are greater than , then
Therefore
and so goes to infinity in the Gromov sense. ∎
Let and be two sequences in which converge to points on the metric boundary. We will say that if these two sequences converge to the same metric boundary point. Similarly, if these sequences converge to points on the Gromov boundary, we will say that . Note that despite the notation is not necessarily an equivalence relation.
Corollary 4.3.
Let be a proper geodesic metric space. Then implies .
Proof.
Let and both converge to . Using the lemma, we can find a point so that is arbitrarily large, and since both and converge to , for any number we can find such that both and are greater then for all sufficiently large.
Hence
for all sufficiently large, and so
and so . ∎
These two corollaries mean that we have a welldefined relation on given by iff given any and , we have . Furthermore, if is an equivalence relation (as it is for hyperbolic spaces), then is an equivalence relation on , and moreover as sets. As usual, we will denote the equivalence class of a point in the metric boundary by .
What we want is to show that we in fact have as topological spaces. In other words, we need to show that the quotient map is continuous.
Lemma 4.4.
Let be a proper hyperbolic metric space. If in , then in .
Proof.
Let be the hyperbolic constant from (1), and . We know that we can find such that is arbitrarily large, and since we have , we can choose such that is also arbitrarily large, for all sufficiently large. Indeed, as in the previous corollaries, we have, given any we can find a number such that for all , there is a number such that
for all .
Now if and , we know that we can find a subsequence of each sequence such that
Furthermore, since we conclude that for any there is some such that
for all , and similarly that there is some such that
for all .
Hence, given any , and fixing some , then we have
for all . But then
for all . Therefore, for any ,
and so
for all . And since does not depend on the choice of sequences converging to and , we therefore have that
for all .
Therefore
and so in . ∎
So we have proved the following result.
Theorem 4.5.
Let be a proper, hyperbolic metric space. Then there is a natural continuous quotient map from onto .
5. Boundaries of WordHyperbolic Groups
We observe that if is a hyperbolic group, then the group acts on either boundary by taking a sequence and letting
This is a continuous action on either boundary. Clearly the quotient map is equivariant for these two actions, since if , we can easily see that by simply changing the base point of the inner product to .
An action of a topological group on a topological space is amenable if there is a net of continuous maps , where is the set of Borel probability measures on , such that
uniformly on compact subsets of . Such a net of maps is called an approximate invariant continuous mean. It was shown by E. Germain (as discussed in [2, 3]) that the action of a wordhyperbolic group on its Gromov boundary is amenable. Rieffel pointed out that if there were a continuous, equivariant surjection from to the Gromov boundary, then the action of on the metric boundary must also be amenable. This is trivial given the above definition, since if is the quotient map of Theorem 4.5, and are the maps in an approximate invariant continuous mean for the action of on , then are an approximate invariant continuous mean for the action of on .
Corollary 5.1.
If is wordhyperbolic group with a finite generating set, and is the wordlength metric, then the group action on the metric boundary is amenable.
This would seem to open the possibility of replicating Rieffel’s work on the metric boundary of in the setting of hyperbolic groups. However, Rieffel’s procedure relied on the fact that the action of on its metric boundary always has finite orbits, and it seems unlikely that this criterion holds with any frequency for general hyperbolic groups.
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