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Boundary rigidity of Gromov hyperbolic spaces

We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we show that for a non-compact Gromov hyperbolic complete Riemannian manifold or a Gromov hyperbolic uniform graph, boundary rigidity is equivalent to having positive Cheeger isoperimetric constant and also to being nonamenable. Moreover, several hyperbolic fillings of compact metric spaces are proved to be boundary rigid if and only if the metric spaces are uniformly perfect. Also, boundary rigidity is shown to be equivalent to being geodesically rich, a concept introduced by Shchur (J. Funct. Anal., 2013).


Introduction and main results
An important theme of the theory of Gromov hyperbolic spaces is to determine when certain information of a hyperbolic space X is completely captured by its Gromov boundary ∂X. For example: When X has a pole (see Definition 2.8), the quasi-isometry type of X is completely captured by the quasisymmetry type of ∂X, see e.g. [7,10,19]. In this paper, we consider the following natural question: When can a self quasi-isometry f of X be determined up to a bounded error by its induced map ∂f on ∂X? More precisely, we want to decide which Gromov hyperbolic spaces have the following property: Definition 1.1 (Boundary rigid). A Gromov hyperbolic metric space (X, d) is called boundary rigid if it has the following property: For any quasi-isometry f : X → X whose induced map on ∂X is the identity map, we have sup x∈X d(x, f (x)) < ∞. We call sup x∈X d(x, f (x)) the displacement of f . Boundary rigid has the following equivalent definition: Let QI(X) be the group of self quasi-isometries of X, i.e., QI(X) = {f | f : X → X is a quasi-isometry}/ ∼, where f ∼ g if and only if sup x∈X d(f (x), g(x)) < ∞. It is known that every self quasi-isometry f of a geodesic hyperbolic space X induces a self homeomorphism ∂f of ∂X and quasi-isometries uniformly close to each other induce the same homeomorphism of ∂X, see [9,10]. Therefore, there is a natural homomorphism ∂ : QI(X) → Homeo(∂X) which sends f to ∂f . Definition 1.2 (Boundary rigid). A Gromov hyperbolic metric space X is called boundary rigid if the natural homomorphism ∂ : QI(X) → Homeo(∂X) is injective. Remark 1.3. As indicated in Proposition 2.7, boundary rigidity of geodesic Gromov hyperbolic spaces is a quasi-isometric invariant.
Let PH be the class of proper geodesic Gromov hyperbolic spaces with a pole. PH includes many important examples of Gromov hyperbolic spaces: infinite hyperbolic groups [16], Gromov hyperbolic domains in R n [5,35], Gromov hyperbolic manifolds with a pole [11,21,23], a lot of negatively curved solvable Lie groups [14,15,25,32,33], and hyperbolic fillings [7,10] etc. Our main theorem decides exactly which spaces in PH are boundary rigid. Theorem 1.4. Suppose that X is a proper geodesic Gromov hyperbolic space with a pole. Then the following conditions are equivalent: (1) X is boundary rigid.
(2) ∂X is uniformly perfect. There are several equivalent definitions for uniformly perfect spaces, see [23,27]. The following was introduced by Martínez-Pérez and Rodríguez in [23]. Definition 1.6. Let S ≥ 1, A metric space (X, d) is called S-uniformly perfect, if there is some r 0 > 0 such that for each x ∈ X and every 0 < r ≤ r 0 there exists a point y ∈ X such that r/S < d(x, y) ≤ r. A metric space is called uniformly perfect if it is S-uniformly perfect for some S ≥ 1.
Remark 1.7. The above definition of uniformly perfect spaces is equivalent to the standard definition ( [27]) when X is bounded. Since we only consider this property in bounded spaces, all our results work as well with the standard definition, see [23].
Examples of uniformly perfect spaces include all connected spaces, many disconnected fractals such as the Cantor ternary set, Julia sets and the limit sets of nonelementary, finitely generated Kleinian groups of R n , see [27] and the references therein. Spaces with isolated points are not uniformly perfect. Uniform perfectness is connected to several important concepts in function theory, geometry, group theory and geometric group theory. See [11,23,27,30]. Hence, by Theorem 1.4, boundary rigidity is also connected to these concepts: Theorem 1.8. Assume that X is a non-compact complete Riemannian manifold or a uniform graph which has bounded local geometry. If X is Gromov hyperbolic and admits a pole, then the following conditions are equivalent: (1) X is boundary rigid.
Proof. First, one observes from [23, Proposition 5.2] that every non-compact complete Riemannian manifold with bounded local geometry is quasi-isometric to a uniform graph. It is known that the all properties mentioned in the theorem are all quasi-isometric invariants, such as Gromov hyperbolicity, uniformly perfectness, positive Cheeger isoperimetric constant and nonamenability, see [23]. Therefore, without loss of generality, we may assume that X is a uniform graph. The equivalence of (1) and (2) follows from Theorem 1.4. The equivalence of (2) and (3) is recently established by Martínez-Pérez and Rodríguez in [23]. The equivalence of (3) and (4) is a well known theorem of Folner, see [12,Theorem 51] or [20,Proposition 2.3]. Remark 1.9. For precise definitions of the concepts appeared in Theorem 1.8 we refer to [11,23]. The Cheeger isoperimetric constant is related to the work of Ancona [1,2] about potential theory on hyperbolic manifolds and graphs. For such a space with boundary rigidity, the Dirichlet problem at infinity is solvable and the Martin boundary is homeomorphic to the Gromov boundary, see [18,23]. Amenability is an important property in the study of Brownian motion, harmonic analysis and random walks on graphs and manifolds, see [20]. There are numerous equivalent conditions for this notation in geometric group theory, see [12,22,24] and the references therein.
We now make several remarks on the requirement we put on the space X in Theorem 1.4: Remark 1.10.
(1) Since ∂X equipped with all visual metrics are power quasisymmetrically equivalent to each other and since uniform perfectness is preserved by power quasisymmetries ([23, Theorem 5.9 and Proposition 5.10]), one may state that ∂X is uniformly perfect without specifying certain visual metric.
(2) Without requiring that the space has a pole, it is easy to construct Gromov hyperbolic spaces with uniformly perfect boundaries that are not boundary rigid. Here is an example: Let H be the hyperbolic plane and x ∈ H. Attach a sequence of segments {I n = [0, n]} n∈N to x by identifying 0 with x. Let X be the resulting space endowed with the induced length metric. Then ∂X = ∂H = S 1 , which is uniformly perfect. However, X is not boundary rigid since it admits quasi-isometries to itself that fix H and "stretch" and "shrink" the attached segments. In fact, it is clear from the above construction that every Gromov hyperbolic space with a uniformly perfect boundary can be embedded into a Gromov hyperbolic space with the same boundary that is not boundary rigid. (3) We expect Theorem 1.4 to be true if one only assumes that X has quasigeodesic between any two points or quasi-isometric to such a space. However for spaces with isolated points being very far apart, Theorem 1.4 is not true as the following example shows: Let X = {±x n | x n = 2 (2 n ) , n ∈ Z} endowed with the Euclidean metric | · | of R. Let f : X → X be a (K, C)quasi-isometry such that its induced map ∂f fixes ∂X. One finds that there exists n 0 depending only on K and C, such that if m > n ≥ n 0 , then we have |x m − x m+1 | > K|x n − x n+1 | + C. One can show that f has to fix x m for all m > n 0 . Hence X is boundary rigid but ∂X is not uniformly perfect.
Boundary rigidity was previously investigated in [35], where the fact that the boundary being uniformly perfect implies boundary rigidity was proved by the second author and Rasila. They were motivated by Teichmuller's displacement problem for the class of quasiconformal maps with identity boundary values and their proof uses tools from complex analysis. See [28,35] for the connections between boundary rigidity, Teichmuller's displacement problem and quasiconformal maps. We provide a different proof by using only Gromov hyperbolic geometry.
Boundary rigidity was also investigated in [26], where geodesically rich hyperbolic spaces (see Definition 4.1) were introduced and showed to be boundary rigid. As an application of our main theorem we prove: Theorem 1.11. Let X be a proper geodesic Gromov hyperbolic space with a pole. Then the following are equivalent: (1) X is boundary rigid.
[26, Question 1] asks if a hyperbolic space can always be isometrically embedded into a geodesically rich hyperbolic space with an isomorphic boundary. While there are spaces without such embedding (See Section 4 for an example), the next corollary, together with Theorem 1.11, completely characterizes all spaces admitting such an embedding. Corollary 1.12. Let X be a proper geodesic Gromov hyperbolic space. If X quasiisometrically embeds into a geodesically rich Gromov hyperbolic space Y with an isomorphic boundary, then X is boundary rigid and ∂X is uniformly perfect.
A hyperbolic filling H(Z) of a bounded complete metric space (Z, d) is a Gromov hyperbolic space whose Gromov boundary (endowed with a visual metric) can be identified with the space (Z, d) via a bilipschitz map. Hyperbolic fillings are useful in the study of asymptotic geometry and function space, see [6,11,23]. Several different constructions of hyperbolic fillings have been introduced ([7, 8, 10]), all of which produces a proper geodesic Gromov hyperbolic space with a pole provided Z is compact. Hence Theorem 1.4 immediately yields: Corollary 1.13. Let (Z, d) be a compact metric space, and let X be its hyperbolic filling in the sense of Bonk-Schramm [7], Bourdon-Pajot [8], and Buyalo-Schroeder [10]. Then X is boundary rigid if and only if Z is uniformly perfect.
Here is the outline of this paper: In Section 2, we recall the definitions and results needed in the proof of Theorem 1.4. Then we prove Theorem 1.4 in Section 3. In Section 4, we prove Theorem 1.11 and Corollary 1.12.

Preliminaries and auxiliary results
In this section, we recall the basic definitions and some basic results that we need in the proof of Theorem 1.4.
be a map between metric spaces X and Y , and let K ≥ 1 and C ≥ 0 be constants. If for all x, y ∈ X we have then f is called a (K, C)-quasi-isometric embedding. If in addition, every point y ∈ Y has distance at most C from the set f (X), then f is called a (K, C)-quasiisometry. In this case, there exists a quasi-isometry g : We say that X quasi-isometrically embeds into Y if there is a quasi-isometric embedding from X to Y . We say that X is quasi-isometric to Y if there is a quasiisometry between X and Y .
We now recall the definitions of Gromov hyperbolic spaces and their Gromov boundaries, see [5,7,9,10]. Let (X, d) be a metric space. Fix a base point w in X.
(1) The space X is called geodesic, if each pair of points x, y ∈ X can be joined by a geodesic [x, y]; that is, a curve whose length is precisely the distance between x and y.
if each point on the edge of any geodesic triangle is within distance δ of the other two edges. If X is δ-hyperbolic for some δ ≥ 0, then we say that it is Gromov hyperbolic. (3) For x, y ∈ X, the Gromov product of x, y with respect to w is The Gromov boundary ∂X of X is defined to be the set of equivalence classes of Gromov sequences, andX = X ∪ ∂X is called the Gromov closure of X. (d) For x ∈ X and ξ ∈ ∂X, the Gromov product (x|ξ) w of x and ξ is defined by (e) For ξ, ζ ∈ ∂X, the Gromov product (ξ|ζ) w of ξ and ζ is defined by Following [9], we now recall the definition of the δ-thin condition. Given any three positive numbers a, b, c, we can consider the metric tree T (a, b, c) that has three vertices of valence one, one vertex of valence three, and edges of length a, b and c. Such a tree is called a tripod. We call the valence three vertex the center of the tripod. Extend the definition of tripod in the obvious way to cover the cases where a, b and c are allowed to be zero and ∞. When a, b, c are all ∞, we call T (a, b, c) an infinite tripod. Given any three points x, y, z in a metric space, the triangle equality tells us that there exist unique non-negative numbers a, b, c such that d(x, y) = a+b, d(x, z) = a + c and d(y, z) = b + c; Note that a = (y|z) x , b = (x|z) y and c = (x|y) z . There is an isometry from {x, y, z} to a subset of the vertices of T (a, b, c) (the vertices of valence one in the non-degenerate case); we label these vertices v x , v y , v z in the obvious way. Given a geodesic triangle, ∆ = ∆(x, y, z), we define the comparison tripod T ∆ of ∆ by T ∆ := T (a, b, c). The above map {x, y, z} → {v x , v y , v z } extends uniquely to a map χ ∆ : ∆ → T ∆ whose restriction to each side of ∆ is an isometry.
Definition 2.4. Let ∆ be a geodesic triangle in a metric space (X, d) and consider the map All triangles in a δ-hyperbolic space are δ ′ -thin for some constant δ ′ depending only on δ (See [9, Proposition 1.17]). Enlarging δ if necessary, we assume all triangles in all δ-hyperbolic spaces we consider in this paper are δ-thin.
When (X, d) is proper and geodesic, ∂X can be identified with the geodesic boundary ∂ G X. By definition ∂ G X is the set of all equivalence classes of geodesic rays, where two geodesic rays are equivalent if the Hausdorff distance between them is finite.
We now recall the definition of visual metrics on ∂X. See [9,10] for more details.
Definition 2.5. Let X be a δ-hyperbolic space with δ > 0 and w ∈ X a base point.
Then (∂X, d w,ε ) is a metric space and we call d w,ε the visual metric on ∂X with respect to w ∈ X and the parameter ε.
The following is a useful estimate of the metric d w,ε .
Then f is a quasi-isometry. Thus, f induces a boundary map Hence X is boundary rigid. We call o a pole of X. We say that X has a pole if X has an L-pole for some L ≥ 0.
Remark 2.9. The terms "roughly starlike" and "quasi-pole" in the literatures [5,11,23] are used for spaces having a pole. In spaces where geodesics do not always exist, one can talk about similar properties using the concept of "L-visual" (See [10]).
The following lemma shows that X has a pole in the space if and only if it has a pole in ∂X. Note that pole constants depend on the locations of poles in X. (1) X has a pole ξ ∈ ∂X.
(2) Each point of X is a pole.
(3) X has a pole w ∈ X.
In the remaining part of this section, we assume that (X, d) is a proper geodesic δ-hyperbolic space. We cite some standard facts about Gromov hyperbolic spaces. See [3,9,10,17,26,29] for proofs. The proof of the following easy lemmas are left to the readers. Note that the constants may not be optimal.
The following facts are parts of [3, Lemma 3.1] although [3] only considered centroids of triples in X instead ofX.
(1) For any has non empty intersection with any geodesic between any two points in Hence (1) follows.
where M depends only on K, C and δ. It follows from [13,Lemma 11.87] that the collection of points within ρ + M of each of the edges of an ideal triangle has diameter bounded by a constant depending only on ρ + M and δ. Hence the diameter of ∆ (K,C,ρ) ({x 1 , x 2 , x 3 }) is also bounded by the same constant.
Hence we have . By Lemma 2.6 and the general fact that ( for all x 1 , x 2 , x 3 ∈X, (3.1) and (3.2) imply: Let v i be a point on [ξ i , ξ] closest to o. Lemma 2.11 ensures that Let v i,i+2 be a point on [ξ i , ξ i+2 ] closest to o. By Lemma 2.11, it follows that Hence we have: By Lemma 2.14, both v i+2 and v i,i+2 are in the 10δ-neighborhood of [o, ξ i+2 ]. Hence (3.5) and (3.6) imply that d(v i+2 , v i,i+2 ) ≤ D 1 , where D 1 depends on S, r 0 , ǫ and δ.
Let u i be a point on [o, ξ] closest to v i . Then by Lemma 2.14, we have d( Let u be a point on [o, ξ] closest to x. Then d(u, x) ≤ L. By the above properties where D 3 = 3D 2 + 10δ + D 1 depends on S, ǫ, r 0 , L and δ. Hence we have Therefore, by Lemma 2.17, we know that d(x, f (x)) is bounded above by a constant D depending only on K, C and D 3 . Proof. Suppose on the contrary that (∂X, d p,ε ) is not uniformly perfect, where d p,ε is the visual metric based at p with some parameter ǫ. Then for any S > 1, there exists ξ ∈ ∂X with the following property: For any ǫ 0 , there exists ǫ < ǫ 0 such that there is no ξ 0 ∈ ∂X satisfying ǫ/S < d p,ε (ξ, ξ 0 ) < ǫ. Hence there are t and s on the geodesic ray [p, ξ] such that t ∈ [p, s] and for any ξ ′ ∈ ∂X, we have Moreover s and t can be chosen so that d(s, t) = ϕ(S) for some increasing function ϕ of S satisfying ϕ(S) → ∞ as S → ∞. Let u be the midpoint of a geodesic [s, t]. Since ∆ (K,C,ρ) (∂X) is M-roughly full in X, there exist {ξ 1 , ξ 2 , ξ 3 } ⊂ ∂X such that u ∈ ∆ (K,C,ρ+M ) ({ξ 1 , ξ 2 , ξ 3 }). Now we consider two cases.
The aim of the rest of this section is to prove the following proposition, which, together with Proposition 3.2 and Proposition 3.1, completes the proof of Theorem 1.4.  (1) Each γ i is a subsegment of a geodesic [o, ξ i ] for some ξ i ∈ ∂X; (2) Elements of Proof. The construction of γ i = [y i , z i ] is as follows. Since ∆ (K,C,ρ) (∂X) is not Mroughly full in X for any M > 0, there are closed balls B i = B(x i , r i ) with radius r i → ∞ such that B i is disjoint from the set ∆ (K,C,ρ) (∂X) for all i ∈ N + . Without loss of generality we may assume that d(B i , B j ) ≥ 10L + 10δ + 1 for any i = j and r i ≥ 100D + L.
Since o is an L-pole ofX, for each i there is a geodesic [o, This, together with Lemma 2.17, tells us that ∆ (1,0,6δ) In the following, we prove some technique lemmas about the above sets and then use these results to construct the desired quasi-isometries with increasing displacements.
Proof. Suppose on the contrary that there is some point p ∈ X y (3D) ∩ X z (3D). By the definitions of X y (3D) and X z (3D), there are p y and p z in proj [y,z] (p) such that d(y, p y ) ≤ 3D and d(z, p z ) ≤ 3D. Lemma 2.17 gives that d(p y , p z ) ≤ D. Thus Note that d(w, u) ≤ d(w, y). Hence d(y, u) ≤ 6δ ≤ D. Then Lemma 2.17 yields that w ∈ X y (2D), contradiction.
On the other hand, we obtain

contradiction.
Therefore, we know that p ′ ∈ [o, q ′ ]. Since p is the closest point on [o, ξ] to p ′ , Lemma 2.13 ensures that d(p, p ′ ) ≤ 3δ. Hence Lemma 3.10. Let x ∈ X y (10D) ∪ X z (10D). Then we have the following: Proof. We only need to prove (1), because (2) follows from a similar argument. Choose points p x and p x ′ on [o, ξ] closest to x and x ′ , respectively. We require p x to be outside of the 10D-neighborhood of y. Then p x and p x ′ are at least 7D apart. Consider a geodesic quadrilateral [x, p x , p x ′ , x ′ ]. We claim that Let w 1 and w 2 be the pre-images of the center of the comparison tripod of the geodesic triangle [x ′ , p x ′ , p x ] in [x ′ , p x ′ ] and [p x ′ , p x ], respectively. Then d(w 1 , w 2 ) ≤ δ. This implies that d( Let w 3 and w 4 be the pre-images of the center of the comparison tripod of the geodesic triangle [x, x ′ , p x ] in [x, p x ] and [x ′ , p x ], respectively.
We will prove by contradiction that Then there is a point w 6 on [p x , x ′ ] such that d(w 5 , w 6 ) ≤ δ and d(p x , w 6 ) < 2δ + 1.
Since p x ′ is in the 3D-neighborhood of y, we obtain Let (X, d) be a proper geodesic δ-hyperbolic space with a L-pole. Let γ = [y, z] be a geodesic in X such that proj γ (∂X) is contained in the D-neighborhood of {y, z}. Define Φ [y,z] : X → X as follows.
Define Φ [y,z] : X → X as follows: If x ∈ X y (3D) ∪ X z (3D), then we define If x / ∈ X y (3D)∪X z (3D), then pick a point p x on [y, z] closest to x and we require that p x is not in the 3D-neighborhood of {y, z}. Let [y ′ , z ′ ] be the largest subsegment of [y, z] outside of the 3D-neighborhood of {y, z}. Let l = d(y ′ , z ′ ) and let ϕ : [y ′ , z ′ ] → [0, l] be an isometry. Then we define Next we prove some properties of the map Φ [y,z] for later use. (1) The displacement of Φ [y,z] is at least l/3.
(1) By the definition of Φ [y,z] , the point on [y ′ , z ′ ] l/3 away from y ′ is moved by Φ [y,z] by l/3. Hence the displacement of Φ [y,z] is at least l/3.
(4) We now show that Φ [y,z] is a quasi-isometry. Let and let By the definition of X y (R) (see (3.8)), the sets X 1 , X 2 and X 3 are pairwise disjoint that cover X. Let x, x ′ ∈ X. By (3), it suffices to show that Again by Lemma 3.9, X 3 is contained in the (20D + K 0 )-neighborhood of {y, z}. Thus by the definition of Φ [y,z] , it is not difficult to see that 40D) and similarly, Hence it follows from Lemma 3.9 that Φ [y,z] moves x ′ by at most 40D + K 0 . On the other hand, the map Φ [y,z] fixes x. Hence we obtain (3.10) for C ′ ≥ 40D + K 0 and K ′ ≥ 1.
Let [x, x ′ ] be a geodesic between x and x ′ . Without loss of generality, we may assume that x ∈ X y (3D). Lemma 3.10 shows that [x, x ′ ] passes through the 5Dneighborhood of y. Let x ′′ ∈ B(y, 5D) be a point on [x, x ′ ]. Note that x ∈ X 1 and x ′′ ∈ X 3 . By Case 3.3, the inequality (3.10) holds for x, x ′′ with C ′ = 40D + K 0 and K ′ = 1. By Case 3.2, the inequality (3.10) holds for x ′′ , x ′ with C ′ = 4K 0 and K ′ = 2. Since x ′′ ∈ [x, x ′ ], these two facts imply that Since x ∈ X y (3D), Lemma 3.10 shows that there is a point This implies that Since y ∈ X 1 and x ′′ ∈ X 3 , a similar argument as Case 3.3 gives that Since x ′ ∈ X 2 and x ′′ ∈ X 3 , the assertion in Case 3.2 yields that Hence this together with (3.11) guarantees that which completes the proof.
Proof of Proposition 3.3. Assume that (X, d) is a proper geodesic δ-hyperbolic space with a pole. Suppose ∆ (K,C,ρ) (∂X) is not M-roughly full in X for any M > 0. To prove that X is not boundary rigid, it suffices to show that there exist K 1 , C 1 and a (K 1 , C 1 )-quasi-isometry Φ : X → X such that Φ fixes ∂X but Φ has infinite displacement.
Let γ i = [y i , z i ] be the sequence of segments given by Lemma 3.6. For each i, we may define the map Φ i = Φ [y i ,z i ] as in Definition 3.11. We claim that Suppose that there is some x moved by Φ i and Φ j with i = j. By the definition of Φ i , we obtain It follows from Lemma 3.9 that d(x, [y i , z i ]) ≤ L + 3δ and d(x, [y j , z j ]) ≤ L + 3δ, which implies that d(γ i , γ j ) ≤ 2L + 6δ. On the other hand, by (2) of Lemma 3.6, we have d(γ i , γ j ) ≥ 10L + 10δ + 1, contradiction. Hence Claim 3.2 is true. Now define Φ = Φ 1 • · · · • Φ i • · · · . Claim 3.2 ensures that Φ is well defined and moreover, Φ is a (K 1 , C 1 )-quasi-isometry with K 1 = K ′2 and C 1 = K ′ C ′ + C ′ by using (4) of Lemma 3.13.
However, by (1) and (2) of Lemma 3.13, Φ fixes ∂X and the displacement of Φ is not finite because d(y i , z i ) → ∞ as i → ∞. This shows that X is not boundary rigid.
Proof of Theorem 1.4. It follows from Propositions 3.1, 3.2 and 3.3.

Geodesically rich spaces
In [26], geodesically rich Gromov hyperbolic spaces were introduced and shown to be boundary rigid. The first goal of this section is to prove Theorem 1.11, which says that, within the class of proper geodesic hyperbolic spaces with a pole, being boundary rigid is equivalent to being geodesically rich. We now recall the definition of geodesically rich spaces and show that it can be simplified. Definition 4.1. A geodesic metric space (X, d) is said to be geodesically rich if there are constants r 0 , r 1 , r 2 , r 3 , and r 4 such that (1) for every pair of points p and q with d(p, q) ≥ r 0 , there exists a bi-infinite geodesic γ such that d(p, γ) < r 1 and |d(q, γ) − d(p, q)| < r 2 and (2) for any bi-infinite geodesic γ and any p ∈ X, there exists a bi-infinite geodesic γ ′ such that d(p, γ ′ ) < r 3 and |d(p, γ) − d(γ ′ , γ)| < r 4 .
Lemma 4.2. Assume that (X, d) is a proper geodesic δ-hyperbolic space. If X satisfies the first condition of Definition 4.1, then X is geodesically rich.
Proof. Assume X satisfies the first condition of Definition 4.1 with constants r 0 , r 1 and r 2 . Let r 3 = 2r 1 + r 2 + r 0 + 1000δ + 1 and r 4 = 2r 1 + 3r 2 + r 0 + 1000δ + 1 ≥ r 3 . Let γ be a bi-infinite geodesic and p ∈ X. We check that there exists a bi-infinite geodesic line γ ′ such that the second condition of Definition 4.1 holds with the above r 3 and r 4 . Let q be a point on γ closest to p. If d(p, γ) < r 3 , then take γ ′ = γ. Thus d(p, γ ′ ) < r 3 and d(p, γ) differs from the distance between γ ′ and γ by not more than r 4 .
Proof. Let X be a geodesically rich space with the constants r 0 , r 1 , r 2 in the first condition. Since X is proper and geodesic, any two points inX can be connected by a geodesic. Let o ∈ X and q ∈ X.
Suppose first that d(o, q) > r 0 . Since X is geodesically rich, there exists a geodesic γ joining a, b ∈ ∂X such that d(q, γ) < r 1 . Let α 1 and α 2 be geodesics joining o to a and b, respectively. Then by the δ-hyperbolicity of X, we have d(q, α 1 ∪α 2 ) ≤ r 1 + δ.
For the other case that d(o, q) ≤ r 0 , we have d(q, α 1 ∪ α 2 ) ≤ r 0 . Hence o is an L-pole of X, where L = max{r 1 + δ, r 0 }. Proposition 4.4. Let (X, d) be a proper geodesic δ-hyperbolic space with a pole. If ∂X is uniformly perfect, then X is geodesically rich.
[26, Question 1] asks if a hyperbolic space can always be isometrically embedded into a geodesically rich hyperbolic space with an isomorphic boundary. The real line does not admit such an embedding: since all the bi-infinite geodesics of a hyperbolic space whose boundary consists of two points fellow travel, the first condition of geodesically rich can not be satisfied. Corollary 1.12 completely characterizes all geodeisc hyperbolic spaces admitting such an embedding.
Proof of Corollary 1.12. Let f : X → Y be a quasi-isometric embedding which induces a bijection ∂f from ∂X and ∂Y . It suffices to show that f (X) is M-roughly full in Y . Let y ∈ Y . Since Y is geodesically rich, by (1) of Definition 4.1, there is a bi-infinite geodesic γ such that y is in the r 1 -neighborhood of γ. Let ξ 1 and ξ 2 on ∂Y be the endpoints of γ. Since ∂f : ∂X → ∂Y is bijective, there are ξ ′ 1 and ξ ′ 2 on ∂X such that ∂f (ξ ′ i ) = ξ i for i = 1, 2. Since X is a proper geodesic Gromov hyperbolic space, we know that there is a geodesic γ ′ connecting ξ ′ 1 and ξ ′ 2 . Let γ ′′ = f (γ ′ ). Then γ ′′ is a quasi-geodesic whose constants depends only on f . Using Lemma 2.12, one observes that γ ′′ is in the r ′ -neighborhood of γ, where r ′ depends on f and δ. Hence y is in the M = (r 1 + r ′ )neighborhood of γ ′′ = f (γ ′ ) ⊂ f (X). Therefore, we know that X is quasi-isometric to Y . By Lemma 4.3, Y has a pole. So Theorem 1.11 ensures that Y is boundary rigid. The second statement now follows from Proposition 2.7.
Combining the above proposition with Theorem 1.11, we have the the following: Corollary 4.5. Let X be a proper geodesic δ-hyperbolic space with a pole. Then the following are equivalent: (1) X quasi-isometrically embeds into a geodesically rich δ ′ -hyperbolic space Y inducing a bijection between ∂X and ∂Y . (2) X is geodesically rich.