Interplay between interior and boundary geometry in Gromov hyperbolic spaces

Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2010

Interplay between interior and boundary geometry in Gromov hyperbolic

spaces

Jordi, Julian

Abstract: We show that two visual and geodesic Gromov hyperbolic metric spaces are roughly isometric if and only if their boundaries at infinity, equipped with suitable quasimetrics, are bilipschitz-quasimoebius equivalent. Similarly, they are quasi-isometric if and only if their boundaries are power quasimoebius equivalent

DOI: https://doi.org/10.1007/s10711-010-9472-0

Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-156456

Journal Article Published Version

Originally published at:

Jordi, Julian (2010). Interplay between interior and boundary geometry in Gromov hyperbolic spaces. Geometriae Dedicata, 149(1):129-154.

DOI 10.1007/s10711-010-9472-0 O R I G I NA L PA P E R

Interplay between interior and boundary geometry

in Gromov hyperbolic spaces

Julian Jordi

Received: 21 July 2009 / Accepted: 1 February 2010 / Published online: 12 February 2010 © Springer Science+Business Media B.V. 2010

Abstract We show that two visual and geodesic Gromov hyperbolic metric spaces are roughly isometric if and only if their boundaries at infinity, equipped with suitable quasimet-rics, are bilipschitz-quasimoebius equivalent. Similarly, they are quasi-isometric if and only if their boundaries are power quasimoebius equivalent.

Keywords Hyperbolic spaces·Boundary at infinity·Quasimetric·Quasisymmetric maps·Quasimoebius maps

Mathematics Subject Classification (2000) 51F99·53C23·54C20·20F67·20F69

1 Introduction

Given a Gromov hyperbolic metric spaceX, one has associated to it the boundary at infinity, or ideal boundary,∂∞X. Via the Gromov product(·|·), one obtains in canonical fashion a family of quasimetricsa−(·|·) on the set∂∞X. It is a very natural question to ask to what extent the structure of the boundary determines the space itself, and what kind of correspon-dence exists between maps of spaces and maps between their associated boundaries. Previous results in this direction were obtained by Paulin [6], Bonk and Schramm [1], and Buyalo and Schroeder [3], among others. These all differ somewhat among one another in the class of spaces and maps they are valid for. The goal of this work was to find a general setting that systematically explores the relationship of a Gromov hyperbolic space to its boundary and vice versa. We are then able to deduce the cited results as special cases within this general framework, cf. Corollaries4and5.

This research has been supported by Swiss National Science Foundation Project Grant 200020-119907/1. J. Jordi (

B

Institute of Mathematics, University of Zurich, Winterthurerstr. 190, 8057 Zurich, Switzerland e-mail: [email protected]; [email protected]

At the heart of this work lie the extension theorems for bilipschitz, power quasisymmetric and power quasimoebius boundary maps, which we can subsume in the

Theorem 1 (Cf. Theorems4,5,8)Let X,X′be hyperbolic metric spaces with X visual and X′roughly geodesic.

If f : ∂∞a,oX → ∂a,o

′

∞ X′is a bilipschitz map, then there exists a rough isometric map F:X→X′such that its naturally associated boundary map∂∞F equals f, ∂∞F= f .

If f :∂o

∞X →∂o

′

∞X′is a power quasisymmetric map, then there exists a power quasi-isometric map F:X→X′such that∂∞F= f .

If f : ∂∞X → ∂∞X′is a power quasimoebius map, then there exists a power quasi-isometric map F:X→X′such that∂∞F= f .

See Definitions 7,8 and its subsequent paragraph for the definition of power quasi-isometric and quasimoebius/quasisymmetric maps.

For spaces which are both visual and roughly geodesic one then obtains the following characterization of rough isometry and PQ-isometry classes.

Theorem 2 Let X,X′be visual, roughly geodesic hyperbolic metric spaces. The following are mutually equivalent.

(I) X and X′are roughly isometric.

(II) There is a map F :X →X′and a D≥0such that for all quadruples Q⊂X cd(Q)−D≤cd(F(Q))≤cd(Q)+D.

(III) For any a > 1there is a bilipschitz-quasimoebius homeomorphism f : ∂_∞a X →

∂_∞a X′.

Also the following are equivalent. (i) X and X′are quasi-isometric. (ii) X and X′are power quasi-isometric.

(iii) For any a,a′>1, ∂∞a X is power quasimoebius equivalent to∂a

′

∞X′.

Note(I I)⇒(I)and(i i)⇒(i)are trivial, as is(I)⇒(I I). The implication(i)⇒(i i) is due to Buyalo and Schroeder ([3], Theorem 4.4.1). The bilipschitz and the power quasi-symmetric extension theorems were proved in the metric setting by Bonk and Schramm in [1], Theorem 7.4.

The main contributions of this paper are the quasimetric extension theorems for power quasisymmetric maps (Theorem5) and inversions (Theorem7), which are combined to give the extension for power quasimoebius maps (Theorem8).

This article is organized as follows. Section2recalls basic notions on Gromov hyperbolic spaces and gives definitions on quasimetric spaces and the various classes of morphisms between (quasi) metric spaces we consider in this article. Section3summarizes the technique of producing a Gromov hyperbolic space to a given boundary via hyperbolic approximation. Section4recalls the well-known theorem on extension of bilipschitz boundary maps, while Sects.5and6contain the proofs for the extension theorems for power quasisymmetric and inversion maps, respectively. Section7combines them to prove the the extension theorem for power quasimoebius maps. Section8combines the pieces to prove Theorems 1and2.

I thank Prof. Viktor Schroeder for his interest in this work and many helpful discussions. I also thank the referee for the many suggestions that have helped the readability of this article.

2 Preliminaries and notation

2.1 Some notation

The notationa≍K bis shorthand fora/K ≤b≤K a,a=.Cbstands fora−C ≤b≤a+C.

For example, saying that|F(x)F(y)|≍K |x y| ∀x,y∈X, or|F(x)F(y)|=.C |x y| ∀x,y∈X

is another way of saying that the mapF:X→Yis bilipschitz or roughly isometric, respec-tively. If we do not specify the constantsK orCand just writea≍b,a=. b, it is understood that there is auniformsuch constant which works for allaandbin the given context.

At some point we will also usea≥˙b, which will analogously mean that there is a uniform Csuch thata≥b−C.

For metric spaces we find it convenient to denote the metric by| · |, that is the distance fromxtoyis written as|x y|.

2.2 Gromov hyperbolic spaces

Definition 1 Givenδ≥0 andT =(x0,x1,x2)a triple of real numbers, we say thatT is a δ-tripleif the two smaller numbers differ by no more thanδ, or equivalently if theδ-inequality

xi≥min{xi−1,xi+1} −δ ∀i∈Z/3Z is satisfied.

Definition 2 Let(X,| · |)be a metric space andx,y,o∈XTheGromov product of x and y with respect to o,(x|y)ois defined as

(x|y)o:=

1

2(|ox| + |oy| − |x y|). (X,| · |)is calledδ-hyperbolicif for allx,y,z,o∈Xthe triple

((x|y)o, (x|z)o, (y|z)o)

is aδ-triple.

(X,| · |)is calledGromov hyperbolicif it isδ-hyperbolic for someδ≥0.

Ageodesicin a metric space X is an isometric mapγ : I → X from a real interval (possibly infinite) intoX.Xis calledgeodesicif for any two pointsx,y ∈Xthere exists a geodesicγ : [a,b] →Xwithγ (a)=xandγ (b)=y.

2.3 Boundary at infinity

A sequence(xi)in a metric space is said toconverge to infinityif

lim

i,j→∞(xi|xj)o= ∞ for one, and hence any, base pointo∈X.

Two sequences(xi), (xi′)are said to be equivalent if

lim

i→∞(xi|x ′

i)o = ∞.

For Gromov hyperbolic spaces, this defines an equivalence relation and we define the set calledboundary at infinity of X, ∂∞X, to be the set of equivalence classes of sequences converging to infinity.

Letξ, ξ′∈∂∞Xando∈ X. We extend the Gromov product to the boundary at infinity by setting

(ξ|ξ′)o :=inf lim inf i→∞(xi|x

′

i)o,

where the infimum is taken over all sequences(xi)∈ξ, (xi′)∈ξ′. It is a fact ([3], Lemma

2.2.2(2)) that with this definition, theδ-inequality extends to the boundary at infinity. That is,

((ξ|ξ′)o, (ξ|ξ′′)o, (ξ′|ξ′′)o)

is aδ-triple for allξ, ξ′, ξ′′∈∂∞X. 2.4 Busemann functions

With the help of the Gromov product for boundary points defined above we can give the set ∂∞Xthe structure of aboundedquasimetric space, see below. To get nonbounded (quasi)met-rics on the boundary, however, we need to introduce Busemann functions.

Forω∈∂∞Xando∈Xdefine the function bω,o:X→IR

x →bω,o(x):=(ω|o)x−(ω|x)o,

where the Gromov product(ω|o)x, with one argument in∂∞X, is defined as inf lim inf(wi|o)x

in analogy to the case with both arguments in∂∞X.

bω,o is the prototype of a Busemann function based atω∈ ∂∞X. It corresponds to the

Busemann function associated to a geodesic ray fromotoω in case X is a Riemannian manifold of pinched negative curvature. Any function that is equal tobω,oup to a constant

and a uniformly bounded additive error shall be called aBusemann function. More precisely:

Definition 3 Letω∈∂∞X. The setB_(ω)of allBusemann functions based atωconsists of all those functionsb:X→IR for which there existso∈Xand a constantc∈IR such that b=.2δbω,o+c.

Forb∈B_(ω)a Busemann function based atω, we define the Gromov product(x|y)bfor

x,y∈(X,| · |)by

(x|y)b:=

1

2(b(x)+b(y)− |x y|). Note that(·|·)b, in contrast to(·|·)o, can be negative.

We extend the Gromov product to∂∞Xby (ξ|ξ′)b :=inf lim inf

i→∞(xi|x ′

i)b,

where the infimum is taken over all sequences(xi)∈ξ, (xi′)∈ξ′.

Proposition 1 ([3], Lemma. 3.2.4(2))For X aδ-hyperbolic space andξ, η, ζ, ω ∈ ∂∞X arbitrary, the numbers(ξ|η)b, (ξ|ζ )b, (η|ζ )bform a22δ-triple for any b∈B(ω).

2.5 Quasimetric spaces

Definition 4 AK-quasimetric spaceis a setZtogether with a mapρ: Z×Z → [0,∞]

such that

I. ρ(z,y)≥0∀z,y∈Z, with equality iffy=z. II. ρ(z,y)=ρ(y,z)∀z,y∈Z.

III. ρ(z, w)≤Kmax{ρ(z,y), ρ(y, w)} ∀w,y,z∈Z.

IV. There is at most onez∈Zsuch thatρ(z,y)= ∞for ally∈Z\{z}.

If no pointzas inIV exists,Zis said to benon-extended, while it isextendedif there is such azand thiszis then called theinfinitely remote point. By convention, a one-point space Z= {z}is never extended.

IfX is aδ-hyperbolic space,a>1,o∈ Xand(·|·)odenotes the Gromov product with

respect to the base pointo, thena−(·|·)o _{is ana}δ_{-quasimetric on the set∂∞X. Similarly,} a−(·|·)b_{, for some Busemann functionb, defines ana}22δ_{-quasimetric.}

In particular, the boundary at infinity of a 0-hyperbolic space (i.e. a subset of a tree) is K-quasimetric withK =1. Such spaces are usually called ultrametric spaces.

A quasimetricρon a space Z induces a topology by declaring a setA⊂ Zto be open if for everya∈ A\{∞}there existsr >0 such that Brρ(a)⊂ A, and if∞ ∈A, then there

existsy∈Zandr>0 such thatA⊂Br(y)c. This topology is metrizable and in particular

first-countable and Hausdorff. This follows from the fact that if(Z, ρ)is K-quasimetric, then(Z, ρs_)isKs_{-quasimetric (and the two topologies are clearly equivalent), and a result}

of Frink’s ([4]) whereby a K-quasimetric with 1 ≤ K ≤ 2 is bilipschitz equivalent to a metric (extended ifρis extended).

Here and in the future we always denoteBrρ(x) := {z∈ Z|ρ(z,x) <r}. Note, though,

that in contrast to the metric setting this need not be an open set.

Definition 5 A quasimetric space(Z, ρ)is calledcompleteif every Cauchy sequence in Z\{∞}converges and ifρis extended in case it is unbounded.

For example, the circleS1with the induced metric from IR2 is complete. IR is not com-plete but IR∪ ∞is. Note that IR∪ ∞is obtained fromS1 via stereographic projection, a Moebius map. It is true in general that quasimoebius maps (see below) send complete spaces to complete spaces.

Boundaries of hyperbolic spaces are always complete, cf. [1], Proposition 6.2.

Definition 6 The symbol∂∞Xdenotes thesetof boundary points of a Gromov hyperbolic space. The symbols∂∞a,oand∂∞a,b, wherea>1,o∈X,b∈B(ω), denote thequasimetric spaces(∂∞X,a−(·|·)o_{)and(∂∞X,a}−(·|·)b_{), respectively.}

Remark 1 In fact, the bilipschitz class of∂∞a,oXdoes not depend ono∈Xand the quasimoe-bius class depends on neither of the parameters. Thus we may suppress one or both of them and just write∂∞a X, or∂∞X. Whenever we do this it is to be understood that the statement holds for any admissible choice of the omitted parameter(s).

Note that∂∞o Xis always bounded, while∂∞b X, forb ∈B(ω), is always extended with infinitely remote pointω.

2.6 Various classes of maps

A map f : X→Y between metric spaces is calledroughly isometric, or more specifically C-roughly isometricif there existsC such that|x y|=.C |f(x)f(y)|for allx,y ∈ X. f is

calledquasi-isometric, or (c,d)-quasi-isometric, if there existc,d such that 1_c|x y|−d ≤ |f(x)f(y)|≤c|x y|+d. If there exists a roughly isometric (quasi-isometric) mapg:Y →X such thatdX(g◦ f(x),x) ≤ D for some uniformD, then f is called a rough isometry

(quasi-isometry) andXis said to beroughly isometric (quasi-isometric) to Y.

A metric spaceX is calledroughly geodesicif there exists for anyx,y ∈ XaC-rough geodesicjoiningx andy, where aC-rough geodesic is aC-roughly isometric map from an intervalI ⊂IR intoX.

If a mapF:X→X′between Gromov hyperbolic spaces maps sequences going to infin-ity inXto sequences going to infinity inX′and equivalent sequences to equivalent sequences, thenFinduces a map between boundaries, which we denote∂∞F:∂∞X →∂∞X′.

For example, every roughly isometric map F : X → X′ induces an injection∂∞F :

∂∞X →∂∞X′. A quasi-isometric map F : X →X′betweengeodesichyperbolic spaces induces a boundary map by the stability of geodesics (cf. [2], Theorem III.H.1.7). However, the mapF: {10i|i ∈N} →IR,F(10i):=(−1)i10iis quasi-isometric, but does not induce a boundary map in any reasonable sense. This is one of the reasons why quasi-isometric maps are in general not the right maps to look at in the setting of hyperbolic metric spaces. In fact, for non-geodesic spaces, quasi-isometric maps need not even preserve Gromov hyperbolicity. The following definition gives the class of maps with all desired properties.

Definition 7 ForQ=(x,y,z, w)an ordered quadruple of points in a metric space(X,|·|) denote bycd(Q)theircross-difference,

cd(Q):= 1

2(|x z|+|yw|−|x y|−|zw|)=(x|y)o+(z|w)o−(x|z)o−(y|w)o. A map F : X → X′ between metric spaces is called (c,d)-power quasi-isometric (PQ-isometric) if

1

ccd(Q)−d≤cd(F(Q))≤c cd(Q)+d.

Sincecd(x,x,y,y)= |x y|, every power quasi-isometric map is quasi-isometric. More-over, every power quasi-isometric mapF : X →X′between hyperbolic spaces induces a boundary map∂∞F:∂∞X→∂∞X′. This follows fromcd(x,y,o,o)=(x|y)o.

A quasi-isometric map betweengeodesic hyperbolic spaces is automatically PQ-iso-metric, cf. [3], Theorem 4.4.1 (what we call PQ-isometric is calledstronglyPQ-isometric in [3]).

The multiplicative analog of a PQ-isometric map is apower quasimoebius map.

Definition 8 ForQ = (x,y,z, w)an ordered quadruple of points in a quasimetric space (Z, ρ)denote bycr(Q)theircross-ratio

cr(Q)= ρ(x,z)ρ(y, w)

ρ(x,y)ρ(z, w).

Ifθ : [0,∞) → [0,∞)is a homeomorphism, a map f : Z → Z′between quasimetric spaces is calledθ-quasimoebius (θ-QM) if 1/θ (cr(Q)−1)≤ cr(f(Q))≤ θ (cr(Q)). f is calledpower quasimoebius(P-QM) if it isθ-QM for aθof the formθ (t)=qmax{t1/p,tp}. It is calledbilipschitz quasimoebiusifθcan be taken of the formθ (t)=λt.

Closely related to QM maps arequasisymmetric(QS) maps, which are the ones which preserve the ordinary ratiosrof a triple(x,y,z),sr(x,y,z):= |x z|/|x y|, in an analogous way.

We refer to [10] and [3], Chapter 5, for more information on quasimoebius and quasisym-metric maps. Quasimoebius maps are called “strictly quasimoebius” in [3].

3 Hyperbolic approximation

The goal of hyperbolic approximation is to find to a given (quasi)metric spaceZa hyperbolic spaceX(with nice properties) such that∂∞X=Z. The procedure we use was developed by Buyalo and Schroeder, cf. Chapter 6 of [3]. However, the idea of constructing a hyperbolic space with prescribed boundary is itself not new. The usual approach has been to mimic the upper half plane or the unit disk situation by crossing the given space with IR≥0(or a finite interval in the case of bounded boundary) and equipping the product with a suitable metric which turns out to be hyperbolic. The oldest such method may be the hyperbolic cone over a metric case, cf. [3] §6.4.4, originally due to Berestovskii. Similar constructions were also used by Gromov, Trotsenko and Väisälä (cf. [8]), as well as Bonk and Schramm (cf. [1]). Buyalo and Schroeder’s method has the advantage that it is very intuitive and produces a particularly nice geodesic space, namely a graph, which is easily recognized to be visual. Basically, only the verification of hyperbolicity needs some work. Furthermore, it is straight-forward to adapt it to the setting of quasimetric boundary spaces, which is crucial for this work.

Let(Z, ρ)be a completeK-quasimetric space. Letr<1/K3. The procedure now goes as follows. For everyk ∈ZletVk be a maximalrk-separated subset of Z (such exist by

Zorn), whererk-separated meansρ(v, v′)≥rkfor allv, v′∈Vk. Denote byVthe set of all

ordered pairs(k,z)withk∈Zandz∈Vk. The projectionℓ:V→Zto the first coordinate

is calledlevel function, andℓ(v)thelevelofv, while the projectionπ:V→Zto the second coordinate sendsvto itscenterπ(v)∈Z.

Remark 2 Sometimes the notationπ(v)becomes too cumbersome so that we often identify a pairv ∈ Vk with its centerπ(v)∈ Z. The notationρ(v, w)is thus interpreted to mean

ρ(π(v), π(w)).

The hyperbolic approximation with parameterr < 1/K3is then defined to be the sim-plicial graph with vertex setV_{, where two verticesv, w}∈V_{are joined by an edge exactly}

when

– ℓ(v)=ℓ(w)and the sets B(v):= B_{K r}l(v)(π(v))andB(w):= B_{K r}l(w)(π(w))intersect inZ, or

– ℓ(v)=ℓ(w)+1 andB(v)is contained inB(w).

It follows from [3], Theorems. 6.3.1, 6.4.1, (cf. Theorems.3below) that then∂∞1/rHypr

(Z, ρ)is bilipschitz equivalent to (Z, ρ). So far this only holds forr < 1/K3. Now the boundaries at infinity come equipped with a family of quasimetricsa−(·|·)fora >1. The corresponding situation for hyperbolic approximations is that they should be taken for a fam-ily of parametersr∈(0,1), not just forr∈(0,1/K3_{). Even though it should intuitively be} possible to make a similar construction with balls as above, it seems the resulting graph is too difficult to control. For this reason, we resort to a scaling trick.

Definition 9 Let(Z, ρ)be a completeK-quasimetric space andr∈(0,1). IfZis extended and|Z|≥3 withξ the infinitely remote point, define Hyp_r(Z, ρ)to be the graph obtained from(Z\{ξ}, ρ1/s)as above whenr <1/K3, and define it to be the graph obtained for an r′<1/K3scaled byln_lnr_r′ whenr≥1/K3. If|Z|=2 define Hyp_r(Z):=IR.

If(Z, ρ)is not extended and hence bounded, then for|Z|≥2,Hyp_r(Z, ρ)is defined in the same way except that it is understood to be truncated (cf. [3] §6.4.1). For|Z|=1 define Hyp_r(Z):=IR≥0.

It is not difficult to show that, up to a rough isometry, the resulting graph does not depend on the choice of vertex systemV, nor on the quasimetricity constantK used for ρ(note a K-quasimetric is also aK′-quasimetric for K′≥ K). Moreover, the rough isom-etry class of Hyp_r(Z, ρ)does not depend on the choice ofr′in Definition9, meaning one has lnr1

lnr2Hypr1(Z, ρ) .

=Hyp_r2(Z, ρ). This can be proved directly with Lemma7although forboundedρit also follows from the bilipschitz extension Theorem4.

We remark that by a Zorn-type argument there existhereditaryvertex systemsV= {Vk}k,

meaning thatπ(Vk) ⊂ π(Vk+1). Working with such hereditary systems often simplifies arguments and we will use them without reservation when it suits us.

In the extended case, Hyp(Z)has a distinguished boundary pointωcorresponding to the infinitely remote pointξofZ, while in the non-extended case the rootoof the approximation will serve as distinguished base point.

The crucial theorem about hyperbolic approximation is

Theorem 3 ([3], Theorems 6.3.1, 6.4.1)Let(Z, ρ)be a complete quasimetric space, r ∈

(0,1). The hyperbolic approximationHyp_r(Z)is a visual geodesic hyperbolic space and there is a canonical identification∂∞Hyp_r(Z)=Z of sets. Moreover, if(Z, ρ)is extended then for any b∈B_{(ω), ∂∞}1/r,b_Hyp

r(Z, ρ)and(Z, ρ)are bilipschitz equivalent. If(Z, ρ)is

not extended, then∂∞1/r,oHypr(Z, ρ)and(Z, ρ)are bilipschitz equivalent.

The moral of the story is that, given a complete quasimetric space(Z, ρ), there is for everya>1 exactly one (up to rough isometry) visual geodesic hyperbolic spaceXsuch that ∂_∞a Xis bilipschitz-quasimoebius to(Z, ρ), and the “functor” Hyp_1/a spits out exactly this spaceXwhen applied to(Z, ρ).

4 Extension of bilipschitz maps

We recall [3], Theorem 7.1.2, stated here for quasimetric boundary spaces. The proof is exactly the same as in the metric setting of [3].

Theorem 4 Let X be a visual and X′be a geodesic hyperbolic space, o ∈ X,o′ ∈ X′. Then to every bilipschitz map f :∂∞a,o,X →∂a,o

′

∞ X′, there exists a roughly isometric map F:X→X′with∂∞F= f .

Corollary 1 ([3], Corollaries 7.1.5,7.1.6 and [1], Theorem 8.2)Let X be a visual hyperbolic space and o∈X,a>1,r∈(0,1).

X embeds roughly homothetically intoHyp_r∂∞a,oX . If X is also roughly geodesic, then there is a rough homothety of X ontoHyp_r∂∞a,oX .

In addition, X embeds roughly isometrically intoHyp_1/a∂∞a,oX . If X is also roughly geo-desic, then X is roughly isometric toHyp_1/a∂∞a,oX .

When we are only concerned about the quasi-isometry class of the approximation, it is thus not necessary to specify the parameterrin Hyp_r(Z). Whenever we write only Hyp(Z) in a statement, it is to be understood that the statement is true forevery r ∈(0,1).

5 Extension of PQ-symmetric maps

Theorem 5 (Compare [1], Theorem 7.4) Let (Z, ρ), (Z′, ρ′) be two bounded complete quasi-metric spaces andHyp(Z),Hyp(Z′)be their hyperbolic approximations. Suppose f :

Z→Z′is a power quasisymmetric homeomorphism, i.e.η-QS withη(t)=Cmax{tα,t1/α}

for some C>0, α≥1. Then there exists a power quasi-isometry F :Hyp(Z)→Hyp(Z′) with∂∞F= f .

This theorem is trivial forZ= {z}, so we assume|Z|≥2. For convenience we shall also assume throughout this section that both spaces areK-quasimetric and that the approxima-tions of both spaces are done w.r.t thesameparameterr =1/(2K3_{). This poses no loss of} generality by Theorem3, Corollary1and indepence ofK of the hyperbolic approximation. We assume the vertex systemV = {Vk}is hereditary. We will split up the vertices into

two disjoint subsets. Recall that ifv∈Vk, then tovis associated the ballB(v)= Bk(v):=

B_{K r}k(π(v))⊂Z.

Definition 10 A vertexv∈ Vkis calledregularif the annulusBK rk(π(v))\B_{K r}k+1(π(v)) is non-empty. It is calledsingularif it is not regular.

The rootoof a truncated hyperbolic approximation is always regular unless Z = {z}, which we assume is not the case.

Lemma 1 Ifv∈Vkis singular and connected radially to a vertexw∈Vk+1andπ(w)= π(v), thenwis regular and so is(π(v),k+1)∈Vk+1.

Moreover, ifwis a horizontal neighbour ofv∈Vk, then at least one ofv, wis regular.

Proof B(w)⊂ B(v)by definition of radial edges. Sincevis singular, this meansB(w)⊂

B_{K r}k+1(π(v)). On the other hand,ρ(π(v), π(w))≥rk+1 > K rk+2, which meanswand (π(v),k+1)are regular.

Ifv, w ∈ Vk are both singular, then ρ(v, w) ≥ rk and for anyz ∈ B(v), ρ(v,z) <

K rk+1by singularity ofv. Henceρ(w,z)≥rk/K > K rk+1andzis not inB(w), hence

B(v)∩B(w)= ∅. ⊓⊔

Remark 3 Ifv ∈VkandBK rk−1(v) B_{K r}k(v), (π(v),k−1)may or may not be inVk−1. At any rate, we know by maximality ofVk−1 that there existsw ∈ Vk−1which is radially connected tov∈Vkandρ(π(w), π(v)) <rk−1.

We will now define the mapFof Theorem5. The idea is to define it first on all regular vertices and then “fill in” the rest. First of all note the

Lemma 2 For any vertexv∈Vkof a hereditary vertex systemVexactly one of the following

holds.

I. vis regular

II. vis singular and so arev∈Vk+l for0≤l<N , whilev∈Vk+N is regular. N ≥1.

III. vis singular in Vk+lfor all l≥0.

Proof The notationv∈Vk+lis meant to denote the element(π(v),k+l)ofVk+l. The cases

are mutually exclusive and exhaustive, so the lemma is evident. ⊓⊔

We will refer to the numbers in Lemma2as thetypesof a given vertexv ∈Vk, typeI

vertices being the regular vertices and so on, cf. Fig.1.

Definition 11 Ifv ∈V_{is regular,F(v)is defined to be a vertexv}′ ∈Hyp(Z′)of highest level such thatB(v′)⊃ f(B(v)).

Fig. 1 v1, v2, v3are vertices of typeI,I I,I I I, respectively

This definesFon the set of regular vertices up to an error of at most 1, as any two such verticesv′are evidently connected by an edge.

Lemma 3 Ifv∈Vkis regular, then so is F(v).

Proof Denote by m the level l(F(v)) of F(v) in Hyp(Z′). If F(v) were singular, BK′_rm(π(F(v)))\B_K′_rm+1(π(F(v)))would have to be empty. This would mean that all of f(B(v))would already be contained inB_K′_rm+1(π(F(v))), contradicting the maximality of

the level ofF(v)among all vertices containing f(B(v)). ⊓⊔

Now supposev ∈ Vk is of type I I. As noted before,vis not the root of Hyp(Z). In

particular, there will be anm ∈Nand aw∈Vk−msuch thatwis regular,v∈Vk−m+1and singular, andw∈Vk−mis radially connected tov∈Vk−m+1.π(w)may or may not be equal toπ(v), confirm Remark3. Trivially, all thev’s on adjacent levels are radially connected. We define the following terms.

Definition 12 A geodesic segment in Hyp(Z)through verticesv0, . . . , vNis calledsingular

if the verticesv1up to and includingvN−1are all singular.

By Lemma1and the paragraph following this definition, we may assume that all edges v0v1, . . . , vN−1vNare radial and thatπ(v1)= · · · =π(vN). It follows that the level function

is monotonous along the geodesic and, after possibly reversing the order, we may assume k−m=ℓ(v0)≤ℓ(v1) < ℓ(v2)· · ·< ℓ(vN−1)≤ℓ(vN)=k+l.

Ifv0is regular, we call it thelower endof the singular geodesic and ifvN is regular, it is the

upper end, respectively.

Every singular geodesic segment has a lower end since the root is regular. A singular geodesic with no upper end is called asingular ray. The lower end of a singular ray is also called itsroot.

Lower and, if they exist, upper ends are uniquely determined by the singular segment up to error 1. In particular, ifvN ∈VNis an upper end we may assumeπ(vN)=π(vN−1)because ifvN−1is singular, then(π(vN−1),N)is connected tovNand both are regular. Similarly, ifv0 is a lower end andℓ(v0)=ℓ(v1), then anyw∈Vℓ(vo)−1withρ(w, v1) <rℓ(v0)

Fig. 2 Singular geodesicw . . . vk+2associated tovk=v∈Vk

and we can replacev0withw. We may thus supposeℓ(v0) < ℓ(v1) <· · ·< ℓ(vN)=k+l

With these assumptions, a vertexv ∈ Vk of typeI I thus gives rise to a singular geodesic

segmentwvk−m+1· · ·vk· · ·vk+lwith lower endw∈Vk−mand upper endv∈Vk+l, where

π(vk−m+1)= · · · =π(vk+l)and every edge of which is radial. Cf. Fig.2.

The hope is now thatF(w)andF(vk+l)will be joined in Hyp(Z′)by a singular segment

whose length is in bilipschitz correspondence to|wvk+l|=m+l. This turns out to be roughly

true, cf. Lemmata5and 6.

Lemma 4 Supposevk ∈ Vkis of type I I andvk+l ∈ Vk+l, w ∈Vk−m are the upper and

the lower ends of the singular geodesic associated tovk∈Vk. There is a C1 =C1(η,K,r) such that if l+m>C1, then B(F(vk+l))= f(Bk+l(v)).

More informally; the smallest ball containing f(Bk+l(v)) contains nothing besides

f(Bk+l(v)).

Proof Suppose f(z)∈Im f(Z)=Z′is outside of f(B(vk+l)). Soz∈/ B(vk+l)and

there-foreρ(v,z)≥K rk−m+1_{. Consequently, for allZ}′_∈B(v

k+l) ρ(vk+l,Z′) ρ(vk+l,z) <r l+m−1_, ρ′(f(vk+l),f(Z′)) ρ′_(f(v k+l),f(z)) <Cr 1 α(l+m−1), diamf(B(vk+l)) ρ′_(f(v k+l),f(z)) <K Cr 1 α(l+m−1)_, r(B(F(vk+l))) ρ′_(f(v k+l),f(z)) <Cr 1 α(l+m) _{by regularity ofvk+l.}

SinceCis a uniform constant depending onη,K andr only, there is aC1 such that if l+m>C1, we will have r(B(F(vk+l))) < 1 Kρ ′_(f(v k+l), f(z)). (1) But of course ρ′(f(vk+l),f(z))≤Kmax{ρ′(f(vk+l),F(vk+l)), ρ′(f(z),F(vk+l))} ≤Kmax{r(B(F(vk+l))), ρ′(f(z),F(vk+l))}.

This and (1) imply

Corollary 2 The centerπ(F(vk+l))of F(vk+l)is in f(B(vk+l)).

Now we want to verify that the image of the upper end of a singular geodesic is the upper end of a singular geodesic with comparable length.

Lemma 5 (Upper Ends go to Upper Ends) Supposevk ∈ Vk is of type I I andvk+l ∈

Vk+l, w ∈ Vk−m are the upper and the lower ends, respectively of the singular

geode-sic associated tovk. There exists a uniform constant C2 = C2(C,C1,K, η,r) such that F(vk+l)is the upper end of a singular geodesic inHyp(Z′)whose length L′satisfies

1

α(m+l)−C2≤L

′_≤α(m+l)+C 2.

Proof Letz∈Z\Bk+l(v). Thenρ(z, π(v))≥K rk−m+1. First of all takeC2≥C1. Then by Corollary 2,∃ ˆv ∈Bk+l(v)such that f(v)ˆ =π(F(vk+l)). Now for allz1 ∈Bk+l(v),z2 ∈ Z\Bk+l(v)we have ρ(v,ˆ z1) <K2rk+l, ρ(v,ˆ z2)≥rk−m+1. Thus ρ(v,ˆ z1) ρ(v,ˆ z2) < K2rl+m−1, whence ρ′(f(v),ˆ f(z1)) ρ′(f(v),ˆ f(z2)) <C K2/αr1α(l+m−1)_,

which, sincer(B(F(vk+l)))≍r diam(f(Bk+l(v)))≍K′ sup_z

1ρ ′_{(f(v),ˆ f(z} 1)), gives r(B(F(vk+l))) ρ′_{(f(v),ˆ f(z} 2)) <Dr1α(l+m−1)=_C2rα1(l+m)_. From this it follows that(f(v),ˆ p−q)∈V_p′_−q for all 0 ≤q ≤ 1

α(l+m)−C2, and it is

obviously singular on all these levels.

On the other hand,vk+lis regular, meaning there exists az3 ∈Bk+l(v)withρ(v,ˆ z3)≥ rk+l+1. With z2 ∈ Z\Bk+l(v) such thatρ(v,ˆ z2) ≤ K2rk−m (exists since w ∈ Vk−m is

regular andvˆ∈B(vk+l)⊂B(w)), we have

ρ(v,ˆ z2) ρ(v,ˆ z3) ≤(K2/r)·r−(m+l), that is, ρ′(f(v),ˆ f(z2)) ρ′(f(v),ˆ f(z3)) ≤C(K2/r)α·r−α(m+l),

which bounds the length of the singular geodesic descending fromF(vk+l)byα(m+l)+C2.

SettingC2 :=max{C1,C2,C2}proves the lemma. ⊓⊔

So ifwvk−m+1· · ·vk· · ·vk+lis a singular geodesic in Hyp(Z), thenF(vk+l)is the upper

end of a singular geodesic in Hyp(Z′)with controlled length. Now we want to know how F(w)and the lower end of the image singular geodesic are related, cf. Fig.3.

Lemma 6 (Lower Ends go roughly to Lower Ends)Supposevk ∈ Vk is singular. Ifvk is

of type I I , letwvk−m+1· · ·vk· · ·vk+lbe the singular segment inHyp(Z)determined byvk

Fig. 3 The distance betweenw′andF(w)is uniformly bounded

singular segment inHyp(Z′)associated to F(vk+l)∈Vp′ according toLemma5. Ifvkis of

type I I I andwvk−n· · ·vk· · ·the associated singular ray inHyp(Z), denote byw′the root

of the singular ray inHyp(Z′)associated to f(π(vk)).

There is a uniform constant C3=C3(η,K,r)such that|w′F(w)|≤C3.

Proof We show it first forvof typeI I. We may assume thatl+m>C1, for if not, Lemma5 says thatw′is uniformly close to F(vk+l), and the fact that diamf(B(vk+l))is uniformly

comparable to diamf(B(w))(and the sets intersect) shows that F(w)uniformly close to F(vk+l).

Now F(w) is by definition the smallest ball containing f(Bk−m(w)). In particular

f(Bk+l(v))⊂B(F(w)), so thatB(F(w))∩B(w′)= ∅. Now the distance between vertices

whose associated balls intersect is roughly equal to their level distance (cf. [3], Lemma 6.2.7). Hence we must show thatl(w′)=. l(F(w)), which is the case iffr(B(w′))≍r(F(w)), iff

diam f(Bk−m(w))≍r(B(w′)). (2) Now, r(B(w′))≍r inf Z′_∈Z′_\f(B k+l(v)) ρ′(Z′, π(F(vk+l))).

But we know (Corollary2) that the center of the ballF(vk+l)is given by f(v)ˆ for some

ˆ

v∈Bk+l(v). Since f is bijective we can write

r(B(w′))≍r inf z∈Z\Bk+l(v)

ρ′(f(z),f(v)).ˆ (3)

For the l.h.s. of (2) we have

diam f(Bk−m(w))≍K sup z∈Bk−m(w)

ρ′(f(v),ˆ f(z)) (4) becausevˆ∈Bk+l(v)⊂B(w).

With (3) and (4), (2) becomes sup

z∈Bk−m(w)

ρ′(f(v),ˆ f(z))≍ inf

z∈Z\Bk+l(v)

ρ′(f(v),ˆ f(z)) (5)

Simplifying further, for any z ∈ B(w)\B(vk+l) we have ρ(v,_ρ(v,v)_zˆ₎ < 1, whence by

quasi-symmetry sup z∈B(w) ρ′(f(v),ˆ f(z))≤D· sup z∈B(w)\B(vk+l) ρ′(f(v),ˆ f(z)),

for a uniformD. This gives sup z∈B(w) ρ′(f(v),ˆ f(z))≍ sup z∈B(w)\B(vk+l) ρ′(f(v),ˆ f(z)). (6) Likewise we get inf z∈Z\B(vk+l) ρ′(f(v),ˆ f(z))≍ inf z∈B(w)\B(vk+l) ρ′(f(v),ˆ f(z)), (7) for pick azˆ∈B(w)\B(vk+l)such that for some uniformE

ρ(v,ˆ z)ˆ ρ(v,ˆ z) ≤E ∀z∈Z\B(vk+l). Then η(E)· inf z∈Z\B(vk+l) ρ′(f(v),ˆ f(z))≥ρ′(f(v),ˆ f(z))ˆ ≥ inf z∈B(w)\B(vk+l) ρ′(f(v),ˆ f(z)), from which (7) follows immediately.

With (6) and (7), (5) follows if we prove inf

z∈B(w)\B(vk+l)

ρ′(f(v),ˆ f(z))≍ sup

z∈B(w)\B(vk+l)

ρ′(f(v),ˆ f(z)). One direction is trivial and we just have to show

sup

z∈B(w)\B(vk+l)

ρ′(f(v),ˆ f(z))≤H· inf

z∈B(w)\B(vk+l)

ρ′(f(v),ˆ f(z)), (8) for some uniform constantH. But in fact, for anyz∈ B(w)\B(vk+l)we haverk−m+1 ≤

ρ(v,ˆ z)≤K2rk−m, thus there is a uniformHsuch that ρ(v,ˆ z1) ρ(v,ˆ z2) ≤H∀z1,z2∈B(w)\B(vk+l), and hence ρ′(f(v),ˆ f(z1)) ρ′_{(f(v),ˆ f(z} 2)) ≤η(H). This implies (8) and thereby the lemma forvkof typeI I.

The argument forvkof typeI I I is analoguous.B(w′)andB(F(w))again intersect, so

we must estimate their level difference. Denoteˆz:=π(vk).F(w)is the smallest ball

con-taining f(B(w)), while the radius ofB(w′)is determined by when a ball around f(ˆz)starts to contain points inZ′\{f(z)ˆ }.

In formulas

r(B(F(w)))≍C(K,r)diam f(B(w)) r(w′)≍D(K,r) inf

z∈Z\{ˆz}ρ

′_{(f(z), f(z)),ˆ}

whereC(K,r) and D(K,r) are appropriate expressions involving only K andr. Since diam f(B(w))≍_E(r,K′₎sup_z∈B(w)ρ′(f(z),ˆ f(z)), the claim follows once we show

sup

z∈B(w)

ρ′(f(z),ˆ f(z))≍_C

4(K,r)_z∈inf_Z\{ˆz}ρ

′_{(f(z),ˆ f(z)). (9)}

Now the same steps as in the proof of (5) yield the lemma forvkof typeI I I. ⊓⊔

So far we have only defined whereFmaps regular vertices. We are now in a position to extend the domain ofFto all of Hyp(Z).

v∈Vkis of typeI F(v)∈Hyp(Z′)is defined to be a vertex of highest levelw′such that

f(B(v))⊂B(w′)′.

v∈Vkis of typeI I v=vk∈Vklies on a singular geodesicwvk−m+1· · ·vk+lwith lower

and upper endsw∈Vk−m, v=vk+l ∈Vk+l. In casel+m < αC2, setF(v):= F(w). Ifl+m ≥αC2, thenF(vk+l)∈Vp′ is the upper

end of a singular geodesic whose lengthL′satisfies_α1(l+m)−C2≤ L′≤α(l+m)+C2(Lemma5) and ifw′∈V_p′−Ldenotes the lower

end of this singular geodesic, then |F(w)w′|≤ C3 (Lemma6). Let L =l+m. In this case define F(v ∈ Vk)to be a vertexv′on the

singular geodesic fromw′toF(vk+l)for which|w′v′|=.1 L

′ L|wv|.

v∈Vkis of typeI I I v ∈ Vk lies on a singular ray in Hyp(Z)going toπ(v)∈ Z. Since

|Z| ≥2, this singular ray has a regular lower endw ∈Vk−m. Since

f is a homeomorphism, f(v)is isolated inZ′, thus there is a singular ray in Hyp(Z′)starting at some regularw′ ∈V_p′.F(v)is defined as the (unique) vertexv′∈V_p′_+mon this ray. Equivalently, F(v)is the vertexv′on the singular ray in Hyp(Z′)same distance fromw′asv has fromw.

This definesFon the whole vertex setV_{, and up to a rough isometry,Fis then well-defined}

on all of Hyp(Z).

Theorem 6 The map F : Hyp(Z) →Hyp(Z′)described above is a quasi-isometry, and ∂∞F= f .

Proof We first show thatFis Lipschitz. Since Hyp(Z)is geodesic, this follows if we show that the distance|F(v)F(w)|is uniformly bounded for neighboringv, w∈Hyp(Z). Now if v, ware both of type I, it follows by standard arguments (such as those used in the proof of Theorem 7.2.1 in [3]) that the level difference ofF(v)andF(w)is uniformly bounded. If, w.l.o.g.vis of typeIandwof typeI I, Lemmas5and 6(or the definition ofFifwis not on a long enough singular geodesic) imply that|v′w′|uniformly bounded. Ifvof typeIand wof typeI I I, Lemma6does the job. A vertex of typeI I never neighbors a vertex of type I I I. This proves thatFis Lipschitz.

Next define a mapG:Hyp(Z′)→Hyp(Z)corresponding to f−1:Z′→Zin the same wayFwas defined (and with the same choice of vertex systemsV_,V′_{). Of courseGis then}

also Lipschitz. We showG◦F=. idHyp(Z).

vof type I By definition B(G ◦ F(v)) ⊃ B(v). In particular, the balls intersect. Their distance is uniformly bounded iff the diameters of these sets are uniformly comparable. But this follows from the facts that f(B(v)) ⊂

B(F(v)),diam f(B(v))is uniformly comparable to diamB(F(v)), and that f−1is quasisymmetric. The doubtful reader is referred to [9], Theo-rem 2.5, which describes exactly this situation.

vof type I I We have a singular geodesicwvk−m+1· · ·v=vk· · ·vk+lwith lower end

w ∈ Vk−m and upper endvk+l. By Lemma5applied twice to F and

then G, there is a uniform constant C5 such that ifl+m > C5, not onlyF(vk+l)is an upper end of a singular geodesic in Hyp(Z)but even

G(F(vk+l))is still the upper end of singular geodesic in Hyp(Z).F(vk+l),

as usual, is a smallest ball containing f(Bk+l(v)). But by Lemma 4

B(F(vk+l))= f(Bk+l(v)). In particular,G(F(vk+l)), being the

G(F(vk+l)) = vk+l. By definition of F andG it is now obvious that

G(F(v))is uniformly close tov.

If the singular geodesicwvk−m+1· · ·v=vk· · ·vk+l is shorter thanC5, thenvis in particular uniformly close to a type I vertex, namelyw(or vk+l). The Lipschitz property ofFandGand the fact thatG(F(w))is

uniformly close towimply thatG(F(v))is uniformly close tov. vof type I I I π(v) = z, an isolated point in Z. f(z)is an isolated point in Z′ and

by definition of F, the ray in Hyp(Z)associated toz, on whichvlies, is mapped one-to-one onto the ray in Hyp(Z′)associated to f(z). But thenGmaps this ray back in one-to-one fashion to the ray associated to

f−1(f(z))=z. So in this case we have in factv=G(F(v)).

This provesG◦F=. idHyp(Z). Since the domain ofGis all of Hyp(Z′),

it follows thatF(Hyp(Z))is cobounded in Hyp(Z′), thusFis a quasi-isometry.

It remains to show that∂∞F= f. By [3], Theorem 5.2.17, we know that F doesinduce a homeomorphism∂∞F : Z → Z′. So take a sequence

{vi}of vertices converging toz ∈ Z. We haveπ(vi) → zin(Z, ρ).

Since the limit of the sequence{F(vi)}does not depend on the

repre-sentative{vi} ∈ z, we may take the latter such that B(vi+1) ⊂ B(vi)

(cf. [3], Lemma 6.3.2) Then{F(vi)}converges to some Z′∈Z′. In

par-ticular,l′(F(vi))i

→∞

−→ ∞. Sinceρ′(f(π(vi)), π(F(vi)))≤ K rl ′_(F(v

i)) and f(π(vi))→ f(z), we getπ(F(vi))→ f(z)inZ′and this implies

∂∞F(z)= f(z). ⊓⊔

6 Extension for inversions

There is a good reason why one would not be satisfied with describing the quasisymmet-ric structure of the boundary, but would rather have a result on its quasimoebius struc-ture. Namely, there is in general no uniform constantLsuch that id :∂∞a,oX →∂a,o

′

∞ Xis L-bilipschitz for anyo,o′∈X. However, thereisa uniformL(depending ona, δ) such that it isL-bilipschitz-quasimoebius. In other words, the ratio of a triple of boundary points is not a uniform quantity, whereas the cross-ratio of a quadruple is. For more on this we refer to [7], Theorem 8.1. This motivates us to look for an extension theorem for quasimoebius maps in the spirit of the Poincaré extension theorems for classical hyperbolic space.

In this section we prove that the hyperbolic approximation of a bounded quasimetric space (Z, ρ)is roughly isometric to the hyperbolic approximation (with the same parameters) of the extended quasimetric space(Z, ρ′)whereρ′is the inversion at a point in Zofρ. This result will be combined with theorem5to give the desired Moebius extension.

Theorem 7 Let(Z, ρ)be a bounded complete quasi-metric space andρ′the quasi-metric obtained fromρby inversion in a pointω∈Z ,

ρ′(a,b):= ρ(a,b)

ρ(a, ω)ρ(b, ω).

Then the (truncated) hyperbolic approximation of(Z, ρ)is roughly isometric to the hyper-bolic approximation of(Z, ρ′). More precisely, for every r ∈ (0,1)there exists a rough isometry F:Hyp_r(Z, ρ)→Hyp_r(Z, ρ′)that induces the identity in∂∞Hyp(Z)=Z .

Remark 4 The proof of this theorem basically consists of a series of uniform comparabil-ity statements,· ≍ ·, all of which remain true if the boundary quasimetrics are replaced by ones that are bilipschitz equivalent to them. In particular, the theorem allows us to con-clude, via the bilipschitz extension Theorem4, that Hyp_1/a(∂a,b1(ω)

∞ X)is roughly isometric to Hyp_1/a(∂a,b2(ω)

∞ X), whereb1(ω),b2(ω)are two arbitrary Busemann functions atω. This fact will be needed in the proof of(I I I)⇒(I)in Theorem10.

Note that if(Z, ρ) is K-quasimetric, then (Z, ρ′)is K2-quasimetric. Throughout this section we assume that both approximations Hyp(Z, ρ),Hyp(Z, ρ′)are done with respect to the sameK. Since the rough isometry class of the approximations does not depend on theK used, this poses no danger. Moreover, we may assumer<1/K3, since for all other values ofr,Hyp_r is obtained by scaling the graphs Hyp_r′(Z, ρ),Hyp_r′(Z, ρ′), wherer′ <1/K3,

by the same factor.

In addition, it turns out to be advantageous to work with a special choice of vertex system

V_{for Hyp(Z, ρ). Namely we require that}V_{be hereditary and the rootobe centered at the}

inversion pointω, π(o)=ω. In particular, we then have a canonical “ray toω” in Hyp(Z, ρ), namely the radial geodesic ray consisting of all vertices centered atω. We will often refer to this ray asthe ray oω.

The idea of the definition forFis to do the same as for quasi-symmetric maps whenever ωis not involved, and “invert the orientation” on the rayoω. This corresponds to the fact that the inversion restricted toZ\O, whereOis any neighborhood ofω, is a PQ-symmetry onto its image because it is a Moebius map between bounded spaces (cf. Lemma12).

We define the mapF.

Definition 13 I Ifvis regular withπ(v)=ωandv=o, setF(v):=any vertexwof highest level in Hyp(Z, ρ′)such thatBρ′

K rl(w)(w)containsB

ρ

K rl(v)+1(π(v))c. I I Ifvis a horizontal neighbor to a vertexv˜as in I, setF(v):=F(v).˜

I I I Ifv=ois regular and neither as in Inor II, setF(v):=any vertexwof highest level in Hyp(Z, ρ′)such thatBρ′(w)⊃Bρ_{K r}l(v)(π(v)).

I V For the rooto, if the immediate radial successorvtooon the rayoωis regular, set F(o) := F(v). If this v is not regular, then Z\B(v) is separated from the rest of Z (in the sense that the two sets have positive distance) and the same is the case in(Z, ρ′). Furthermore, there is a branch point (cf. [3], p. 72) in Hyp(Z, ρ′) for

{B:=Bρ

K rl(o)+1(ω),Z\B}. In this case setF(o)=such a branch point.

V Ifvis singular and lies on a singular segmentw1w2in Hyp(Z, ρ), map it to an appro-priate vertex on the singular segment associated tow1w2in Hyp(Z, ρ′), cf. Lemma9. V I Ifvis singular and lies on a singular raywzin Hyp(Z, ρ), mapvto an appropriate

vertex on the singular ray in Hyp(Z, ρ′)associated to the raywz, cf. Lemma10. The verification thatFis a rough isometry is straightforward but a bit tedious. We first show|F(v)F(w)|= |. vw|forv, wfrom a cobounded subset of the set of regular vertices, Lemma8. Then we can extend it to allv, wregular. Afterwards we show well-behavedness of singular segments and rays, Lemmata9and 10, respectively.

Lemma 7 Letv, wbe any regular vertices inHyp(Z, ρ). Then

|vw|=. log_r diam(B(v))diam(B(w)) supρ(zv,zw)2 .

Proof There is a geodesic connectingvtowthat has either exactly one or exactly two points of lowest level (cf. [3], Lemma 6.2.6). In either case, there is a branch pointufor{v, w}with

distance at most one from any lowest level vertex. Then

|vw|=.1(l(v)−l(u))+(l(w)−l(u)).

Butl(v)=. log_r(diamB(v))by regularity ofv(the error constant depending onK,r), and the same forw.

Now B(v)∪ B(w) ⊂ B(u) by definition. On the other hand, any vertext such that B(v)∪B(w)⊂B(t)is uniformly close (error 1) to a cone point by [3] Lemma 6.2.1. Take t to be any vertex of highest level satisfying B(v)∪ B(w) ⊂ B(t), thent is uniformly close to a (and hence, any) branch point. But then diam(B(t))≍supρ(zv,zw). The lemma

follows. ⊓⊔

Lemma 8 Letv, wbe regular vertices inHyp(Z, ρ)which if centered atωor horizontally connected to oωare at least two levels above the root. Then|F(v)F(w)|= |. vw|.

Proof For the proof we show that

diamρ′(Bρ′(F(v)))diam_ρ′(Bρ′(F(w))) supρ′(zF(v),zF(w))2 ≍ diamρ(B ρ_(v))diam ρ(Bρ(w)) supρ(zv,zw)2 , (10)

which implies the claim by Lemma7. Here the notationzv,zwis supposed to suggest that

the sup is taken over allzv∈B(v),zw∈B(w)and likewise forF(v),F(w).

Ifπ(v)=π(w)=ω, we have diamρ′(B(F(v)))≍ 1

diamρ(B(v)). (10) simplifies to

supρ′(zF(v),zF(w))≍

supρ(zv,zw)

diam(B(v))diam(B(w)),

both sides of which compare uniformly to 1/diam(B(v))(if w.l.o.g.l(v)≥l(w)). ByI I of Definition13, the same argument gives (10) for vertices horizontally connected to the ray oω. So supposeπ(v)=ωandwnot horizontally connected to the ray. Then

ρ′(z, w)= ρ(z, w) ρ(z, ω)ρ(w, ω) ≍ ρ(z, w) ρ(w, ω)2 ∀z∈B(w), whence diamρ′(B(w))≍ diamρ(B(w)) ρ(w, ω)2 . (10) then becomes diamρ′_(B(F(v))) supρ′_(z vc +1,zw) 2_{ρ(w, ω)}2 ≍ diamρ(B(v)) supρ(zv,zw)2 , (11)

where zv+c1 suggests elements in B(v+1)

c _{:= B}ρ K rl(v)+1(v)c. Since diamρ′(B(F(v))) ≍ 1/diamρ(B(v)), (11) is equivalent to supρ′(zvc +1,zw)ρ(w, ω)≍ supρ(zv,zw) supρ(zv, ω) . Thus we must show

sup ρ(zv c +1,zw) ρ(zv+c1, ω)ρ(zw, ω) ·ρ(w, ω)≍ supρ(zv,zw) supρ(zv, ω)

which, sinceρ(zw, ω)≍ρ(w, ω), finally becomes supρ(zv c +1,zw) ρ(zvc +1, ω) ≍ supρ(zv,zw) supρ(zv, ω) . (12)

We prove (12), which implies the lemma in casevon the ray,wnot horizontally connected to the ray. We show first that the l.h.s. of (12) is≥ _K1. Sincevis regular∃z1∈B(v+1)cwith ρ(ω,z1) <K rk, and sincevis at least 2 from the root, there also existsz2 ∈B(v+1)cwith ρ(ω,z2)≥rk−1. Now suppose for allz1withρ(ω,z1) <K rk, wherek=ℓ(v).

ρ(z1,zw) <

1

Kρ(z1, ω)∀zw∈B(w).

Thenρ(z1, ω)≍K ρ(zw, ω). But nowz2is much farther fromωthanz1, hence ρ(z2, ω)≍K ρ(z2,zw) ∀zw∈B(w).

This shows that the l.h.s. of (12) is≥1/K in any case. Next suppose supρ(zv c +1,zw) ρ(zvc+1, ω) >K4. (13)

Then since necessarilyρ(zvc

+1,zw)≍K ρ(zw, ω)forzv+c1,zwsuch that the sup is (almost) attained,

ρ(zw, ω) >K3ρ(zvc₊₁, ω). (14)

That is, whenzw,zvc+1are taken so that the sup is (almost) attained,zwwill be much farther away fromωthanzvc+1. We want to know that thenzv+c1may as well be taken inB(v), thus we must show that ifz2∈B(v)is arbitrary, then the quantity

ρ(z2,zw)

ρ(z2, ω) ,

where thezw is the same as above, is not smaller (or at least not by much) than whenz2 is replaced byzvc+1. So pickz2 ∈ B(v)arbitrary. We may supposeρ(z2, ω) < ρ(zvc+1, ω), otherwisezv₊c₁would already be inB(v)and we are done. So then

ρ(zvc +1,z2)≤Kρ(zvc+1, ω) (14) < 1 K2ρ(zw, ω)≤ 1 Kρ(zv+c1,zw), whence ρ(zvc +1,zw)≍K ρ(z2,zw). Sinceρ(zvc₊₁, ω) > ρ(z2, ω)it thus follows that

ρ(z2,zw) ρ(z2, ω) > 1 K ρ(z_vc +1,zw) ρ(z_vc +1, ω) . It follows that the claimed uniform comparability of (12) holds.

It remains to prove (12) when 1 K ≤sup

ρ(zv+c1,zw) ρ(zv+c1, ω)

In fact we show more, namely supρ(zv c +1,zw) ρ(z_vc +1, ω) ≍_K41 ⇒ supρ(zv,zw) supρ(zv, ω) ≍1. (15)

The assumption on the l.h.s. means in particular that for any choice ofzvc

+1, everyzw lies rather close tozvc

+1. Quantitatively speaking we have

ρ(zw, ω)≤K5min{ρ(zvc₊₁,zw), ρ(zvc₊₁, ω)} ∀zw,zv₊c₁. (16)

Now sincevis regular, there is azvc₊₁withρ(zv₊c₁, ω)≤K rl(v). Then by (16),ρ(zw, ω)≤

K6rl(v).

On the other handB(w)must not containω, soρ(zw, ω)≥K rl(w). It thus follows that

l(w)˙≥l(v)up to a uniform error, or in words thatB(w)is smaller thanB(v)up to a uniform factor.

But then

supρ(zv,zw)≍supρ(zv, ω).

In addition,

supρ(zv, ω)≍supρ(zv, w),

sinceB(w)is contained within the ball of radiusK6_rl(v)_{aroundω. This proves (15).}

It remains to prove the lemma forv, wboth not horizontally connected to nor on the ray. We start again with (10),

diamρ′(B(F(v)))diam_ρ′(B(F(w))) supρ′_(z F(v),zF(w))2 ≍ diamρ(B(v))diamρ(B(w)) supρ(zv,zw)2 . Sincevis not connected to the ray, we get, just as in the case above

ρ′(z, v)= ρ(z, v) ρ(z, ω)ρ(v, ω)≍K ρ(z, v) ρ(v, ω)2 ∀z∈B(v) and thus diamρ′(B(v))≍ diamρ(B(v)) ρ(v, ω)2 . The same estimate also holds for diamρ′(B(w)). (10) becomes

diamρ(B(v))diamρ(B(w)) ρ(v, ω)2_{ρ(w, ω)}2_supρ′_(z v,zw)2 ≍ diamρ(B(v))diamρ(B(w)) supρ(zv,zw)2 , which is equivalent to supρ′(zv,zw)ρ(v, ω)ρ(w, ω)≍supρ(zv,zw).

This follows if we can show that

ρ′(zv,zw)ρ(v, ω)ρ(w, ω)≍Cρ(zv,zw) ∀zv,zw (17)

for some uniform constantC. But (17) is equivalent to ρ(zv,zw)

ρ(zv, ω)ρ(zw, ω)

It thus suffices to show

ρ(zv, ω)≍ρ(v, ω)

ρ(zw, ω)≍ρ(w, ω),

and these estimates hold becauseρ(zv, ω) >K rl(v), so in {ρ(zv, ω), ρ(v, ω), ρ(v,zv)}

the minimum will always, that is, for allzv∈B(v), beρ(v,zv), thusρ(zv, ω)≍K ρ(v, ω).

The same holds forw. The lemma follows. ⊓⊔

Corollary 3 Letv, warbitrary regular vertices. Then|F(v)F(w)|= |. vw|.

Proof This follows from Lemma8. ⊓⊔

Lemma 9 Let v, w be the top and lower ends, respectively of a singular segment in Hyp(Z, ρ). Then F(v),F(w) are uniformly close to the ends of a singular segment in Hyp(Z, ρ′)of roughly the same length.

Proof First assume thatvis not on the rayoωand not horizontally connected to it. Consider z0,z1∈B(v)andz2∈B(v)c. Then ρ′(z0,z1) ρ′(z0,z2) =ρ(z0,z1)ρ(z2, ω) ρ(z1, ω)ρ(z0,z2) .

This cannot be (much) larger thanρ(z0,z1)/ρ(z0,z2), which implies thatF(v)is the top end of a singular segment of length≥|˙ vw|.

If we can prove thatℓ(F(w)) < ℓ(F(v)), then a geodesic joiningF(w)toF(v)will reach F(v)from below, thus has to go through the singular segment. Since|F(v)F(w)|= |. vw|by Lemma8, the lemma follows.

Now ifwis neither on the rayoωnor horizontally connected to it, then ρ′(v,zv) ρ′_(w,z w) = ρ(v,zv) ρ(w,zw) ·ρ(w, ω)ρ(zw, ω) ρ(zv, ω)ρ(v, ω) ≍_K2 ρ(v,zv) ρ(w,zw) ,

whencel(w)≤· l(v). Similar estimates hold in casewis connected to or on the rayoω, that is, theρ′-diameter ofB(w+1)cis much larger than that ofB(v), where , again, the notation B(w+1)means the ball associated to(π(w),l(w)+1), i.e.BK rl(w)+1(π(w)). This proves the lemma in casevis not horizontally connected to, nor on the ray.

Finally, ifπ(v)=ω, then alsoπ(w)=ωorF(w)= F(w)˜ withπ(w)˜ =ω(w˜ being a horizontal neighbor towon the ray). It follows immediately by definition ofρ′that there is a singular segment of roughly the same length betweenF(v)andF(w)(as long asw=o,

but in this case simply apply the definition ofF). ⊓⊔

Now we show that a root of a singular ray in Hyp(Z, ρ)is mapped uniformly close to the root of a singular ray in Hyp(Z, ρ′).

Lemma 10 There is a one-to-one correspondence between singular rays inHyp(Z, ρ)and Hyp(Z, ρ′)and a root of a singular ray inHyp(Z, ρ)is mapped uniformly close to a (hence, any) root of the associated singular ray inHyp(Z, ρ′), with the exception of a singular ray in Hyp(Z, ρ)going toω, which is mapped to a singular ray “downwards” to∞inHyp(Z, ρ′).

Proof That there is a one-to-one correspondence is clear because every singular ray corre-sponds to an isolated point in the boundary, and id|Z\{ω}is a homeomorphism onto its image, so maps isolated points to isolated points, and if there is a singular ray toωthen(Z\{ω}, ρ′) is bounded, so there will be an associated singular ray descending to∞in Hyp(Z, ρ′). We just need to argue that the root of a ray associated tozin Hyp(Z, ρ)is mapped close to the root of the ray associated tozin Hyp(Z, ρ′). Assume first that ifvis a root of the ray associated toz, then eithervis not connected to nor on the rayoω, or if it is on the ray, then it is at least two levels aboveo.

Now noteB(F(v))containszby definition. It therefore suffices to show that the level of F(v)is roughly the same as that of the rootqof the ray associated tozin Hyp(Z, ρ′). Now ifvis not connected to nor on the rayoω, then diamρ′(B(v))≍diam_ρ(B(v))/ρ(v, ω)2and

similarly infZ′_=zρ′(z,z′)≍infρ(z,z′)/ρ(v, ω)2, hence the levels ofqandF(v)agree up

to uniform error. If on the other handvis centered atω inf z′ ρ ′ (z,z′)=inf ρ(z,z ′₎ ρ(z, ω)ρ(z′, ω) ≥ 1 min{ρ(z, ω), ρ(z′, ω)},

and sincevis at least two levels above the root, there existsz′such thatρ(z′, ω) >Kρ(z, ω), i.e.ρ(z, ω)≍K ρ(z,z′). It follows that infz′ρ′(z,z′)≍1/ρ(z, ω). The same argument yields

that diamρ′_(B(v+1))c≍1/ρ(z, ω).

For the exceptional cases where the rootvis equal too, to(π(o), ℓ(o)+1), or horizontally connected to the latter, one shows with similar arguments that ifR1,R2are two singular rays with thesameexceptional rootv, then the rootsq1,q2 of the associated singular rays in Hyp(Z, ρ′)are uniformly close to each other. Since there are only 3 types of exceptional roots, it follows that the distance between the imageF(v)of the root and the rootqof the

ρ′-ray associated tozis uniformly bounded,|F(v)q|=. 0. ⊓⊔

It follows readily that a roughly isometric map between geodesic spaces which induces a surjective boundary map is a rough isometry. The only thing left to show in the proof of Theorem7, then, is that∂∞F =i dZ. That a sequence converging toωis mapped to∞ ∈

(Z, ρ′)is clear by definition ofF. If{vi}is a sequence converging to infinity, say{vi} ∈z,z=

ω, we may suppose by [3] Lemma 6.3.2 that thevi form a radial geodesic in Hyp(Z, ρ).

SinceF is a rough isometry,{F(vi)}converges to a pointz′ ∈ (Z, ρ′). But F(vi)is the

smallestρ′-ball containingBρ(vi), which containsz. Sinceρ(π(vi), π(F(vi))) i→∞

→ 0 (the levels ofF(vi)go to infinity), we haveρ′(π(F(vi)),z)

i→∞

→ 0, i.e.∂∞F(z) =z∀z∈ Z. This completes the proof of Theorem7.

7 Extension for P-QM maps

In this section we prove

Theorem 8 Let f : (Z, ρ) → (Z′, ρ′)a power quasimoebius homeomorphism between complete quasimetric spaces. Then there exists a power quasi-isometry F : Hyp(Z) →

Hyp(Z′)with∂∞F= f .

The idea of the proof is to factor f as a composition of inversions and a P-QS map. We follow 3.15 of [10], where this factorization is explained in the metric setting.

Lemma 11 (Cf. [10], Theorem 2.1)Let(X, ρ), (Y, ρ′)be bounded quasimetric spaces and f : X →Y beθ-quasimoebius. Let z1,z2,z3 ∈ X andλ > 0be such thatρ(zi,zj) ≥