Sphericalization with its applications in Gromov hyperbolic spaces
aa r X i v : . [ m a t h . M G ] S e p SPHERICALIZATION WITH ITS APPLICATIONS IN GROMOVHYPERBOLIC SPACES
QINGSHAN ZHOU, YAXIANG LI ∗ , AND XINING LI Abstract.
In this paper, we study certain applications of sphericalization inGromov hyperbolic metric spaces. We first show that the doubling property re-garding two classes of metrics on the Gromov boundary of hyperbolic spaces arecoincided. Next, we obtain a characterization of unbounded Gromov hyperbolicdomains via metric spaces sphericalization. Finally, we investigate the topolog-ical equivalence of Gromov hyperbolic ϕ -uniform domains between the Gromovboundary and the inner metric boundary. Introduction and main results
The sphericalization of a locally compact metric space was first introduced byBonk and Kleiner [3] in defining a metric on the one point compactification ofan unbounded space. It is a natural generalization of the deformation from theEuclidean distance on R n to the chordal distance on S n . In [29], Wang and Yangintroduced a chordal distance on p -adic numbers which is an ultra-metric. Recently,the authors generalized this notation to ultra-metric spaces and provided a newproof for a recent work of Heer [13] concerning the quasim¨obius uniformization ofCantor set in [31].Inspired by [3], Balogh and Buckley in [2] defined a flattening transformation on abounded metric space. It was shown in [8] that these two conformal transformationsare dual in the sense that if one starts from a bounded metric space, then performsa flattening transformation followed by a sphericalization, then the object space isbilipschitz equivalent to the original space. This duality comes from the idea thatthe stereographic projection between Euclidean space and the Riemann sphere canbe realized as a special case of inversion. Sphericalization and flattening have a lotof applications in the area of analysis on metric spaces and asymptotic geometry,such as [2, 8, 9, 14, 20].Recently, Wildrick investigated the quasisymmetric parametrization of unbounded2-dimensional metric planes by using the sphericalization (named by a warping File: arxiv.tex, printed: 2020-9-30, 0.39
Mathematics Subject Classification.
Primary: 30C65, 30F45; Secondary: 30C20.
Key words and phrases.
Sphericalization, Gromov hyperbolic space, quasihyperbolic metric, ϕ -uniform domain. ∗ Corresponding author.The first author was supported by NNSF of China (No. 11901090), and by Department of Ed-ucation of Guangdong Province, China (No. 2018KQNCX285). The second author was supportedby NNSF of China (Nos. 11601529, 11671127), and the third author was supported by NNSF ofChina (No. 11701582). process in [30]). It was shown in [17] that two visual geodesic Gromov hyper-bolic spaces are roughly quasi-isometric if and only if their Gromov boundaries arequasim¨obius equivalent by virtue of the flattening and sphericalized deformations.In [22], Mineyev studied the metric conformal structures on the idea boundaries ofhyperbolic complexes via sphericalization.In this paper, we investigate certain applications of sphericalization in Gromovhyperbolic metric spaces. In [11], Gromov observed that the essential large scalegeometry of classical hyperbolic spaces H n is determined by a δ -inequality concerningquadruples of points, and meanwhile introduced the concept of δ -hyperbolicity forgeneral metric spaces. Since its appearance, the theory of Gromov hyperbolicityhas been found numerous applications in the study of geometric group theory andgeometric function theory, for instance [1, 4, 7, 18] and the references therein.We briefly review the theory of Gromov hyperbolic spaces. For a more completeexposition, see [7, 9] or Subsection 2.4. A geodesic metric space X is called Gromovhyperbolic if there is a constant δ ≥ δ from some point on one of the other two sides.The Gromov boundary of X , denoted by ∂ ∞ X , is defined as the set of equivalenceclasses of geodesic rays, with two geodesic rays γ i , i = 1 ,
2, being equivalent if theHausdorff distance dist H ( γ , γ ) < ∞ . There is a cone topology on X ∪ ∂ ∞ X so thatit is metrizable, see [7]. In [9], Buyalo and Schroeder systematically investigatedtwo different kinds of metrics (namely, Bourdon metric and Hamenst¨adt metric, forthe definitions see Section 2) on ∂ ∞ X which are respectively based at a point in X and ∂ ∞ X (cf. [6, 12]).As the first aim of this paper, we show that the doubling property of these twoconformal gauge on the Gromov boundary of a Gromov hyperbolic space are coin-cided. Theorem 1.1.
Let X be a Gromov hyperbolic space and ∂ ∞ X its Gromov boundary.Then ∂ ∞ X is doubling for any Bourdon metric if and only if it is doubling for anyHamenst¨adt metric. The terminology used in Theorem 1.1 and in the rest of this section will beexplained in Section 2.
Remark 1.2.
In [20], the third author and Shanmugalingam showed that the pro-cess of sphericalization preserves the Ahlfors regular and doubling measures in metricspaces. With the aid of this quantitative result, Heer [13] proved the quasim¨obiusinvariance of doubling metric spaces, which is needed in the proof of Theorem 1.1.In this note, we provide a quite different but direct proof for Heer’s result by meansof Assouad’s embedding Theorem (see [5, Theorem 8.1.1]).The doubling property of metric spaces plays an important role in the area ofanalysis on metric spaces. For instance, Herron [15] demonstrated that a Gromovhyperbolic abstract domain with doubling Gromov boundary carries a uniformizingvolume growth density. Recently, Wang and Zhou [28] studied the equivalence ofweakly quasim¨obius maps and quasim¨obius maps on doubling quasi-metric spaces.A metric space X is said to be of bounded growth at some scale , if there existconstants r and R with R > r >
0, and N ∈ N such that every open ball of radius hou et al. 3 R in X can be covered by N open balls of radius r . In [5], Bonk and Schrammproved that a Gromov hyperbolic space with bounded growth at some scale embedsrough isometrically into the classical hyperbolic spaces with higher dimension. In[9], Buyalo and Schroeder provided a different proof of this result. According to[5, Theorem 9 .
2] every Gromov hyperbolic geodesic space X of bounded growth atsome scale has a boundary ∂ ∞ X of finite Assouad dimension (which is equivalent tothe doubling condition, see [23]). Here ∂ ∞ X is equipped with any visual (Bourdon)metric. Recently, Herron corroborated that a locally Ahlfors regular abstract domainis bounded growth at some scale associated to the quasihyperbolic metric, see [15,Proposition 3.6]. Combining these results with Theorem 1.1 we obtain the followingtwo corollaries; for the related definitions see [5, 15]. Corollary 1.3.
Let X be a Gromov hyperbolic geodesic metric space with boundedgrowth at some scale. Then the Gromov boundary equipped with any Bourdon orHamenst¨adt metric is doubling. Corollary 1.4.
Let Ω be a locally quasiconvex locally Ahlfors Q -regular abstractdomain with a Gromov hyperbolic quasihyperbolization. Then the Gromov boundaryequipped with any Bourdon or Hamenst¨adt metric is doubling. The second goal of this paper is to explore the characterization of Gromov hy-perbolic domains; here and hereafter, a Gromov hyperbolic domain always meansan incomplete metric space which is Gromov hyperbolic in its quasihyperbolic met-ric k (see Definition 2.2). In [18], Koskela, Lammi and Manojlovi´c proved that if(Ω , d ) is a bounded abstract domain, then (Ω , k ) is Gromov hyperbolic if and onlyif the length space (Ω , l d ) satisfies both the Gehring-Hayman condition and the ballseparation condition. It is natural to ask whether this result holds for unboundeddomains. In this work, we consider the unbounded case via sphericalization andestablish the following result similar to [18, Theorem 1.2]. Theorem 1.5.
Let
Q > and let ( X, d, µ ) be an Ahlfors Q -regular metric measurespace with ( X, d ) an annular quasiconvex, proper and geodesic space. Let Ω ( X bea domain ( an open connected set ) , and let l d be the length metric on Ω associated to d . Then (Ω , k ) is Gromov hyperbolic if and only if (Ω , l d ) satisfies both the Gehring-Hayman condition and the ball separation condition. In [4], Bonk, Heinonen and Koskela proved that a bounded domain in R n is uni-form if and only if it is Gromov hyperbolic with respect to the quasihyperbolicmetric and there is a natural quasisymmetric identification between the Euclideanboundary and the Gromov boundary. Recently, Lammi [19] showed that the innerboundary of a Gromov hyperbolic domain with a suitable growth condition in thequasihyperbolic metric is homeomorphic to the Gromov boundary. Note that thequasihyperbolic boundary condition stated in [19] implies the boundness of the do-main. So it is natural to ask whenever the Gromov boundary and the inner metricboundary of an unbounded Gromov hyperbolic domain are homeomorphic.Motivated by these considerations, we focus on the study of the boundary behaviorof Gromov hyperbolic domains, particularly, for unbounded domains. It was shownin [4, 10] that there is a characterization of uniform domains (see Definition 2.6) in Sphericalization with its applications in Gromov hyperbolic spaces terms of two metrics, the quasihyperbolic metric k and the distance ratio metric j (see (2.3)). That is, a domain D is uniform in R n if and only if there exist constants c ≥ c ≥ x, y ∈ D ,(1.6) k D ( x, y ) ≤ c j D ( x, y ) + c . Subsequently, Vuorinen observed that the additive constant c on the right hand sideof (1.6) can be chosen to be 0. This observation leads to the definition of ϕ - uniformdomains introduced in [27]. Definition 1.7.
Let X be a rectifiably connected, locally compact and completemetric space, and let D ( X be a domain with d D ( x ) = dist( x, ∂D ) for all x ∈ D .Let ϕ : [0 , ∞ ) → [0 , ∞ ) be a homeomorphism. We say that D is ϕ - uniform if for all x , y in D , k D ( x, y ) ≤ ϕ ( r D ( x, y )) where r D ( x, y ) = d ( x, y ) d D ( x ) ∧ d D ( y ) . Here and hereafter, r ∧ s = min { r, s } for all r, s ∈ R .In [24], V¨ais¨al¨a has also investigated this class of domains, and pointed out thatthese two classes of domains are the same provided ϕ is a slow function (i.e. afunction ϕ satisfying lim t →∞ ϕ ( t ) /t = 0). We remark that every convex domain is ϕ -uniform with ϕ ( t ) = t . However, in general, convex domains need not be uniform.For example, D = { ( x , x ) ∈ R : 0 < x < } is ϕ -uniform with ϕ ( t ) = t , but it isnot uniform.The third purpose of this paper is to study whether or not the ϕ -uniformity condi-tion is sufficient for the homeomorphism equivalence between the Gromov boundaryand metric boundary of Gromov hyperbolic domain. Our result in this direction isas follows. Theorem 1.8.
Let
Q > and let (Ω , d, µ ) be a locally compact, c -quasiconvex,Ahlfors Q -regular incomplete metric measure space. Assume that (Ω , k ) is roughlystarlike Gromov hyperbolic and (Ω , d ) is ϕ -uniform, where k is the quasihyperbolicmetric of Ω . (1) If Ω is bounded and R ∞ dtϕ − ( t ) < ∞ , then the Gromov boundary and themetric boundary of Ω are homeomorphic; (2) If Ω is unbounded and R ∞ dt √ ϕ − ( t ) < ∞ , then the Gromov boundary of Ω and ∂ Ω ∪ {∞} are homeomorphic. Here and hereafter, Ω ∪ {∞} is the one pointcompactification of the space (Ω , d ) and ∂ Ω ∪ {∞} = Ω ∪ {∞} \ Ω . Note that Theorem 1.8 is new even for Gromov hyperbolic domains in R n . Itfollows from [4, Theorem 1.11] that (inner) uniform domains in R n are Gromovhyperbolic. However, ϕ -uniform domains need not be Gromov hyperbolic. Let G = { ( x , x , x ) ∈ R : 0 < x < } . Thus G is a convex domain which is ϕ -uniform with ϕ ( t ) = t . Moreover, it is not difficult to check that G is LLC (see[4, Chapter 7]) but not c -uniform for any c ≥
1. Hence we see from [4, Proposition7.12] that G is not δ -hyperbolic for any δ ≥ hou et al. 5 We note that a Gromov hyperbolic ϕ -uniform domain in R n is not necessarilyuniform. For instance, Ω = { ( x , x ) ∈ R : 0 < x < } is a plane simply connectedconvex domain, and therefore it is a Gromov hyperbolic ϕ -uniform domain. But itis not uniform. Therefore, Theorem 1.8 is a generalization of the results in [4, 19].Moreover, the unbounded case is also in our considerations by using the sphericalizedtransformation.The organization of this paper is as follows. In Section 2, we recall some definitionsand preliminary results. In Section 3, we will prove the main results.2. Preliminary and Notations
Metric geometry.
In this paper, we always use (
X, d ), ( X ′ , d ′ ), ( Y, d ) etcto denote metric spaces. For (
X, d ), its metric completion and metric boundaryare denoted by X and ∂X := X \ X , respectively. A domain D ⊂ X is an openconnected set.The open (resp. closed) metric ball with center x ∈ X and radius r > B ( x, r ) = { z ∈ X : d ( z, x ) < r } (resp. B ( x, r ) = { z ∈ X : d ( z, x ) ≤ r } ) . We say that X is incomplete if ∂X = ∅ . X is called proper if all its closed balls arecompact.By a curve, we mean a continuous function γ : I = [ a, b ] → X . The length of γ isdenoted by ℓ ( γ ) = sup n n X i =1 d (cid:0) γ ( t i ) , γ ( t i − ) (cid:1)o , where the supremum is taken over all partitions a = t < t < t < . . . < t n = b .The curve is rectifiable if ℓ ( γ ) < ∞ . We also denote the image γ ( I ) of γ by γ , andthe subcurve of γ with endpoints x and y by γ [ x, y ]. A curve γ is called rectifiable ,if the length ℓ d ( γ ) < ∞ . A metric space ( X, d ) is called rectifiably connected if everypair of points in X can be joined with a rectifiable curve γ .Suppose the curve γ is rectifiable. The length function s γ : [ a, b ] → [0 , ℓ ( γ )] isdefined by s γ ( t ) = ℓ ( γ [ a, t ]) . Then there is a unique curve γ s : [0 , ℓ ( γ )] → X such that γ = γ s ◦ s γ . Obviously, ℓ ( γ s [0 , t ]) = t for each t ∈ [0 , ℓ ( γ )]. The curve γ s is called the arc-lengthparametrization of γ .For a rectifiable curve γ in X , the line integral over γ of each Borel function ̺ : X → [0 , ∞ ) is defined as follows: Z γ ̺ds = Z ℓ ( γ )0 ̺ ◦ γ s ( t ) dt. Sphericalization with its applications in Gromov hyperbolic spaces
For c ≥
1, a curve γ ⊂ X , with endpoints x, y , is c -quasiconvex , c ≥
1, if its lengthis at most c times the distance between its endpoints; i.e., if γ satisfies ℓ ( γ ) ≤ cd ( x, y ) .X is c -quasiconvex if each pair of points can be joined by a c -quasiconvex curve.Let c ≥ < λ ≤ /
2. An incomplete metric space (
D, d ) is said to be locally ( λ, c ) -quasiconvex, if for all x ∈ D , each pair of points in B ( x, λd D ( x )) canbe joined with a c -quasiconvex curve. A geodesic γ joining x to y in X is a map γ : I = [0 , l ] → X from an interval I to X such that γ (0) = x , γ ( l ) = y and d ( γ ( t ) , γ ( t ′ )) = | t − t ′ | for all t, t ′ ∈ I. If I = [0 , ∞ ) or R , then γ is called a geodesic ray or a geodesic line . A metric space X is said to be geodesic if every pair of points can be joined by a geodesic arc.Every rectifiably connected metric space ( X, d ) admits a length (or intrinsic)metric, its so-called length distance given by ℓ d ( x, y ) = inf ℓ d ( γ ) , where the infimum is taken over all rectifiable curves γ joining x to y in X . Definition 2.1.
Let X be a metric space with p ∈ X , and let c ≥
1. We say that X is c - annular quasiconvex with respect to p , if for all r >
0, every pair of pointsin the annular B ( p, r ) \ B ( p, r ) can be joined by a c -quasiconvex curve lying in B ( p, cr ) \ B ( p, r/c ). X is called c - annular quasiconvex if it is c -annular quasiconvexat each point. Definition 2.2.
Let (
X, d ) be a rectifiably connected, locally compact and completemetric space, and let D ( X is a domain. The quasihyperbolic length of a curve γ ⊂ D is defined as ℓ k ( γ ) = ℓ k D ( γ ) = Z γ | dz | d D ( z ) . For any x, y ∈ D , the quasihyperbolic distance k ( x, y ) between x and y is defined by k ( x, y ) = k D ( x, y ) = inf ℓ k ( γ ) , where the infimum is taken over all rectifiable curves γ joining x to y in D .We remark that the resulting space ( D, k ) is complete, proper and geodesic when-ever D is locally quasiconvex (cf. [4, Proposition 2 . j as follows:(2.3) j D ( x, y ) = log (cid:16) d ( x, y ) d D ( x ) ∧ d D ( y ) (cid:17) . Definition 2.4.
Let (
X, d ) be a rectifiably connected, locally compact and completemetric space, and let D ( X is a domain. Let C gh ≥ D satisfies the C gh -Gehring-Hayman inequality , if for all x , y in D and for eachquasihyperbolic geodesic γ joining x and y in D , we have ℓ ( γ ) ≤ C gh ℓ ( β x,y ) , hou et al. 7 where β x,y is any other curve joining x and y in D . In other words, quasihyperbolicgeodesics are essentially the shortest curves in D . Definition 2.5.
Let (
X, d ) be a rectifiably connected, locally compact and completemetric space, and let D ( X is a domain. Let C bs ≥ D satisfies the C bs -ball separation condition , if for all x , y in D and for eachquasihyperbolic geodesic γ joining x and y in D , we have for every z ∈ γ , B ( z, C bs d D ( z )) ∩ β x,y = ∅ , where β x,y is any other curve joining x and y in D . Definition 2.6.
Let (
X, d ) be a rectifiably connected, locally compact and completemetric space, and let D ( X is a domain. We say that D is c - uniform providedthere exists a constant c with the property that each pair of points x , y in D canbe joined by a rectifiable arc γ in D satisfying(1) min { ℓ ( γ [ x, z ]) , ℓ ( γ [ z, y ]) } ≤ c d D ( z ) for all z ∈ γ , and(2) ℓ ( γ ) ≤ c d ( x, y ),where γ [ x, z ] the part of γ between x and z .2.2. Mappings on metric spaces.
In this part, we recall certain definitions formappings between metric spaces. Here primes always denote the images of pointsunder f , for example, x ′ = f ( x ).A quadruple in X is an ordered sequence Q = ( x, y, z, w ) of four distinct pointsin X . The cross ratio of Q is defined to be the number r ( x, y, z, w ) = d ( x, z ) d ( y, w ) d ( x, y ) d ( z, w ) . Observe that the definition is extended in the well known manner to the casewhere one of the points is ∞ . For example, r ( x, y, z, ∞ ) = d ( x, z ) d ( x, y ) . Suppose that η and θ are homeomorphisms from [0 , ∞ ) to [0 , ∞ ), and that f :( X , d ) → ( X , d ) is an embedding between two metric spaces. Then(1) we call that f is L -bilipschitz for some L ≥ x, y ∈ X ,L − d ( x, y ) ≤ d ( x ′ , y ′ ) ≤ Ld ( x, y ) . (2) f is said to be η -quasisymmetric if for all x , y and z in X , d ( x, y ) ≤ td ( x, z ) implies that d ( x ′ , y ′ ) ≤ η ( t ) d ( x ′ , z ′ ) . (3) f is called θ -quasim¨obius if for all x, y, z, w in X , r ( x, y, z, w ) ≤ t implies that r ( x ′ , y ′ , z ′ , w ′ ) ≤ θ ( t ) . For a set A in a metric space X , it is called c -cobounded in X for c ≥
0, if everypoint x ∈ X has distance at most c from A . If A is c -cobounded for some c ≥ A is cobounded in X [5]. Sphericalization with its applications in Gromov hyperbolic spaces
Definition 2.7.
Let λ ≥ c ≥
0. A mapping f : ( X , d ) → ( X , d ) is said tobe a roughly ( λ, c ) -quasi-isometry , if f ( X ) is c -cobounded in X and for all x, y ∈ X ,1 λ d ( x, y ) − c ≤ d ( x ′ , y ′ ) ≤ λd ( x, y ) + c. Sphericalization of metric measure spaces.
In this subsection, we recallsome materials concerning metric measure spaces and sphericalization, for which werefer to standard references [1, 14, 20].Let (
X, d ) be a metric space. X is said to be doubling if there is a constant C such that every (metric) ball B in X can be covered with at most C balls of half theradius of B . A positive Borel measure µ on X is doubling if there is a constant C µ such that µ ( B ( x, r )) ≤ C µ µ ( B ( x, r ))for all x ∈ X and r >
0. Moreover, X is said to be Ahlfors Q -regular if it admits apositive Borel measure µ such that C − R Q ≤ µ ( B ( x, R )) ≤ CR Q for all x ∈ X and 0 < R < diam( X ) (it is possible that the diameter of X satisfiesdiam( X ) = ∞ ), where C ≥ Q > R n with Lebesguemeasure satisfies the Ahlfors n -regularity.Given an unbounded locally compact metric space ( X, d ) and a non-isolated point a ∈ X , we consider the one point compactification ˙ X = X ∪ {∞} and define d a : ˙ X × ˙ X → [0 , ∞ ) as follows d a ( x, y ) = d a ( y, x ) = d ( x, y )[1 + d ( x, a )][1 + d ( y, a )] , if x, y ∈ X,
11 + d ( x, a ) , if y = ∞ and x ∈ X, , if x = ∞ = y. In general, d a is not a metric on X , but a quasimetric. However, there is a standardprocedure, known as chain construction , to construct a metric from a quasimetricas follows. Define b d a ( x, y ) := inf n k X j =0 d a ( x j , x j +1 ) o , where the infimum is taken over all finite sequences x = x , x , ..., x k , x k +1 = y from˙ X .Then ( ˙ X, b d a ) is a metric space and it is said to be the sphericalization of ( X, d )associated to the point a ∈ ˙ X . Moreover, by [9, Lemmma 2.2.5], we have for all x, y ∈ ˙ X (2.8) 14 d a ( x, y ) ≤ b d a ( x, y ) ≤ d a ( x, y ) . hou et al. 9 In the case that (
X, d ) is a rectifiably connected unbounded metric space, wedefine a Borel function ρ a : X → [0 , ∞ ) by ρ a ( x ) = 1[1 + d ( a, x )] . A similar argument as [8, 4.1 and (4.2)], we obtain that for any rectifiable curve γ joining x and y ,(2.9) ℓ b d a ( γ ) = Z γ ρ a ( z ) | dz | . Suppose that (
X, d ) is a proper space equipped with a Borel-regular measure µ such that the measures of non-empty open bounded sets are positive and finite. Wedefine the spherical measure µ a associated to the point a ∈ X as µ a ( A ) = Z A \{∞} µ ( B ( a, d ( a, z ))) dµ ( z ) . Gromov hyperbolic spaces.
In this subsection, we give some basic infor-mation about Gromov hyperbolic spaces, for which we refer to standard references[5, 7, 9, 11, 26].We begin with the definition of δ -hyperbolic spaces. We say that a metric space( X, d ) is
Gromov hyperbolic , if there is a constant δ ≥ δ -inequality ( x | y ) w ≥ min { ( x | z ) w , ( z | y ) w } − δ for all x, y, z, w ∈ X , where ( x | y ) w is the Gromov product with respect to w definedby ( x | y ) w = 12 [ d ( x, w ) + d ( y, w ) − d ( x, y )] . Definition 2.10.
Suppose that (
X, d ) is a Gromov δ -hyperbolic metric space forsome constant δ ≥ { x i } in X is called a Gromov sequence if ( x i | x j ) w → ∞ as i,j → ∞ . (2) Two such sequences { x i } and { y j } are said to be equivalent if ( x i | y i ) w → ∞ .(3) The Gromov boundary or the boundary at infinity ∂ ∞ X of X is defined to bethe set of all equivalence classes.(4) For a ∈ X and η ∈ ∂ ∞ X , the Gromov product ( a | η ) w of a and η is definedby ( a | η ) w = inf (cid:8) lim inf i →∞ ( a | b i ) w : { b i } ∈ η (cid:9) . (5) For ξ, η ∈ ∂ ∞ X , the Gromov product ( ξ | η ) w of ξ and η is defined by( ξ | η ) w = inf (cid:8) lim inf i →∞ ( a i | b i ) w : { a i } ∈ ξ and { b i } ∈ η (cid:9) . Let X be a proper, geodesic δ -hyperbolic space, and let w ∈ X . We say that X is K -roughly starlike with respect to w if for each x ∈ X there is some point η ∈ ∂ ∞ X and a geodesic ray γ = [ w, η ] emanating from w to η withdist( x, γ ) ≤ K. This concept was introduced by Bonk, Heinonen and Koskela [4], see also [5]. Theyproved that bounded uniform spaces and every Gromov hyperbolic domain in R n are roughly starlike.For 0 < ε < min { , δ } , define ρ w,ε ( ξ, ζ ) = e − ε ( ξ | ζ ) w for all ξ, ζ in the Gromov boundary ∂ ∞ X of X with convention e −∞ = 0.We now define d w,ε ( ξ, ζ ) := inf n n X i =1 ρ w,ε ( ξ i − , ξ i ) : n ≥ , ξ = ξ , ξ , ..., ξ n = ζ ∈ ∂ ∞ X o . Then ( ∂ ∞ X, d w,ε ) is a metric space with ρ w,ε / ≤ d w,ε ≤ ρ w,ε , and we call d w,ε the Bourdon metric of ∂ ∞ X based at w with the parameter ε .Following [9], we say that b : X → R is a Busemann function based at ξ , denotedby b ∈ B ( ξ ), if for some w ∈ X , we have b ( x ) = b ξ,w ( x ) = b ξ ( x, w ) = ( ξ | w ) x − ( ξ | x ) w for x ∈ X. We next define the Gromov product of x, y ∈ X based at the Busemann function b = b ξ,w ∈ B ( ξ ) by ( x | y ) b = 12 ( b ( x ) + b ( y ) − d ( x, y )) . Similarly, for x ∈ X and η ∈ ∂ ∞ X \ { ξ } , the Gromov product ( x | η ) b of x and η isdefined by ( x | η ) b = inf (cid:8) lim inf i →∞ ( x | z i ) b : { z i } ∈ η (cid:9) . For points ξ , ξ ∈ ∂ ∞ X \ { ξ } , we define their Gromov product based at b by( ξ | ξ ) b = inf (cid:8) lim inf i →∞ ( x i | y i ) b : { x i } ∈ ξ , { y i } ∈ ξ } . According to [9, (3 .
2) and Example 3 . . x | y ) b . = δ ( x | y ) w,ξ := ( x | y ) w − ( ξ | x ) w − ( ξ | y ) w , where x . = a y means that | x − y | ≤ a for some real number a ≥ . .
3] that ( x | y ) b , ( y | z ) b , ( x | z ) b form a 22 δ -triplefor every x, y, z ∈ X .Now we recall the definition of the Hamenst¨adt metric of ∂ ∞ X based at ξ or aBusemann function b = b ξ,w ∈ B ( ξ ). For ε > e εδ ≤
2, define ρ b,ε ( ξ , ξ ) = e − ε ( ξ | ξ ) b for all ξ , ξ ∈ ∂ ∞ X. Then for i = 1 , , ξ i ∈ ∂ ∞ X \ { ξ } , we have ρ b,ε ( ξ , ξ ) ≤ e εδ max { ρ b,ε ( ξ , ξ ) , ρ b,ε ( ξ , ξ ) } . That is, ρ b,ε is a K ′ -quasi-metric on ∂ ∞ X \ { ξ } with K ′ = e εδ ≤
2. We now define σ b,ε ( x, y ) := inf n n X i =1 ρ b,ε ( x i − , x i ) : n ≥ , x = x , x , ..., x n = y ∈ ∂ ∞ X \ { ξ } o . hou et al. 11 By [9, Lemma 3 . . ∂ ∞ X \ { ξ } , σ b,ε ) is a metric space with ρ b,ε / ≤ σ b,ε ≤ ρ b,ε . Then σ b,ε is called the Hamenst¨adt metric on the punctured space ∂ ∞ X \ { ξ } basedat ξ with parameter ε .To conclude this part, we note that ∂ ∞ X equipped with any Bourdon metric isbounded. However, the punctured space ∂ ∞ X \ { ξ } equipped with any Hamenst¨adtmetric σ b,ε is unbounded.3. Proofs of the main results
Theorem 3.1. ([9, Theorem 8 . . Let ( Z, d ) be a doubling metric space. Then forevery s ∈ (0 , there is a bilipschitz embedding ϕ : ( Z, d s ) → R N , where N ∈ N depends only on s and the doubling constant of the metric d . Secondly, the following auxiliary result concerns the quasim¨obius invariance ofdoubling metric spaces. We note that this lemma has been proved by Heer [13].His proof was based on a recent work of the third author and Shanmugalingam[20]. However, our arguments are quite different and based on Assouad’s embeddingTheorem. We think it is interesting and thus present a proof here.
Lemma 3.2.
Let ( X, d ) and ( X ′ , d ′ ) be metric spaces, and let f : X → X ′ be aquasim¨obius embedding. If X ′ is a doubling metric space, then X is doubling.Proof. Without loss of generality, we may assume that X ′ = f ( X ) and f is a homeo-morphism because the subspace of a doubling metric space is also doubling, see [23,Remark 2.8]. Then according to Theorem 3.1, we see that there is a quasisymmetricembedding ϕ : ( X ′ , d ′ ) → Z := ϕ ( X ′ ) ⊂ R N , where N ∈ N depends only on the doubling constant of the metric d ′ . Moreover, weobtain a quasim¨obius homeomorphism g = ϕ ◦ f : X → Z. In order to show that X is doubling, we consider two cases. Case 3.3. X is unbounded. If g ( x ) → ∞ as x → ∞ , by [25, Theorem 3 . g is quasisymmetricand thus the claim follows from [23, Theorem 2.10].If g ( x ) → p = ∞ as x → ∞ , we may assume without loss of generality that g ( ∞ ) = 0 ∈ Z . Let u ( x ) = x | x | be the reflection about the unit sphere centered at the origin in R N ∪ {∞} . Clearly, u is a M¨obius transformation. It follows that u ◦ g : X → u ( Z ) is quasim¨obius with u ◦ g ( x ) → ∞ as x → ∞ . Again by [25, Theorem 3.10] we see that u ◦ g isquasisymmetric. Therefore, it follows from [23, Theorem 2.10] that X is a doublingmetric space. Case 3.4. X is bounded. In this case, we consider the spherical metric σ on R N ∪ {∞} which is determinedby the length element | dz | σ = 2 | dz | | z | , where | dz | is the Euclidean length element and | z | is the Euclidean norm of a point z ∈ R N . Then g : X → ( Z, σ ) is a quasim¨obius map between two bounded metricspaces, which is actually quasisymmetric. By [23, Theorem 2.10], we see that X isdoubling. (cid:3) Next, we show that the Gromov boundary of a hyperbolic space equipped withany Bourdon metric and Hamenst¨adt metic are quasim¨obius equivalent.
Lemma 3.5.
Let X be a Gromov δ -hyperbolic space and ∂ ∞ X its Gromov boundary.Then the identity map ( ∂ ∞ X \ { ξ } , d w,ε ) → ( ∂ ∞ X \ { ξ } , σ b,ε ′ ) is θ -quasim¨obius with θ depending only on δ, ε and ε ′ , where d w,ε is the Bourdon metric based at w ∈ X with parameter ε > and σ b,ε ′ is the H¨amenstadt metric based at the Busemannfunction b = b ξ,o with parameter ε ′ > for o ∈ X and ξ ∈ ∂ ∞ X .Proof. For any distinct points x i ∈ ∂ ∞ X \ { ξ } , i = 1 , , ,
4, by [9, Lemmas 2.2.2and 3.2.4] we may assume that all of them lie in X . Thus by (2 . x | x ) w + ( x | x ) w − ( x | x ) w − ( x | x ) w = ( x | x ) o + ( x | x ) o − ( x | x ) o − ( x | x ) o . = δ ( x | x ) b + ( x | x ) b − ( x | x ) b − ( x | x ) b . Therefore, we have σ b,ε ′ ( x , x ) σ b,ε ′ ( x , x ) σ b,ε ′ ( x , x ) σ b,ε ′ ( x , x ) ≤ e − ε ′ [( x | x ) b +( x | x ) b − ( x | x ) b − ( x | x ) b ] ≤ e − ε ′ δ e − ε ′ [( x | x ) w +( x | x ) w − ( x | x ) w − ( x | x ) w ] ≤ e − ε ′ δ ε ′ /ε h d w,ε ( x , x ) d w,ε ( x , x ) d w,ε ( x , x ) d w,ε ( x , x ) i ε ′ /ε . This proves Lemma 3.5. (cid:3)
Proof of Theorem 1.1.
This follows from Lemmas 3.2 and 3.5. (cid:3)
Lemma 3.6.
Let f : ( X, d ) → ( Y, d ′ ) be a rough ( λ, c ) -quasi-isometry between twoproper geodesic Gromov hyperbolic spaces. If X is K -roughly starlike with respect tosome point w ∈ X , then Y is K ′ -roughly starlike with respect to the point f ( w ) . hou et al. 13 Proof.
Since f : X → Y is a rough ( λ, c )-quasi-isometry, we see that for any x ′ ∈ Y there is some point x ∈ X such that(3.7) d ′ ( f ( x ) , x ′ ) ≤ c. By the assumption that X is K -roughly starlike with respect to w ∈ X , we see thatthere is a geodesic ray γ emanating from w to some ξ ∈ ∂ ∞ X and satisfying(3.8) dist( x, γ ) ≤ K. Moreover, we see from [5, Proposition 6.3] that there is an extension f : ∂ ∞ X → ∂ ∞ Y of f with f ( ξ ) = ξ ′ ∈ ∂ ∞ Y . Then take another geodesic ray γ ′ emanating from w ′ = f ( w ) to ξ ′ . Because f ( γ ) is a rough quasi-isometric ray, it follows from theextended stability theorem [26, Theorem 6.32] that the Hausdorff distance(3.9) dist H ( f ( γ ) , γ ′ ) ≤ C. Hence we obtain from (3.7), (3.8) and (3.9) thatdist( x ′ , γ ′ ) ≤ c + C + λK + c = K ′ , as desired. (cid:3) We now pause to recall certain auxiliary definitions that we shall need. Supposethat (
X, d ) is a C -annular quasiconvex, geodesic and proper metric space, and Ω ( X is a domain. For any x ∈ Ω, denote d Ω ( x ) = dist( x, ∂ Ω). Let 0 < λ ≤ /
2. Following[4, Chapter 7] or [14], a point x in Ω is said to be a λ -annulus point of Ω, if thereis a point a ∈ ∂ Ω such that t = d ( x , a ) = d Ω ( x ) , the annulus B ( a, t/λ ) \ B ( a, λt ) is contained in Ω.If x is not a λ -annulus point of Ω, then it is said to be a λ -arc point of Ω. Thenwe have the following lemma. Lemma 3.10.
Suppose that ( X, d ) is a C -annular quasiconvex, geodesic and propermetric space, and Ω ( X is a bounded δ -hyperbolic domain. Then (Ω , k ) is K -roughly starlike with respect to some point w ∈ Ω , where K is a constant dependingonly on C and δ .Proof. Choose a point w such that d Ω ( w ) = max { d Ω ( x ) : x ∈ Ω } . We shall find a constant K depending only on C and δ such that for each x ∈ Ωthere exists a quasihyperbolic geodesic ray α emanating from w satisfyingdist k ( x, α ) ≤ K. Let λ = 1 / (2 C ). Fix x ∈ Ω, we consider two cases.
Case 3.11. x is a λ -arc point. Then by [14, Lemma 7.3], we see that there exist two points a, b ∈ ∂ ∞ Ω anda C -anchor γ connecting a, b with x ∈ γ and C = C ( λ, C ), where ∂ ∞ Ω is theGromov boundary of hyperbolic space (Ω , k ). For the definition of anchor see [14,Definition 7.2]. Moreover, by the definition of anchor, we see that γ is a continuousquasihyperbolic ( C , C )-quasigeodesic, that is, ℓ k ( γ [ x, y ]) ≤ C k ( x, y ) + C for all x, y ∈ γ . On the other hand, we see from [4, Proposition 2.8] that (Ω , k )is a proper geodesic metric space. Thus there is a quasihyperbolic geodesic line γ connecting a and b . It follows from [14, Lemma 3.4] that the quasihyperbolicHausdorff distance dist H ( γ, γ ) ≤ C , where C is a constant depending only on C and δ . This implies that there is apoint y ∈ γ with k ( x, y ) ≤ C Moreover, we may join w to a and b by quasihyperbolic geodesic rays α and α ,respectively. Now by [14, Lemma 3.1], we find thatdist k ( y, α ∪ α ) ≤ δ, and thus, dist k ( x, α ∪ α ) ≤ δ + C = K, as desired. Case 3.12. x is a λ -annular point. Thus there is a point x ∈ ∂ Ω with t = d ( x , x ) = d Ω ( x ) , the annulus B ( x , t/λ ) \ B ( x , λt ) is contained in Ω. Then choose a quasihyperbolicgeodesic ray α emanating from w to x . Since d ( w, x ) ≥ d Ω ( w ) ≥ d Ω ( x ) = t , thereexist a point z ∈ α with d ( z, x ) = t . Because X is C -annular quasiconvex, we seethat there is a curve β ⊂ B ( x , Ct ) \ B ( x , t/C ) connecting x and z with ℓ ( β ) ≤ Cd ( x, z ) ≤ Ct.
Note that B ( x , t/λ ) \ B ( x , λt ) is contained in Ω and λ = 1 / (2 C ), it follows that β ⊂ Ω and for each u ∈ β , we have d Ω ( u ) ≥ t/ (2 C ). Therefore, we have k ( x, z ) ≤ ℓ k ( β ) ≤ C = K, as required. (cid:3) Proof of Theorem 1.5.
Firstly, it follows from [1, Theorem 6.1] that the Gehring-Hayman condition and the ball separation condition imply the Gromov hyperbolicty.It remains to show the necessity. If (Ω , d ) is bounded, then the assertion follows from[18, Theorem 1.2]. If Ω is unbounded, by [18, Corollary 5.2], it suffices to check therough starlikeness of (Ω , k ).Towards this end, take a point a ∈ ∂ Ω. Denote the sphericalization of metric space(
X, d ) associated to the point a by ( ˙ X, b d a ). Since ( X, d ) is annular quasiconvex, it hou et al. 15 follows from [8, Proposition 6.3 ] that (
X, d ) is quasiconvex. Hence, [16, Lemma3.4] implies (Ω , d ) is locally quasiconvex. Since a ∈ ∂ Ω and (Ω , d ) is unbounded, itfollows from [8, Theorem 4.12] that the identity map (Ω , k ) → (Ω , b k a ) is bilipschitz,where b k a is the quasihyperbolic metric of Ω with respect to the metric b d a . It thusimplies that (Ω , b k a ) is Gromov hyperbolic, since (Ω , k ) is Gromov hyperbolic andthe quasihyperbolic metric is a length metric. Hence, by Lemma 3.6, we only needto show that (Ω , b k a ) is roughly starlike.According to [8, Theorem 6.5(a)], it follows that the space ( ˙ X, b d a ) is quasiconvexand annularly quasiconvex. Note that the rough starlikeness and Gromov hyperbol-icity are preserved under bilipschitz mappings. So we may assume that ( ˙ X, b d a )is geodesic, because the quasiconvexity condition implies that the identity map( ˙ X, b d a ) → ( ˙ X, ℓ b d a ) is bilipschitz, where ℓ b d a is the length metric of ( ˙ X, b d a ). Hence,by Lemma 3.10, we get (Ω , b k a ) is roughly starlike, as desired. (cid:3) ϕ -uniformity condition impliesthe quasihyperbolic growth condition as follows. Lemma 3.13.
Let (Ω , d ) be a locally compact, rectifiably connected and incompletemetric space. If (Ω , d ) is bounded and ϕ -uniform, then there is an increasing function φ : [0 , ∞ ) → [0 , ∞ ) such that for all x ∈ Ω k ( w, x ) ≤ φ (cid:16) d ( w ) d ( x ) (cid:17) , where w ∈ Ω satisfies d ( w ) = max x ∈ Ω d ( x ) and k is the quasihyperbolic metric of Ω .In particularly, we can take φ ( t ) = ϕ ( diamΩ d ( w ) t ) . Here and hereafter, we use d ( x ) todenote the distance from x to the boundary of Ω with respect to the metric d . Secondly, we verify that the ϕ -uniformity condition is preserved under spherical-ization. Lemma 3.14.
Let (Ω , d ) be a locally compact, c -quasiconvex and incomplete metricspace with a ∈ ∂ Ω . If (Ω , d ) is unbounded and ϕ -uniform, then there exists a home-omorphism ψ : [0 , ∞ ) → [0 , ∞ ) such that the sphericalized space (Ω , b d a ) associatedto a is ψ -uniform.Proof. Since (Ω , d ) is c -quasiconvex, we observe from [8, Proposition 4.3] that thereare constants λ ∈ (0 , /
2) and c ≥ c such that the spheri-calization (Ω , b d a ) is locally ( λ, c )-quasiconvex. Moreover, we see from [8, Theorem4.12] that the identity map (Ω , k ) → (Ω , b k a ) is 80 c -bilipschitz, where b k a is the quasi-hyperbolic metric of (Ω , b d a ).To prove this lemma, we only need to find a homeomorphism ψ : [0 , ∞ ) → [0 , ∞ )such that(3.15) b k a ( x, y ) ≤ ψ b d a ( x, y ) b d a ( x ) ∧ b d a ( y ) ! for all x, y ∈ Ω, where b d a ( x ) denotes the distance from x to the boundary of Ω withrespect to the metric b d a . To this end, we divide the proof into two cases. Case 3.16. b d a ( x, y ) ≤ λ c b d a ( x ) . A similar argument as in [16, Lemma 3.8] shows that(3.17) b k a ( x, y ) ≤ c b d a ( x, y ) b d a ( x ) ≤ c b d a ( x, y ) b d a ( x ) ∧ b d a ( y ) , as desired. Case 3.18. b d a ( x, y ) > λ c b d a ( x ) . Then we claim that for all x ∈ Ω,(3.19) b d a ( x ) ≤ d ( x )[1 + d ( x, a )] . This can be seen as follows. Take a point x ∈ ∂ Ω with d ( x ) = d ( x, x ). Weconsider two possibilities. If d ( x , a ) ≤ d ( x, a ) − , then we have d ( x ) ≥ d ( x, a ) − d ( x , a ) ≥
12 (1 + d ( x, a )) , and therefore, b d a ( x ) ≤ b d a ( x, ∞ ) ≤
11 + d ( x, a ) ≤ d ( x )[1 + d ( x, a )] . On the other hand, if d ( x , a ) > d ( x, a ) − , thus we have by (2.8) that b d a ( x ) ≤ b d a ( x, x ) ≤ d ( x, x )[1 + d ( x, a )][1 + d ( x , a )] ≤ d ( x )[1 + d ( x, a )] , as needed. This proves (3.19). hou et al. 17 Now by (2.8) and (3.19), we have for all x, y ∈ Ω, b j a ( x, y ) = log (cid:16) b d a ( x, y ) b d a ( x ) ∧ b d a ( y ) (cid:17) ≥
12 log (cid:16) b d a ( x, y ) b d a ( x ) (cid:17)(cid:16) b d a ( x, y ) b d a ( y ) (cid:17) ≥
12 log (cid:16) d ( x, y )(1 + d ( x, a )) d ( x ) (cid:17)(cid:16) d ( x, y )(1 + d ( y, a )) d ( y ) (cid:17) > log (cid:16) d ( x, y )8 p d ( x ) d ( y ) (cid:17) > log p d ( x ) d ( y ) + 2 d ( x, y ) p d ( x ) d ( y ) − ≥
12 log (cid:16) d ( x, y ) d ( x ) (cid:17)(cid:16) d ( x, y ) d ( y ) (cid:17) − ≥ j ( x, y ) − . Because the identity map (Ω , k ) → (Ω , b k a ) is 80 c -bilipschitz and (Ω , d ) is ϕ -uniform,the above inequality implies that b k a ( x, y ) ≤ ck ( x, y )(3.20) ≤ cϕ ( e j ( x,y ) − ≤ cϕ h (cid:16) b d a ( x, y ) b d a ( x ) ∧ b d a ( y ) (cid:17) − i . Set ψ ( t ) = 3 c t whenever 0 ≤ t ≤ λ/ c , and ψ ( t ) = 80 cc ϕ (256(1 + t ) − t ≥ λ/ c . Therefore, by (3.17) and (3.20), we obtain (3.15). (cid:3) Remark 3.21.
In [21], Li, Vuorinen and Zhou proved that quasim¨obius mappingspreserve ϕ -uniform domains of R n . In the proof of Lemma 3.14, we calculate thecontrol function for our needs. Proof of Theorem 1.8. (1) We assume first that (Ω , d ) is bounded. Since (Ω , d, µ ) isAhlfors Q -regular and (Ω , k ) is roughly starlike and Gromov hyperbolic, we see from[18, Theorem 5 .
1] that (Ω , d ) satisfies the Gehring-Hayman condition. Because (Ω , d )is ϕ -uniform, it follows from Lemma 3.13 that (Ω , d ) satisfies the quasihyperbolicgrowth condition with φ ( t ) = ϕ ( diamΩ d ( w ) t ). That is, for all x ∈ Ω we have k ( w, x ) ≤ φ (cid:16) d ( w ) d ( x ) (cid:17) , where w ∈ Ω satisfies d ( w ) = max x ∈ Ω d ( x ) and k is the quasihyperbolic metric of Ω.Note that the conditions Z ∞ dtϕ − ( t ) < ∞ and Z ∞ dtφ − ( t ) < ∞ are mutually equivalent. Therefore, according to [19, Theorem 1 . , d ) is unbounded. Let a ∈ ∂ Ω. Denote by (Ω , b d a , µ a ) thesphericalization of (Ω , d, µ ) associated to the point a . Let ℓ b d a be the length metricof Ω with respect to the metric b d a . In order to show that there is a homeomorphismidentification ∂ Ω ∪ {∞} → ∂ ∞ Ω , we need some preparations.Firstly, we show that Claim 3.22. the identity map ( ∂ Ω ∪ {∞} , d ) → ( ∂ Ω ∪ {∞} , ℓ b d a ) is a homeomor-phism. Indeed, by the definition of ℓ b d a , we see that d ( x n , a ) → ∞ if and only if ℓ b d a ( x n , a ) →
0, as n → ∞ . Thus it suffices to verify that for any sequence { x n } ⊂ Ω, { x n } isCauchy in the metric d if and only if { x n } is Cauchy in the metric ℓ b d a .On one hand, by [20, (2.11)] and the quasiconvexity of (Ω , d ), it follows that ℓ b d a ( x n , x m ) ≤ ℓ d ( x n , x m ) ≤ cd ( x n , x m ) , where ℓ d is the length metric of (Ω , d ).On the other hand, we see from (2.8) that d ( x n , x m ) ≤ d ( x n , a )][1 + d ( x m , a )] ℓ b d a ( x n , x m ) , as required. We have proved Claim 3.22.Secondly, we check that Claim 3.23. (Ω , b k a ) is roughly starlike Gromov hyperbolic and (Ω , ℓ b d a ) satisfies theGehring-Hayman condition, where b k a is the quasihyperbolic metric of Ω with respectto the metric b d a . By [8, Theorem 4 . , k ) → (Ω , b k a ) is 80 c -bilipschitz. Since (Ω , k ) is Gromov hyperbolic, it follows from [7, Page 402, Theorem1.9] that (Ω , b k a ) is Gromov hyperbolic as well. Since (Ω , k ) is roughly starlike, wesee from Lemma 3.6 that (Ω , b k a ) is also roughly starlike. Furthermore, accordingto [20, Proposition 3.1], we immediately find that the sphericalization (Ω , b d a , µ a ) isAhlfors Q -regular. Thus we obtain from [18, Theorem 5.1] that (Ω , ℓ b d a ) satisfies theGehring-Hayman condition, which shows Claim 3.23.Next, we claim that Claim 3.24. there is a natural homeomorphism identification ( ∂ Ω ∪ {∞} , ℓ b d a ) → ( ∂ ∞ Ω a , b ρ ) , where ∂ ∞ Ω a is the boundary at infinity of Gromov hyperbolic space (Ω , b k a ) and b ρ isan arbitrary Bourdon metric on ∂ ∞ Ω a . Since (Ω , d ) is ϕ -uniform, we see from Lemma 3.14 and its proof that both thespaces (Ω , b d a ) and (Ω , ℓ b d a ) are ψ -uniform with ψ ( t ) = 80 cϕ (256(1 + t ) −
1) for hou et al. 19 all t ≥
1. Thus it follows from Lemma 3.13 that there is an increasing function φ : [0 , ∞ ) → [0 , ∞ ) and w ∈ Ω such that b k a ( w, x ) ≤ φ (cid:16) b d a ( w ) b d a ( x ) (cid:17) with φ ( t ) = ψ ( t b d a ( w ) ), since diam b d a (Ω) ≤ ψ and φ that the conditions Z ∞ dt p ϕ − ( t ) < ∞ and Z ∞ dtφ − ( t ) < ∞ are mutually equivalent. Now we observe that the conditions [19, (1.3) and (1.5)]with respect to the metric b k a are verified. Therefore, it follows from Claim 3.23and [19, Theorem 1.1] that there is a natural homeomorphism identification ( ∂ Ω ∪{∞} , ℓ b d a ) → ( ∂ ∞ Ω a , b ρ ), which proves Claim 3.24.Finally, since the identity map (Ω , k ) → (Ω , b k a ) is 80 c -bilipschitz, we see from [5,Propositions 6.3 and 6.5] that there is a natural homeomorphism identification( ∂ ∞ Ω , ρ ) → ( ∂ ∞ Ω a , b ρ ) , where ∂ ∞ Ω is the Gromov boundary of hyperbolic space (Ω , k ) and ρ is a Bourdonmetric defined on ∂ ∞ Ω.Therefore, this together with Claims 3.22 and 3.24 shows that there is a naturalhomeomorphism identification ∂ Ω ∪ {∞} → ∂ ∞ Ω . Hence Theorem 1.8 is proved. (cid:3)
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Qingshan Zhou, School of Mathematics and Big Data, Foshan University, Foshan,Guangdong 528000, People’s Republic of China
E-mail address : [email protected]; [email protected] Yaxiang Li, Department of Mathematics, Hunan First Normal University, Chang-sha, Hunan 410205, People’s Republic of China
E-mail address : [email protected] Xining Li, Department of Mathematics(Zhuhai), Sun Yat-sen University, Zhuhai519082, People’s Republic of China
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