Hyperbolic Distance versus Quasihyperbolic Distance in Plane Domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct an example where the spaces are not quasiisometrically equivalent.


Introduction
Throughout this section Ω denotes a hyperbolic plane domain: Ω ⊂ C is open and connected and C\Ω contains at least two points. Each such Ω carries a unique maximal constant curvature -1 conformal metric λ ds = λ Ω ds usually referred to as the Poincaré hyperbolic metric on Ω. The length distance h = h Ω induced by λ ds is called hyperbolic distance in Ω. There is also a quasihyperbolic metric δ −1 ds = δ −1 Ω ds on Ω, whose length distance k = k Ω is called quasihyperbolic distance in Ω; here δ(z) = δ Ω (z) := dist(z, ∂Ω) is the Euclidean distance from z to the boundary of Ω. See §2.C for more details.
This works continues that begun in [BH21], [Her21a], [Her21b] where we elucidate the geometric similarities and metric differences between the metric spaces (Ω, h) and (Ω, k). Our first result, Theorem A below, characterizes when the metric spaces (Ω, h) and (Ω, k) are quasisymmetrically equivalent. Then Theorem B reveals that, when these spaces are Gromov hyperbolic, their Gromov boundaries are always quasisymmetrically equivalent.
To set the stage, we begin with some preliminary observations. A straightforward, albeit non-trivial, argument reveals that the metric spaces (Ω, h) and (Ω, k) are isometric if and only if Ω is an open half-plane and the isometry is the restriction of a Möbius transformation. Furthermore, these metric spaces are bi-Lipschitz equivalent if and only if the identity map is bi-Lipschitz; see §2.C.3.
It is well-known that the identity map (Ω, k) id − → (Ω, h) enjoys the following properties: • The map id is a 2-Lipschitz 1-quasiconformal homeomorphism.
• For any simply connected Ω, id is 2-bi-Lipschitz. 1 • In general, id is bi-Lipschitz if and only ifĈ \ Ω is uniformly perfect. 2 The above is just an easy to state special case of our more general Theorem 3.9 which provides a large class of plane domains whose hyperbolizations and quasihyperbolizations are quasiisometrically equivalent. This raises the natural question of whether the conclusion of Theorem C could be true in general, and we answer this below. Note that a quasiisometry can have an arbitrarily large additive rough constant and this obstacle must be overcome.
Theorem D. There is a uniform (hence Gromov) hyperbolic plane domain Ω with the property that any quasisymmetric equivalence between ∂ G (Ω, k) and ∂ G (Ω, h), e.g., that given by Theorem B, is not via a power quasisymmetry. In particular, (Ω, k) and (Ω, h) are not quasiisometrically equivalent.

2.A.
Maps, Paths, and Geodesics. An embedding X f − → Y between two metric spaces is a quasisymmetry if there is a homeomorphism η : [0, ∞) → [0, ∞) (called a distortion function) such that for all triples x, y, z ∈ X, when this holds, we say that f is η-QS. These mappings were studied by Tukia and Väisälä in [TV80]; see also [Hei01].
The bi-Lipschitz maps form an important subclass of the quasisymmetric maps; X f − → Y is bi-Lipschitz if and only if there is a constant L such that for all x, y ∈ X, and when this holds we say that f is L-bi-Lipschitz; such an f is η-QS with η(t) := Lt.
More generally, a map X f − → Y is an (L, C)-quasiisometry if L ≥ 1, C ≥ 0 and for all x, y ∈ X, These are often called rough bi-Lipschitz maps, and there seems to be no universal agreement regarding this terminology; some authors use the adjective quasiisometry to mean what we have called bi-Lipschitz, and then a rough quasiisometry satisfies our definition of quasiisometry. A (1, 0)-quasiisometry is simply an isometry (onto its range), and a (1, C)-quasiisometry is called a C-rough isometry. Two metric spaces X, Y are isometrically equivalent (or BL, QS, QC equivalent, respectively) if and only if there is a bijection X → Y that is an isometry (or BL, QS, or QC).
Also, X, Y are quasiisometrically equivalent if and only if there is a quasiisometry X f − → Y with the property that f (X) is cobounded in Y (i.e., the Hausdorff distance between f (X) and Y is finite). More precisely: X, Y are (L, C)-QI equivalent if there is an (L, C)-quasiisometry f : X → Y and for each y ∈ Y there is an x ∈ X with |y − f (x)| ≤ C. An alternative way to describe this is to say that there are quasiisometries in both directions that are rough inverses of each other.
2.1. Example. Suppose X ⊂ Y and for each y ∈ Y there is an x y ∈ X with |x y − y| ≤ C. The maps are both rough isometric equivalences: the "identity" inclusion is a (1, C)-QI equivalence and f is a (1, 2C)-QI equivalence.
Our metric spaces will always be the domain Ω, either in C or inĈ, with either Euclidean distance, chordal distance, spherical distance, an associated length distance, an associated quasihyperbolic distance, or an associated hyperbolic distance.
A path in X is a continuous map R ⊃ I γ − → X where I = I γ is an interval (called the parameter interval for γ) that may be closed or open or neither and finite or infinite. The trajectory of such a path γ is |γ| := γ(I) which we call a curve. When I is closed and I = R, ∂γ := γ(∂I) denotes the set of endpoints of γ which consists of one or two points depending on whether or not I is compact. For example, if I γ = [0, 1] ⊂ R, then ∂γ = {γ(0), γ(1)}.
We call γ a compact path if its parameter interval I is compact (which we often assume to be [0, 1]). We call γ a rectifiable path if its length ℓ(γ) is finite, and then we may assume that γ is parameterized with respect to arclength in which case the parameter interval for γ is [0, ℓ(γ)]. We note that arclength parametrizations are a priori 1-Lipschitz continuous.
When ∂γ = {a, b}, we write γ : a b (in Ω) to indicate that γ is a path (in Ω) with initial point a and terminal point b; this notation implies an orientation: a precedes b on γ.
An arc α is an injective compact path. Every arc is taken to be ordered from its initial point to its terminal point. Given points a, b ∈ |α|, there are unique u, v ∈ I with α(u) = a, α(v) = b and we write α[a, b] := α| [u,v] . Every compact path contains an arc with the same endpoints; see [Väi94].
A characteristic property of geodesics is that the length of each subpath equals the distance between its endpoints. There is a corresponding description for quasi-geodesics: I γ − → X is an L-chordarc path if it is rectifiable and for all s, t ∈ I , ℓ(γ| [s,t] ) ≤ L |γ(s) − γ(t)|.
If we ignore parameterizations, then the class of all quasi-geodesics (in some metric space) is exactly the same as the class of all chordarc paths. More precisely, a K-quasi-geodesic is a K 2 -chordarc path, and if we parameterize an L-chordarc path with respect to arclength, then we get an L-quasi-geodesic.
In this paper we study the metric spaces (Ω, h) or (Ω, k) where Ω is a hyperbolic plane domain and h and k are the hyperbolic and quasihyperbolic distances in Ω. The geodesics and quasi-geodesics in (Ω, h) are called hyperbolic geodesics and hyperbolic quasi-geodesics, and similarly in (Ω, k) we attach the adjective quasihyperbolic.
2.B. Annuli and Uniformly Perfect Sets. Given a point c ∈ C and 0 < r < R < +∞, A := { z ∈ C | r < |z − c| < R } is an Euclidean annulus with center c(A) := c and conformal modulus mod(A) := log(R/r); if r = 0 or R = ∞, A is a degenerate annulus and mod(A) := +∞. We call S 1 (A) := S 1 (c; √ rR) the conformal center circle of A; A is symmetric about this circle. The inner and outer boundary circles of A are, respectively, ∂ in A := S 1 (c; r) and ∂ out A := S 1 (c; R) . 5 One can also consider rough-quasi-geodesics where γ is a quasiisometry; we do not use these here.
It is convenient to introduce the notation Two annuli are concentric if they have a common center, and A ′ is a concentric subannulus of A, denoted by A ′ ⊂ c A, provided c(A ′ ) = c(A) and A ′ is a subannulus of A.
An annulus A separates E if A ⊂Ĉ \ E and both components ofĈ \ A contains points of E; thus when A separates { a, b }, one of a or b lies inside A and the other lies outside A, and if A does not meet nor separate { a, b }, then a and b are on the same side of A. Evidently, if A ′ is a subannulus of A, then A ′ separates the boundary circles of A.
We define Following Pommerenke [Pom79], we say that E ⊂Ĉ is M-uniformly perfect if and only if E is closed, E contains the point at infinity, and Pommerenke [Pom84] established a number of equivalent conditions; see also [Sug01], [HLM89]. Heinonen [Hei01] has a general metric space definition for uniform perfectness (which appears as item (2.3.e) below) which is similar to, but different from, Pommerenke's definition.
Nonetheless, the following result shows that the Pommerenke and Heinonen definitions are equivalent; this must be folklore, but we do not know a reference. Recall that a ring domain is a topological annulus, and a ring domain D ⊂Ĉ separates a set E ⊂Ĉ if and only if D ⊂Ĉ \ E and both components ofĈ \ D contain points of E.
2.3. Proposition. For any closed set E ⊂Ĉ with ∞ ∈ E, the following are quantitatively equivalent: Proof. It is easy to check that (a) and (c) are equivalent. Beardon and Pommerenke [BP78] established the equivalence of (c) and (b), and Pommerenke [Pom79], [Pom84] demonstrated that these are equivalent to (d) (along with several other variations).
That (e) is equivalent to the other conditions is surely well-known, yet does not seem to be explicitly mentioned in the literature, so we outline an explanation for this.
Suppose o ∈ E, R > r > 0, and D : Finally, suppose (e) holds with a constant M ≥ 2. To establish (a), we verify that (2.2) holds with the constant 64M 4 . Let A = { r < |z − c| < R } ⊂ C \ E be an annulus with c ∈ E ∩ C; thus A separates E. Assume R ≥ 2r. Roughly, we exhibit a chordal subannuluŝ A ⊂ a A and useR/r ≤ M to bound R/r.

2.C. Conformal Metrics. A continuous function
and where the infimum is taken over all rectifiable paths γ : a b in X. We describe this by calling ρ ds = ρ(x)|dx| a conformal metric on X. Below we consider the hyperbolic and quasihyperbolic metrics defined on plane domains.
We call γ a ρ-geodesic if d ρ (a, b) = ℓ ρ (γ); these need not be unique. We often write [a, b] ρ to indicate a ρ-geodesic with endpoints a, b, but one must be careful with this notation since these geodesics need not be unique. When z is a fixed point on a given geodesic [a, b] ρ , we write [a, z] ρ to mean the subarc of the given geodesic from a to z.
We note that the ratio ρ ds/σ ds of two conformal metrics is a well-defined positive function. We write ρ ≤ C σ to indicate that this metric ratio is bounded above by C.
Wen ρ ds is a conformal metric on Ω, we let Ω ρ := (Ω, d ρ ). The following is surely folklore, but we briefly outline a proof which employs standard techniques.
As above, pick r 0 ∈ (0, δ(a)) and s 0 ∈ (0, δ(a ′ )) so that (2.5) holds for d ρ and its analog holds for d τ and w ∈ D[a ′ ; s 0 ]. Then take r 1 ∈ (0, r 0 ] so that f D[a; r 1 ] ⊂ D(a ′ ; s 0 ). Then for all r ∈ (0, r 1 ) and all z, w ∈ S 1 (a; r), here the inner inequality holds by quasisymmetry and the two outer inequalities follow from repeated applications of (2.5). Selecting such z, w that attain L f (r), l f (r) respectively yields and so letting r → 0 + , then ε → 0 + , we deduce that, with respect to Euclidean distance, f is indeed η(1)-QC.
The careful reader recognizes that in the above, we employ the metric (aka, linear) dilatation for quasiconformal maps, not the geometric dilatation, whereas in Fact 2.12 below K is the geometric dilatation.
2.C.1. The QuasiHyperbolic Metric. The quasihyperbolic metric δ −1 ds is defined for any proper subdomain Ω C; here δ = δ Ω is the Euclidean distance to the boundary of Ω. This metric can be defined in very general metric spaces and has proven useful in many areas of geometric analysis. See [BHK01] and [HRS20].
We remind the reader of the following basic estimates for quasihyperbolic distance, first established by Gehring and Palka [GP76, 2.1]: For all a, b ∈ Ω, where l(a, b) is the (intrinsic) length distance between a and b. The first inequality above is a special case of the more general (and easily proven) inequality which holds for any rectifiable path γ in Ω. See also [BHK01,(2.3),(2.4)]. There are analogous inequalities for k χ and k σ where we replace all the Euclidean metric quantities by the appropriate chordal or spherical metric quantities.
It is well known that the holomorphic covering C exp − − → C ⋆ pulls back the quasihyperbolic metric δ −1 ⋆ ds on C ⋆ to the Euclidean metric on C, which in turn reveals that (C ⋆ , k ⋆ ) is (isometric to) the Euclidean cylinder S 1 ×R 1 with its Euclidean length distance inherited from the standard embedding into R 3 ; here k ⋆ := k C⋆ . See [MO86]. In particular, quasihyperbolic geodesics in C ⋆ are logarithmic spirals and for all a, b ∈ C ⋆ , Note the special cases of the above that arise when |a| = |b| or Arg(b/a) = 0.
Often, (2.7) provides good estimates for quasihyperbolic distances as described next.
As the chordal and spherical distances are bi-Lipschitz equivalent, and σ is the length distance associated with χ (onĈ), it follows that the chordal and spherical quasihyperbolic metrics (and their associated distances) are bi-Lipschitz equivalent with k σ ≤ k χ ≤ π 2 k σ . It is useful to know that Euclidean and spherical quasihyperbolic distances are bi-Lipschitz equivalent. Note, however, that the distortion constant depends on the location of the origin. Essentially, this is because a general Möbius transformation is not a chordal nor spherical isometry, just bi-Lipschitz. One can establish this by using appropriate estimates between δ and χ as in [BHX08, Lemma 3.10] or alternatively appeal to [BHX08, Theorem 4.12]; the interested reader should also peruse [BB03].

Fact. Let Ω
C be a domain. Then (Ω, k), (Ω, k χ ), and (Ω, k σ ) are all bi-Lipschitz equivalent. In particular, 1 An important property of hyperbolic distance is its conformal invariance. While this does not hold for quasihyperbolic distance, it is Möbius quasi-invariant in the following sense; see [GP76, Lemma 2.4, Corollary 2.5]. Modifications to their argument also gives quasiinvariance of the relative distances j, j ′ .
Consequently, for all rectifiable paths γ in Ω, 6 One should view the constant D as depending on diam χ Ω. By examining the distances between 0 and 1 in D(0; R) (or in C \ {R}) (as R → +∞) we see that this bi-Lipschitz constant really does depend on D.
In particular, with a ′ := T (a) and b ′ : Theorem 3], Gehring and Osgood proved the following, which says that quasiconformal homeomorphisms are rough quasihyperbolic quasiisometries and conformal maps are even quasihyperbolically bi-Lipschitz.

2.C.2. The Hyperbolic
Metric. Every hyperbolic domain inĈ carries a unique metric, λ ds = λ Ω ds, which enjoys the property that its pullback p * [λ ds], with respect to any holomorphic universal covering projection p : D → Ω, is the hyperbolic metric λ D (ζ)|dζ| = 2(1−|ζ| 2 ) −1 |dζ| on D. Another description is that λ ds is the unique maximal (or unique complete) metric on Ω that has constant Gaussian curvature −1. In terms of such a covering p, the metric-density λ = λ Ω of the Poincaré hyperbolic metric λ Ω ds can be determined from the above being valid for points z ∈ Ω ∩ C whereas one must use local coordinates in any neighborhood of the point at infinity if Ω ⊂ C. (Alternatively, one can use the chordal hyperbolic metric-densityλ and then the hyperbolic metric isλds whereds denotes the chordal (or spherical) arclength "differential".) For example, the hyperbolic metric λ * ds on the punctured unit disk D * := D \ {0} can be obtained by using the universal covering z = exp(w) from the left-half-plane onto D * and we find that The hyperbolic distance h = h Ω is the length distance h Ω := d λ induced by the hyperbolic metric λ ds on Ω. This is a geodesic distance: for any points a, b in Ω, there is an h-geodesic [a, b] h joining a, b in Ω. These geodesics need not be unique, but they enjoy the property that Except for a short list of special cases, the actual calculation of any given hyperbolic metric is notoriously difficult; computing hyperbolic distances and determining hyperbolic geodesics is even harder. Indeed, one can find a number of papers analyzing the behavior of the hyperbolic metric in a twice punctured plane. Typically one is left with estimates obtained by using domain monotonicity and considering 'nice' sub-domains and super-domains in which one can calculate, or at least estimate, the metric.
The standard technique for estimating the hyperbolic metric and hyperbolic distance is via domain monotonicity, a consequence of Schwarz's Lemma. That is, if Ω in ⊂ Ω ⊂ Ω out , then in Ω in , λ in ds ≥ λ ds ≥ λ out ds and h in ≥ h ≥ h out .
Notice that the largest hyperbolic plane regions are twice punctured planes. We write λ ab ds and h ab for the hyperbolic metric and hyperbolic distance in the twice punctured plane C ab . The standard twice punctured plane is C 01 and its hyperbolic metric has been extensively studied by numerous researchers including [Hem79], [Min87], [SV01], [SV05]. We mention only the following.
Fact 2.13 was first proved by Lehto, Virtanen and Väisälä (see [LVV59]); later proofs were given by Agard [Aga68], Jenkins [Jen81], and Minda [Min87]. For future reference we record the following well-known estimates for hyperbolic distance in D ⋆ , in D ⋆ , and in C 01 . Since we know the hyperbolic metrics in D ⋆ and D ⋆ , it is easy to check the first two estimates. The estimates for h 01 are straightforward consequences of Fact 2.13 above.
(c) For all a, b ∈ C 01 :

2.C.3. The Beardon-Pommerenke Function bp.
We desire upper and lower estimates for the hyperbolic metric in terms of the quasihyperbolic metric. These metrics are 2-bi-Lipschitz equivalent for simply connected hyperbolic plane regions; this is false for any domain with an isolated boundary point (such as the punctured unit disk). The hyperbolic and quasihyperbolic metrics are bi-Lipschitz equivalent precisely whenĈ \ Ω is uniformly perfect (see [BP78], [Pom79], [Pom84]). Beardon and Pommerenke corroborated this latter assertion as an application of their elegant result [BP78, Theorem 1] which says: For any hyperbolic region Ω in C and for all z ∈ Ω, .
Here the domain function Ω bp − → R, introduced by Beardon and Pommerenke, is defined via note that the infimum is restricted to nearest boundary points ζ ∈ B(z) = ∂Ω ∩ ∂D(z) for z (that is, ζ ∈ ∂Ω with δ(z) = |z − ζ|). Also, The definition of bp is motivated by examining the standard lower bound for the hyperbolic metric on a twice punctured plane. The (BP) inequalities follow via domain monotonicity: the upper bound for λ(z) holds because z lies on the conformal center of a certain annulus in Ω, and the lower bound holds because Ω lies in a certain twice punctured plane.
We find that 2bp(z, ζ) is the conformal modulus of the maximal Euclidean annulus that is contained in Ω and symmetric with respect to the circle S 1 (ζ; δ(z)). It follows that 2bp(z) is the minimum of these numbers; so, 2bp(z) is the smallest of these maximal moduli.
Thus we see that whenever bp(z) > 0, there is an annulus is any nearest boundary point for z that realizes bp(z), and We call BP(z) a Beardon-Pommerenke annulus (or briefly, a BP annulus) associated with the point z; it needn't be unique. Typically, the Beardon-Pommerenke inequalities (BP) are employed to give lower estimates for hyperbolic distance, but they can also provide useful upper estimates.
Thus by (BP) In [BH21,Proposition 3.3] we established especially useful estimates for bp; these say that in any annulus in A 2 Ω (8 log 2), the domain function bp decays 'linearly' as we move away from the center circle. The assumption that both boundary circles of the annulus meet ∂Ω is crucial for obtaining the upper bounds; when only one boundary circle has a boundary point, bp can actually increase when we move away from the center circle towards the boundary circle that does not meet ∂Ω. Here is a summary.
Roughly speaking, a metric space is uniform when points in it can be joined by paths which are not "too long" and which "move away" from the region's boundary. More precisely, Ω ⊂ C is C-uniform (for some constant C ≥ 1) provided each pair of points can be joined by a C-uniform arc. Here a rectifiable arc γ : a b is a C-uniform arc if and only if it is both a C-quasiconvex arc and a double C-cone arc; these conditions mean, respectively, that Martio and Sarvas introduced the notion of a uniform domain in [MS79], and this has proven to be invaluable in geometric function theory and especially for the "analysis in metric spaces" program. A simply connected proper subdomain of the plane is uniform if and only if it is a quasidisk. Each uniform domain has the Sobolev extension property, and the BMO extension property characterizes uniformity. See [Geh82] and the many references therein, especially [Jon81,Jon80].
A geodesic metric space X is Gromov hyperbolic if there exists a constant θ ≥ 0 such that every geodesic triangle is θ-thin, meaning that each point on any edge of the triangle is at distance at most θ from the other two edges. See [BHK01, Chapter 3], [BH99], [BBI01], or [Väi05] and the many references in these.
The Gromov boundary ∂ G X of a Gromov hyperbolic space X is the set of equivalence classes of geodesic rays, where two rays are equivalent if and only if their Hausdorff distance is finite. One can also use quasi-geodesic rays, or, Gromov sequences. There is no canonical preferred distance on the Gromov boundary. However, for each ε ∈ (0, ε 0 ] (usually ε 0 = ε 0 (θ) : for all ξ, η ∈ ∂ G X, where (ξ|η) o is the usual Gromov product and o ∈ X a fixed base point. Standard estimates then give where (ξ, η) is any geodesic line in X with endpoints ξ, η ∈ ∂ G X.
The conformal gauge on ∂ G X is the maximal collection of all distance functions on ∂ G X that are quasisymmetrically equivalent to some (hence all) visual distance(s).

3.A.
Proof of Theorem A. We employ the following technical fact about "fat" annuli. We demonstrate that these points possess the asserted properties. Thanks to Fact 2.8 we know that the inequalities in (3.1a) follow.
When −e m ∈ ∂Ω we take a := e m−1 , b := √ a, c := 1 and argue similarly. With this in mind, suppose (Ω, h) f − → (Ω, k) is K-bi-Lipschitz. Let A ∈ A Ω , and let α be the simple loop in Ω whose trajectory is the center circle |α| = S 1 (A). Using hyperbolic distance in A ⊂ Ω, we deduce that
Let A ∈ A Ω . By enlarging A if necessary, we may assume that ∂A ∩ ∂Ω = ∅, and also that m := 1 2 mod(A) > 2C + 1. Let a, b, c be the points in A given by Lemma 3.1 and let a ′ , b ′ , c ′ be their f images. Since k(a, b) ≥ 1 2 (m − 1) ≥ 1, Fact 2.12 tells us that k(a ′ , b ′ ) ≤ C k(a, b).
The same fact, now applied to f −1 , tells us that and thus by quasisymmetry which gives the asserted estimate 2m ≤ M. The above is quantitative, but the constants are somewhat murky! Bonk, Heinonen, and Koskela established a similar result [BHK01, Theorem 3.6] for abstract uniform metric spaces but using quasihyperbolic distance in lieu of hyperbolic distance. We closely follow their proof, but there are significant modifications that we detail.
In particular, we utilize the following information; much of this is either a direct consequence of work in [BHK01], or follows by similar reasoning, the latter being especially true whenever only upper estimates for quasihyperbolic distance are employed (because always, h ≤ 2k). See especially [BHK01, Chapters 2 and 3]. We sketch the ideas.

Proposition.
Let Ω be a hyperbolic domain inĈ with (Ω, l σ ) A-uniform. There are constants θ, B, C (that depend only on the "data") such that the following hold.
(c) Each pair of distinct points in (Ω, l σ ) can be joined by a hyperbolic geodesic which is a B-uniform arc in (Ω, l σ ); when one or both points lie in ∂(Ω, l σ ), we get a hyperbolic geodesic ray or line respectively.
(f) Given ζ, ξ ∈ ∂(Ω, l σ ) and x = x(ξ) ∈ [o, ζ) h as above, we have Proof. There is no harm in rotating the sphereĈ, so we can assume that Ω ⊂ C. To see Item (e) can be established exactly as done in [BHK01, Lemma 3.14] for quasihyperbolic distance. Evidently, (f) follows from (e) and the standard estimates for visual distances given at the end of §2.D.
Item (d) follows mostly as in [BHK01, Proposition 3.12] with one major modification. It is routine to see that each hyperbolic geodesic ray in Ω has an endpoint in ∂(Ω, l σ ), that rays with the same endpoint are equivalent, and that each boundary point is the endpoint of such a ray. It remains to show that equivalent rays have the same endpoint. Suppose α and β are hyperbolic geodesic rays in Ω with ξ := α(∞), η := β(∞) ∈ ∂(Ω, l σ ), and ξ = η. We claim that dist h H (|α|, |β|) = +∞ (so α and β are not equivalent). This is not difficult to check when ξ and η correspond to 8 different points in∂Ω = ∂(Ω, σ), but requires additional effort if these two length boundary points are attached to the same spherical boundary point, which we assume is the origin 0. Since we are "near" the origin, we can work with Euclidean quantities in place of spherical. Since ξ = η, there is "plenty" of ∂Ω "near" the origin. In particular, it is not hard to check that bp ≤ log 10 on |γ a | ∪ |γ b |, so by (BP) λ ds and δ −1 ds are bi-Lipschitz on |γ a | ∪ |γ b |. It follows that for all s > |s a | ∨ |s b |, Thus α and β are indeed non-equivalent hyperbolic geodesic rays :-) We require the following technical information. The upshot of this is that, given two length boundary points, we can always find spherical boundary points at a distance comparable to the length distance between the two given length boundary points. Here ι is as described in footnote 8. Also, we employ the ABC property for hyperbolic geodesics; see §2.C.4.
Let ζ, η, ξ be points in ∂(Ω, l σ ) and put t := l σ (ζ, ξ)/l σ (ζ, η). When t ≥ 1, we can copy the Bonk-Heinonen-Koskela argument as it only uses upper estimates for quasihyperbolic distances. 10 Thus we may, and do, assume that t < 1. Let x = x(ξ), y = y(η) be the points on the hyperbolic geodesic ray [o, ζ) h that are given by Proposition 3.4(e) and associated with ξ, η respectively. Since t < 1, we have ζ < x ≤ y ≤ o where the geodesic is ordered from ζ to o. Then from Proposition 3.4(f) we find that It follows that for any fixed t 0 ∈ (0, 1), To finish the proof, we demonstrate that for all 0 < t < t 0 := c/πB 2 , h(x, y) ≥ H(t) where H(t) → +∞ as t → 0 + . This then gives which in turn confirms that the bijection ∂(Ω, l σ ) → ∂ G (Ω, h) is indeed quasisymmetric. Before immersing ourselves in the details, we explain the idea. Whenever one knows three distinct boundary points, one has a standard lower bound for hyperbolic distance given by looking at the appropriate thrice punctured sphere; Fact 2.14(c) is handy for estimating hyperbolic distance in such a domain. A difficulty here is that we have points in ∂(Ω, l σ ) whereas we need points in∂Ω. To overcome this, we appeal to Lemma 3.5.
3.C. Proof of Theorem C. Below, in §3.C.3, we establish the following general result; this provides a large class of plane domains whose hyperbolizations and quasihyperbolizations are quasiisometrically equivalent. In particular, each finitely connected domain belongs to this class, thus corroborating Theorem C. However, this class also includes many infinitely connected domains such as C \ Z or C ⋆ \ { 1 n | n ∈ N}.
In the above, δ = δ Ω is the Euclidean distance to ∂Ω which is infinite if Ω ⊃ C.
The hypotheses above ensure that Π is discrete in Ω, so Ω Π is a domain. Similarly, ∆ is closed in Ω and Ω ∆ is a domain. We use the subscripts Π and ∆ to denote quantities associated with Ω Π and Ω ∆ . For example, h Π = h Ω Π and k Π = k Ω Π are the hyperbolic and quasihyperbolic distances in Ω Π (respectively), and especially bp Π = bp Ω Π .

3.C.2. Quasiisometric Equivalence of Punctured Disks.
Here we present some technical details that allow us to streamline our proof of Theorem 3.9 and focus on the underlying ideas. Roughly, we show that whenever ∆ * is a punctured disk in a hyperbolic plane domain Ω (with the puncture a point ofĈ \ Ω), then (∆ * , h) and (∆ * , k) are quasiisometrically equivalent.
We use the following numerical result whose proof is left for the diligent reader.
(3.11) ∀ K > 0, L > 0, x > 0, y ≥ L : We require a result similar to Lemma 2.9 but for punctured disks in arbitrary domains; here there are separate cases for finite versus infinite punctures. Since Euclidean similarity transformations are quasihyperbolic isometries, we can normalize as in the following.
is an isometric equivalence and thus Lemma 2.9 gives the first assertion. The second assertion holds because z → z −1 is quasihyperbolically 2-bi-Lipschitz, as per Fact 2.11. Alternatively, it is not hard to check that for |z| ≥ 2, 1 2 |z| ≤ δ * (z) ≤ |z| = δ ⋆ (z). We conclude this subsubsection with a technical result that we employ in our proof of Theorem 3.9. As above there are separate cases for finite versus infinite punctures. Employing the estimates in Facts 2.14 we deduce that for all 0 < |a| ≤ |b| ≤ 1 2 : here (3.11) provides the last inequality. Thus for all a, b ∈ ∆ * , h * (a, b) − π log 2 ≤ |ψ(a) − ψ(b)| ≤ h * (a, b) + log 1 + k log 2 and so ψ is indeed a surjective (1, C 1 )-QI equivalence.
It is now not difficult to confirm that the map is a C 1 -rough isometric equivalence.
Next, we examine an infinite puncture. Suppose 1 ∈Ĉ \ Ω ⊂D with Ω * = Ω \ {∞} hyperbolic. Fix R ≥ 4 and let ∆ * := C \ D(0; R). Here we cannot argue as in the second part of the proof of Lemma 3.12 because we do not know whether the origin lies in Ω or in C \ Ω. Also, unlike the first part above, where we had the three boundary points 0, 1, ∞, here we only know two boundary points. Nonetheless, both (∆ * , h * ) and (∆ * , k * ) are QI equivalent to the infinite ray [0, +∞), |·| as we now corroborate.

Now we check that
First we show that for all q ∈ Π in , ∆ q ⊂ D[c; r ′ ]. Assume Π in = ∅. If c / ∈ Π in , then (3.15) provides the estimate r q ≤ 1 2 r for q ∈ Π in , so ∆ q ⊂ D[c; 3 2 r] as asserted. A similar argument works for the case {c} Π in . Assume {c} = Π in . Suppose A in ∩ (C \ Ω) = ∅, and let ζ be a point in this set. By (3.10a), 2r c ≤ |c − ζ| ≤ r, so ∆ c ⊂ D[c; r] as asserted. When A in ∩(C\Ω) = ∅, we have no estimates for r c , but S 1 (A)∩∆ = ∅ means that ∆ c ⊂ D[c; √ rR].

3.D. Proof of Theorem D.
Here we concoct a plane domain Ω and prove that the assertions of Theorem D hold for Ω. We require the following technical fact about quasisymmetric maps; this must be folklore, but we do not know a reference.
3.16. Lemma. Let (a n ) ∞ 0 be a strictly increasing sequence in R\{∞} with a n+1 /a n → +∞ as n → +∞. Put A := { a n | n ≥ 0 } ∪ {∞} ⊂R. Then every QS homeomorphism f : A → A is "eventually the identity"; i.e., there is an N such that for all n ≥ N, f (a n ) = a n . 12 Proof. To start, note that as f is a homeomorphism, it is a bijection and f (∞) = ∞ (as ∞ is the only non-isolated point of A).
We exhibit p, q such that for all n ≥ 1, f (a p+n ) = a q+n . Then f maps {a 0 , a 1 , . . . , a p } bijectively onto {a 0 , a 1 , . . . , a q }. Hence p = q and our claim is established.
We need only produce a p such that for all n ≥ p, f (a n ) < f (a n+1 ) with f (a n ) and f (a n+1 ) adjacent (meaning that if f (a n ) = a m , then f (a n+1 ) = a m+1 ). Indeed, given such a p we simply let q be the unique integer with a q = f (a p ).
Below we use the quasisymmetry of f to verify that there is an N such that for all n ≥ N, f (a n ) < f (a n+1 ). Let M be the unique integer with a M = f (a N ). Thus for all n ≥ N, a M ≤ f (a n ) < f (a n+1 ).
We claim that there are a finite number of m ≥ M such that a m / ∈ f { a n | n ≥ N } . Indeed, if m is such, then a m ∈ f {a 0 , a 1 , . . . , a N −1 } , so, there are at most N such m. Let ℓ be the largest of all these m.
Thus there is a p > N such that a M ≤ f (a p−1 ) < a ℓ < f (a p ), but for all n ≥ p, f (a n ) and f (a n+1 ) are adjacent. This is the sough after p.
3.17. Proof of Theorem D. Let (a n ) ∞ 1 be a strictly decreasing sequence in (−∞, 0) with a n+1 /a n → +∞ as n → +∞. Put a 0 := 0, A := { a n | n ≥ 0 }, and Ω := C \ A. We verify that the assertions of Theorem D hold for Ω. 13 It is straightforward to check that Ω is a uniform domain; [Her84,Lemma 8.4] provides a convenient criterion here.