Uniform Metric Graphs

Abstract

We prove that every complete metric space “is” the boundary of a uniform length space whose quasihyperbolization is a geodesic visual Gromov hyperbolic space. There is a natural quasimöbius identification of the original space’s conformal gauge with the canonical gauge on the Gromov boundary. All parameters are absolute constants.

1 Introduction

Throughout this section X is a rectifiably connected non-complete locally complete metric space. These are the minimal requirements for X to support a quasihyperbolic distance $k=k_X$, and we dub X a quasihyperbolic metric space; see §2.1.3 for a precise definition.

Roughly speaking, such an X is a uniform metric space provided each pair of points can be joined by a path that moves away from the boundary of X and whose length is comparable to the distance between the points. See §2.3 for a precise definition.

Martio and Sarvas initiated the study of Euclidean uniform domains in [17]; then Bonk, Heinonen, and Koskela [2] introduced these ideas in locally compact metric spaces and established fundamental ties with Gromov hyperbolicity; and, Väisälä [22] developed a similar theory in the Banach space setting. Uniform spaces have proven to be invaluable in geometric function theory, potential theory, geometric group theory, and especially for the “analysis in metric spaces” program as discussed, for instance, in [9, 14, 15, 19]. A finitely connected proper subdomain of the plane is uniform if and only if each boundary component is either a point or a quasicircle, but in general there are no such simple geometric criterion for uniformity.

Given their fundamental importance, it seems worthwhile to record the following. See Sect. 3 for its proof, and Sect. 2 for definitions and other relevant information.

Theorem

Given any metric space Z, there is a uniform length space X whose quasihyperbolization $X_k:=(X,k)$ is a geodesic visual Gromov hyperbolic space. Also, there is a natural bi-Lipschitz equivalence between $\bar{Z}$ and the metric boundary $\partial X$ of X that induces a quasimöbius equivalence between the conformal gauge on $\hat{Z}$ and the canonical gauge on the Gromov boundary $\partial _G X$.

All associated parameters are absolute constants; e.g., X is A-uniform with $A:=49$ and all other constants depend only on A.

A geodesic metric space is hyperbolic if its geodesic triangles are thin; see §2.2. The elegant well studied notion of a hyperbolic filling (see [8, Chapter 6]) can surely be used to establish some version of the above in the locally compact setting, however, to do so one must invoke the uniformization theory of Bonk-Heinonen-Koskela [2, Chapter 4]. Our approach avoids the initial snowflaking of the distance on Z and directly produces a uniform space.

Hyperbolicity of $X_k$ follows via Theorem 2.5 which is a modest generalization of results of Bonk, Heinonen, Koskela [2] in the locally compact setting, and Väisälä [22] in the Banach space setting. It would be useful to have a version valid for arbitrary uniform spaces.

2 Preliminaries

For real numbers r and s, $r\wedge s:= \min \{{r,s}\}$ and $r\vee s:= \max \{{r,s}\}$. Two numerical quantities P and Q are roughly equal if $P{{\,\mathrm{\simeq }\,}}Q$ and quasi equal if $P{{\,\mathrm{\approx }\,}}Q$ where

$$\begin{aligned} P{{\,\mathrm{\simeq }\,}}Q&\iff Q-C \le P \le Q+C \end{aligned}$$

and

$$\begin{aligned} P{{\,\mathrm{\approx }\,}}Q&\iff C^{-1} Q\le P\le C\,Q; \end{aligned}$$

here $C=C(D,\dots )>0$ is some computable constant that depends only on the relevant data $D,\dots$. We emphasize that throughout Sect. 3all constants are absolute.

2.1 Metric Space Notation and Terminology

Throughout this section X is an arbitrary metric space with distance denoted $|x-y|$; this is not meant to imply that X possesses any sort of linear or group structure. In this setting, all topological notions refer to the metric topology; here ${{\,\mathrm{\textsf{cl}}\,}}(A), {{\,\mathrm{\textsf{bd}}\,}}(A), {{\,\mathrm{\textsf{int}}\,}}(A)$ are the topological closure, boundary, interior (respectively) of $A\subset X$.

The open ball, sphere, closed ball of radius r centered at the point $a\in X$ are

$$\begin{aligned}{} & {} \textsf{B}(a;r):=\{{x:|x-a|<r}\} ,\quad \textsf{S}(a;r):=\{{x:|x-a|=r}\} ,\\{} & {} \textsf{B}[a;r]:=\textsf{B}(a;r)\cup \textsf{S}(a;r). \end{aligned}$$

Every metric space can be isometrically embedded into a complete metric space. We let $\bar{X}$ denote the metric completion of the metric space X; thus $\bar{X}$ is the closure of the image of X under such an isometric embedding. We call $\partial X:=\bar{X}\setminus X$ the metric boundary of X.

When X is non-complete, $\delta (x)=\delta _X(x):={{\,\mathrm{\textsf{dist}}\,}}(x,\partial X)$ is the distance from a point $x\in X$ to the boundary $\partial X$ of X. Note that $\partial X$ is closed in $\bar{X}$ if and only if $\delta (x)>0$ for all $x\in X$; e.g., this holds when X is locally compact.

A metric space X is locally complete provided each point has an open neighborhood which is complete. When X is non-complete, this is equivalent to requiring that $\delta (x)>0$ for all $x\in X$, or, $\partial X$ is closed in $\bar{X}$, or, X is open in $\bar{X}$.

The one-point extension of X is $\dot{X}:=X\cup \{{\infty }\}$ together with the topology consisting of all open sets in X and sets U that contain $\infty$ with $\dot{X}\setminus U=X\setminus U$ closed and bounded in X. The one-point extension is most useful when a space is unbounded. In our setting, we employ

$$\begin{aligned} \hat{X}:={\left\{ \begin{array}{ll} \bar{X}=X\cup \partial X&{}\text {when}\, X\,\text { is bounded} \\ \dot{\bar{X}}=X\cup \dot{\partial }X &{}\text {when}\, X\,\text { is unbounded} \end{array}\right. } \end{aligned}$$

where $\dot{\partial }X:=\partial X\cup \{{\infty }\}$, and then

$$\begin{aligned} \hat{\partial } X:=\hat{X}\setminus X={\left\{ \begin{array}{ll} \partial X&{}\text {when}\, X\,\text { is bounded} \\ \dot{\partial }X &{}\text {when}\, X\,\text { is unbounded}. \end{array}\right. } \end{aligned}$$

As with the Riemann sphere $\hat{\mathbb {C}}=\mathbb {C}\cup \{{\infty }\}$ and its chordal distance, for any unbounded space X, there are chordal distances on $\hat{X}$ whose metric topology coincides with the one-point extension topology; see [3, 6]. For any fixed $o\in X$, there is a distance $c_o$ on $\hat{X}$ such that

$$\begin{aligned} \forall \; a,b\in X, \quad \frac{1}{4}\frac{|a-b|}{\bigl (1+|a-o|\bigr )\bigl (1+|b-o|\bigr )} \le c_o(a,b) \le \frac{|a-b|}{\bigl (1+|a-o|\bigr )\bigl (1+|b-o|\bigr )}. \end{aligned}$$

The (absolute) cross ratio of four distinct points $a,b,c,d\in X$ is

$$\begin{aligned} r=|a,b,c,d|:=\frac{|a-b||c-d|}{|a-c||b-d|}. \end{aligned}$$

The 24 permutations of a, b, c, d yield at most six values for their absolute cross ratios, three numbers and their reciprocals; e.g., $|d,b,c,a|=1/r$. Employing continuity of cross ratios allows us to consider the cross ratio for points in $\hat{X}$; more simply, we delete all distances involving the point $\infty$. For example, for any chordal distance $c_o$ on $\hat{X}$ (with X unbounded), we see that for all distinct $a,b,c,d\in \hat{X}$,

$$\begin{aligned} 16^{-1}|a,b,c,d|\le |a,b,c,d|_{c_o}:=\frac{c_o(a,b)c_o(c,d)}{c_o(a,c)c_o(b,d)} \le 16|a,b,c,d|. \end{aligned}$$

An embedding $\hat{X}\supset A\xrightarrow {f}\hat{Y}$ is quasimöbius if there is a homeomorphism $[0,+\infty )\xrightarrow {\theta }[0,+\infty )$ such that for all distinct $a,b,c,d\in A$,

$$\begin{aligned} r'\le \theta (r) \quad \text {where}\, r:=|a,b,c,d|\,\text { and}\, r':=|f(a),f(b),f(c),f(d)|. \end{aligned}$$

See [18, 21, §4.8], [22, §2.28]. For example, if X is unbounded and $c_o$ is any chordal distance on $\hat{X}$, then the identity map

is a 16t-quasimöbius embedding. Moreover, if $c_p$ is a second chordal distance on $\hat{X}$, then the identity map

$$\begin{aligned} (\hat{X},c_o)\xrightarrow {{{\,\mathrm{{\textsf{id}}}\,}}}(\hat{X},c_p) \end{aligned}$$

is a 256t-quasimöbius equivalence.

The conformal gauge associated with a given metric space $X=(X,{{\,\mathrm{|\cdot |}\,}})$ is the collection of all distances d on X such that the identity map $(X,d)\rightarrow (X,{{\,\mathrm{|\cdot |}\,}})$ is quasimöbius. When X is unbounded, the conformal gauge associated with $\hat{X}$ is the maximal collection of all distances on $\hat{X}$ that are quasimöbius equivalent to some (hence all) chordal distance(s) on $\hat{X}$. Evidently, the conformal gauge associated with $\hat{X}$ is quasimöbius equivalent to the conformal gauge associated with $\bar{X}$.

2.1.1 Paths, Arcs, Length, & Geodesics

A path in X is a continuous map $\mathbb {R}\supset I\xrightarrow {\gamma }X$ where $I=I_\gamma$ is the parameter interval for $\gamma$ that may be closed or open or neither and finite or infinite. The trajectory of such a path is $|\gamma |:=\gamma (I)$, a curve; when easily understood in context, we abuse notation and write $\gamma$ in place of $|\gamma |$.

When $I\ne \mathbb {R}$ is closed, $\partial \gamma :=\gamma (\partial I)$ denotes the set of endpoints of $\gamma$ and consists of one or two points depending on whether or not I is compact; e.g., if $I_\gamma =[u,v]$, $\partial \gamma =\{{\gamma (u),\gamma (v)}\}$. When $\partial \gamma =\{{a,b}\}$, we write $\gamma :a\curvearrowright b$ (in X) to indicate that $\gamma$ is a path (in X) with initial point a and terminal point b; this notation is meant to imply an orientation—a precedes b on $\gamma$.

We call $\gamma$ a compact path if its parameter interval is compact. A compact path $\gamma$ is a loop if $\partial \gamma$ is a single point, and then $|\gamma |$ is often dubbed a closed curve. A loop $\gamma :[a,b]\rightarrow X$ is a Jordan loop (aka, a simple closed curve) if $\gamma \vert _{[a,b)}$ is injective.

An arc $\alpha$ is an injective compact path; here $|\alpha |$ is often called a simple curve; again, we sometimes abuse terminology and call $|\alpha |$ an arc.

Given points $a,b\in |\alpha |$, there is a unique subarc $\alpha [a,b]$ of $\alpha$ with endpoints a, b; precisely, there are unique $u,v\in I$ with $\alpha (u)=a$, $\alpha (v)=b$ and $\alpha [a,b]:=\alpha \vert _{[u,v]}$. (Again, sometimes $\alpha [a,b]$ is this map and sometimes it denotes its trajectory.) We also use this notation for a general path $\gamma$, but here $\gamma [a,b]$ denotes the unique subpath of $\gamma$ that joins a, b and is obtained by using the last time $\gamma$ is at a up to the first time $\gamma$ is at b.

When $\alpha :a\curvearrowright b$ and $\beta :b\curvearrowright c$ are paths, $\alpha \star \beta$ denotes the concatenation¹ of $\alpha$ and $\beta$; so $\alpha \star \beta :a \curvearrowright c$. The reverse of $\gamma$ is the path $\tilde{\gamma }$ defined by $\tilde{\gamma }(t):=\gamma (1-t)$ (when $I_\gamma =[0,1]$) and going from $\gamma (1)$ to $\gamma (0)$. Of course, $|\alpha \star \beta |=|\alpha |\cup |\beta |$ and $|\tilde{\gamma }|=|\gamma |$.

Every compact path contains an arc with the same endpoints; see [20].

The length of a compact path $[0,1]\overset{\gamma }{\rightarrow }X$ is defined in the usual way by

$$\begin{aligned} \ell (\gamma ):=\sup \Bigl \{\sum _{i=1}^n |\gamma (t_i)-\gamma (t_{i-1})| \,\Big \vert \, 0=t_0<t_1<\dots <t_n=1 \Bigr \} , \end{aligned}$$

$\gamma$ is rectifiable when $\ell (\gamma )<\infty$, and X is rectifiably connected provided each pair of points in X can be joined by a rectifiable path. Every rectifiably connected metric space X admits a natural intrinsic (inner) length distance distance given by

$$\begin{aligned} l(a,b):= \inf \bigl \{\ell (\gamma )\,\big \vert \,\gamma :a\curvearrowright b\,\text { a rectifiable path in}\, X\bigr \} . \end{aligned}$$

To define a length distance l, we only need to know the lengths of compact paths in X. See, for example, [7, Chapter 2]. A metric space $(X,{{\,\mathrm{|\cdot |}\,}})$ is a length space when $|a-b|=l(a,b)$ for all $a,b\in X$, and we call such a ${{\,\mathrm{|\cdot |}\,}}$ a length (or intrinsic) distance. An l-geodesic $[a,b]_l$ is a shortest path joining a and b, and any shortest path can be parametrized to be an l-geodesic.

2.1.2 Geodesics and QuasiGeodesics

A map $I\xrightarrow {\gamma }X$ is: a geodesic if it is an isometry (for all $s,t\in I$, $|\gamma (s)-\gamma (t)|=|s-t|$), a K-quasi-geodesic if it is K-bi-Lipschitz (aka, a K-quasi-isometry),

$$\begin{aligned} \forall s,t\in I, \quad K^{-1}|s-t| \le |\gamma (s)-\gamma (t)| \le K|s-t|, \end{aligned}$$

and a (K, C)-rough-quasi-geodesic if it is a (K, C)-rough-quasi-isometry,

$$\begin{aligned} \forall \;s,t\in I, \quad K^{-1}|s-t|-C \le |\gamma (s)-\gamma (t)| \le K|s-t| + C; \end{aligned}$$

here $I\subset \mathbb {R}$ is an interval, although for rough-quasi-geodesics it is common to allow I to be the intersection of $\mathbb {Z}$ with an interval. Every quasi-geodesic (so every geodesic too) is an injective path (so, an arc), but rough-quasi-geodesics need not be continuous. Nonetheless, every rough-quasi-geodesic can be “tamed” (see [4, p. 403]) meaning that it can be replaced by a nearby continuous rough-quasi-geodesic.

We use the terms geodesic (or quasi-geodesic or rough-quasi-geodesic) segment, ray, line when $I\subset \mathbb {R}$ is, respectively, a compact interval, a closed semi-infinite interval, or $I=\mathbb {R}$. In a geodesic metric space each pair of points can be joined by a geodesic segment.

A characteristic property of geodesics is that the length of each subarc is the distance between its endpoints. A corresponding description for quasi-geodesics involves chordarc paths: $\gamma$ is an L-chordarc path if it is locally rectifiable and

$$\begin{aligned} \forall s,t\in I, \quad \ell (\gamma \vert _{[s,t]}) \le L\,|\gamma (s)-\gamma (t)|. \end{aligned}$$

Ignoring parameterizations, quasi-geodesics are the same as chordarc paths.²

Similarly, a locally rectifiable path $I\xrightarrow {\gamma }X$ is an (L, C)-rough-chordarc path if

$$\begin{aligned} \forall s,t\in I, \quad \ell (\gamma \vert _{[s,t]}) \le L\,|\gamma (s)-\gamma (t)|+C; \end{aligned}$$

such a path has an arclength parametrization that is an (L, C/L)-rough-quasi-geodesic.

2.1.3 Quasihyperbolic Distance

Recall that X is a quasihyperbolic metric space if it is non-complete, locally complete, and rectifiably connected. For all points x in such a space X, $\delta (x)=\delta _X(x):={{\,\mathrm{\textsf{dist}}\,}}(x,\partial X)>0$, so $\delta ^{-1}ds$ is a conformal metric that we call the quasihyperbolic metric on X. The length distance induced by the quasihyperbolic metric $\delta ^{-1}ds$ is dubbed the quasihyperbolic distance $k=k_X$ in X. In a locally compact quasihyperbolic space, this is a geodesic distance: there are always k-geodesics joining any two points. In general, k-geodesics may not exist, however, this is always a complete distance; see the discussion in [13, §2.A.5].

The following basic estimates for quasihyperbolic distance were established for Euclidean domains by Gehring and Palka [11, 2.1]. For all $a,b\in X$ and any rectifiable $\gamma :a\curvearrowright b$ in X,

$$\begin{aligned}{} & {} k(a,b) \ge \log \left( 1+\frac{l(a,b)}{\delta (a)\wedge \delta (b)}\right) \ge \log \left( 1+\frac{|a-b|}{\delta (a)\wedge \delta (b)}\right) \ge \left| \log \frac{\delta (a)}{\delta (b)}\right| \ \nonumber \\ \end{aligned}$$

(2.1a)

which is a special case of the more general inequality

$$\begin{aligned}{} & {} \ell _k (\gamma ) \ge \log \Bigl (1+\frac{\ell (\gamma )}{{{\,\mathrm{\textsf{dist}}\,}}(|\gamma |,\partial X)} \Bigr ). \end{aligned}$$

(2.1b)

2.2 Gromov Hyperbolic Spaces

A geodesic metric space X is $\theta$-hyperbolic if and only if each edge of any geodesic triangle in X lies in the $\theta$-neighborhood of the union of the other two edges, and X is Gromov hyperbolic if it is $\theta$-hyperbolic for some $\theta \in [0,+\infty )$. When X is not geodesic, one employs the Gromov product (based at some fixed $o\in X$)

$$\begin{aligned} \bigl (x\mid y\bigr )=\bigl (x\mid y\bigr )_o:=\frac{1}{2}\Bigl (|x|+|y|-|x-y|\Bigr ) \quad \text {where}\, |x|:=|x-o| \end{aligned}$$

as explained in, e.g., [2, 4, 7, 8, 21]. For a geodesic $\theta$-hyperbolic X,

$$\begin{aligned} \bigl (x\mid y\bigr )_o\le {{\,\mathrm{\textsf{dist}}\,}}\bigl (o,[x,y]\bigr )\le \bigl (x\mid y\bigr )_o+2\theta ; \end{aligned}$$

see [2, (3.2), p.19] or [4, p.410].

The so-called geodesic stability, that any rough-quasi-geodesic has bounded Hausdorff distance from a geodesic with the same endpoints, is an important characteristic property of geodesic hyperbolic spaces; see [1, 4, 8, 21].

2.2.1 Gromov Boundary and Visual Distances

The Gromov boundary $\partial _G X$ of a Gromov hyperbolic space X is the set of equivalence classes of Gromov sequences. A sequence $\underline{x}=(x_n)_{n\in \mathbb {N}}$ in X is Gromov if and only if $\bigl (x_n\mid x_m\bigr )\rightarrow +\infty$, and $\underline{x}$ is equivalent to a Gromov $\underline{y}=(y_n)_{n\in \mathbb {N}}$ if and only if $\bigl (x_n\mid y_n\bigr )\rightarrow +\infty$. Then

$$\begin{aligned} \partial _G X:=\{{\hat{x} \mid \underline{x}\;\text {Gromov}}\} \quad \text {where}\, \hat{x}\,\text { is the equivalence class of}\, \underline{x}. \end{aligned}$$

We follow [21], but see also [2, 4, 7, 8] and references therein.

The Gromov product extends to $\partial _G X$ via

$$\begin{aligned} \bigl (\hat{x}\mid \hat{y}\bigr ):=\inf \{{\liminf _{m,n\rightarrow \infty }\bigl (x_m\mid y_n\bigr ) \;\big |\; \underline{x}\in \hat{x},\underline{y}\in \hat{y}}\} \end{aligned}$$

and similarly

$$\begin{aligned} \bigl (a\mid \hat{x}\bigr )=\bigl (\hat{x}\mid a\bigr ):=\inf \{{\liminf _{n\rightarrow \infty }\bigl (x_n\mid a\bigr ) \;\big |\; \underline{x}\in \hat{x}}\} \end{aligned}$$

for $a\in X$ and $\hat{x},\hat{y}\in \partial _G X$.

A Gromov hyperbolic space X is said to be visual (with respect to some base point $o\in X$) if and only if there is a constant K such that

$$\begin{aligned} \forall \;x\in X, \quad \exists \;\xi \in \partial _G X\;\text { with } |x|\le \bigl (x\mid \xi \bigr )_o+K. \end{aligned}$$

This is independent of the choice of base point o.

There is no canonical preferred distance on the Gromov boundary. A distance d on $\partial _G X$ is a visual distance (with respect to a base point o and parameter $\upsilon$) if and only if

$$\begin{aligned} \forall \;\xi ,\eta \in \partial _G X, \quad d(\xi ,\eta ){{\,\mathrm{\approx }\,}}\exp \Bigl (-\upsilon \bigl (\xi \mid \eta \bigr )_o\Bigr ); \end{aligned}$$

see [4, Definition 3.20, p.434] or [8, §2.2.3, p.15]. The canonical gauge on $\partial _G X$ is the maximal collection of all distances on $\partial _G X$ that are quasimöbius equivalent to some (hence all) visual distance(s).

For each $\varepsilon \in (0,\varepsilon _o]$ (one can take $\varepsilon _o=\varepsilon _o(\theta ):=1/5\theta$, e.g. see [21, Prop. 5.16]) there is a visual distance $d_\varepsilon =d_{\varepsilon ,o}$ on $\partial _G X$ that satisfies

$$\begin{aligned} \forall \; \xi ,\eta \in \partial _G X, \quad \frac{1}{2}\, \exp \Bigl (-\varepsilon \bigl (\xi \mid \eta \bigr )_o\Bigr ) \le d_\varepsilon (\xi ,\eta ) \le \exp \Bigl (-\varepsilon \bigl (\xi \mid \eta \bigr )_o\biggr ). \end{aligned}$$

In a proper geodesic $\theta$-hyperbolic X, standard estimates then give

$$\begin{aligned} C^{-1} \exp \Bigl (-\varepsilon \, {{\,\mathrm{\textsf{dist}}\,}}\bigl (o,(\xi ,\eta )\bigr ) \Bigr ) \le d_\varepsilon (\xi ,\eta ) \le C \exp \Bigl (-\varepsilon \,{{\,\mathrm{\textsf{dist}}\,}}\bigl (o,(\xi ,\eta )\bigr ) \Bigr ) \end{aligned}$$

where $(\xi ,\eta )$ is any geodesic line in X with endpoints $\xi ,\eta \in \partial _G X$ and $C=C(\theta )$.

The following useful information describes the visual effect of changing a base point; see [21, Prop. 5.28]

Fact 2.2

Let $p,q\in X$. Suppose X is a $\theta$-hyperbolic space and $\varepsilon \in (0,\varepsilon _o]$. Then the identity map $(\partial _GX,d_{\varepsilon ,p})\rightarrow (\partial _G X,d_{\varepsilon ,q})$ is 16t-quasimöbius.

2.3 Quasiconvex and Uniform Spaces

A rectifiable path $\gamma :a\curvearrowright b$ is C-quasiconvex if

$$\begin{aligned} \ell (\gamma )\le C \, |a-b|. \end{aligned}$$

A metric space is C-quasiconvex if each pair of points can be joined by a C-quasiconvex path. A 1-quasiconvex metric space is a geodesic space, and a space is a length space if and only if it is C-quasiconvex for all $C>1$. By cutting out loops, we can always replace a C-quasiconvex path with a C-quasiconvex arc having the same endpoints; see [20].

Roughly speaking, a metric space is uniform when points in it can be joined by paths that are not “too long” and “move away” from the region’s boundary. More precisely, a quasihyperbolic metric space X is C-uniform provided each pair of points can be joined by a C-uniform arc. Here a rectifiable $\gamma :a\curvearrowright b$ is a C-uniform arc if and only if it is both a C-quasiconvex arc and a double C-cone arc; this latter condition means that

$$\begin{aligned} \forall \; x\in |\gamma |\,, \quad \ell (\gamma [x,a])\wedge \ell (\gamma [x,b])\le C \, \delta (x). \end{aligned}$$

(2.3)

Double cone arcs are sometimes called cigar arcs. In [19] Väisälä provides a description of various possible double cone conditions (which he calls length cigars, diameter cigars, distance cigars, and Möbius cigars). The work [16] of Martio should also be mentioned.

We require the following information about uniform spaces; Gehring and Osgood [10] established these in the Euclidean setting; Bonk, Heinonen, and Koskela [2] gave proofs for the locally compact setting; see [5, 12] for the locally complete setting.

Facts 2.4

Suppose X is a C-uniform metric space. Then

• (2.4a) $\forall \; a,b\in X, \quad k(a,b)\le 4C^2 \log \left( 1+\dfrac{|a-b|}{\delta (a)\wedge \delta (b)}\right)$;

• (2.4b) each quasihyperbolic geodesic $[a,b]_k$ is a B-uniform arc where $B=B(C)$.³

Bonk, Heinonen, and Koskela established a version of the following in the locally compact setting; see [2, Theorem 3.6, Prop. 3.12]. Väisälä developed a similar theory in the Banach space setting; see [22]. This result holds if $X_k$ admits quasihyperbolic K-quasi-geodesics for all $K>1$. It would be interesting to better understand the situation where $X_k$ is merely a length space.

Theorem 2.5

Let X be a uniform space. Suppose its quasihyperbolization $X_k=(X,k)$ is geodesic. Then $X_k$ is Gromov hyperbolic and there is a natural bijection $\partial _G X\xrightarrow {\iota }\hat{\partial }X$ with the property that a sequence $\underline{x}$ in X converges to $\xi \in \hat{\partial }X$ if and only if $\underline{x}$ is a Gromov sequence with $\iota (\hat{x})=\xi$.

Proof Sketch

Bonk, Heinonen, Koskela [2, Theorem 3.6] proved that the quasihyperbolization $X_k:=(X,k)$ of a uniform space X is Gromov hyperbolic. They assume local compactness (so that $X_k$ is proper and geodesic), but a careful examination of their proof of hyperbolicity reveals that it only requires the existence of quasihyperbolic geodesics.

In our setting, one can mimic the proofs of Väisälä’s results [22, Lemmas 2.22, 2.25; Prop. 2.26] to obtain the natural map $\iota$ with its asserted properties. $\square$

3 Proof of Theorem

Here Z is a given metric space and we denote distance in Z by $|a-b|$. For each $n\in \mathbb {Z}$, let $Z^{(n)}$ be a maximal $2^{-n}$-separated set in Z. Thus

$$\begin{aligned} \forall x\ne y \text { in } Z^{(n)},\; |x-y|\ge 2^{-n} \quad \text {and}\quad Z=N(Z^{(n)};2^{-n}). \end{aligned}$$

When Z is unbounded, $Z^{(n)}\ne \emptyset$ for all $n\in \mathbb {Z}$; here we set $n_o:=0$, pick any fixed $o\in Z^{(0)}$, and let $\mathbb {Z}_o:=\mathbb {Z}$. If Z is bounded, we set

$$\begin{aligned} n_o:=\max \{{n\in \mathbb {Z}\mid Z^{(n)}=\emptyset }\}, \end{aligned}$$

pick any fixed $o\in Z$, put $Z^{(n_o)}:=\{{o}\}$, and let $\mathbb {Z}_o:=\{{n\in \mathbb {Z}\mid n\ge n_o}\}$.

For each $z\in Z$ and $n\in \mathbb {Z}_o$ there exists (at least one) $z_n\in Z^{(n)}$ such that

$$\begin{aligned} |z_n-z|<\frac{1}{2^n}, \quad \text {and then}, \quad \underline{z}:=(z_n)_{n\in \mathbb {Z}_o} \;\text { is an}\, approximating sequence for \, z. \end{aligned}$$

If $z\in Z^{(n)}$, $z_n=z$. If Z is bounded, $z_{n_o}=o$. Also, $\forall \;z\in Z$, $|z|:=|z-o|$.

3.1 Constructing X

Here we construct a metric graph $X=<\mathscr {V},\mathscr {E},l>$⁴ whose vertex set $\mathscr {V}$ is the disjoint union of the sets $Z^{(n)}$. By assigning lengths to each edge (i.e., making each edge isometric to a compact interval) we can compute the length of a compact path, and then distance in X is the standard length distance l.

3.1.1 Vertices

Let $\mathscr {V}:=\bigcup _{n\in \mathbb {Z}_o} V^{(n)}$ where $V^{(n)}:=Z^{(n)}\times \{{n}\}$. Define $\mathscr {V}\xrightarrow {\pi }Z$ and $\mathscr {V}\xrightarrow {\lambda }\mathbb {Z}$ by

$$\begin{aligned} \pi (v):=z \quad \text {and}\quad \lambda (v):=n \quad \text {for}\, v=(z,n)\in V^{(n)}; \end{aligned}$$

we call $\lambda (v)$ the level of v. For each $v\in \mathscr {V}$, $\textsf{B}_v:=\textsf{B}[\pi (v);2^{-\lambda (v)}]$ is a closed ball in Z.

Given $v=(z,m)\in \mathscr {V}$ and $n\in \mathbb {Z}_o$ there are vertices

$$\begin{aligned} v_n:=(z_n,n) \quad \text {and}\quad v^p:=v_{n-1} \quad (\text {so}~v_m=v). \end{aligned}$$

Then $\underline{v}:=(v_n)_{n\in \mathbb {Z}_o}$ is a vertex sequence for v associated with the chosen approximating sequence $\underline{z}=(z_n)_{n\in \mathbb {Z}_o}$ for $z:=\pi (v)$, and, we call $v^p$ a parent of v.

Also, $\textbf{o}:=(o,n_o)\in V^{(n_o)}$ is a vertex base point; if Z is unbounded, $\textbf{o}=(o,0)$; if Z is bounded, $\textbf{o}$ is the root of X. The vertex base point has associated vertex sequences $(\textbf{o}_n)_{n\in \mathbb {Z}_o}$ corresponding to approximating sequences $(o_n)_{n\in \mathbb {Z}_o}$ for $o\in Z^{(n_o)}$.

3.1.2 Edges

A pair v, w of distinct vertices in $\mathscr {V}$ have an associated edge $[v,w]\in \mathscr {E}$ if and only if

$$\begin{aligned}{} & {} |\lambda (v)-\lambda (w)|\le 1 \quad \text {and}\quad 2\textsf{B}_v\cap 2\textsf{B}_w\ne \emptyset ;\\{} & {} \text {an edge} \ [v,w] \ \text {has length}\\{} & {} \ell ([v,w]):=1/2^{\lambda (v)\vee \lambda (w)}. \end{aligned}$$

Also, [v, w] is a horizontal edge if $\lambda (v)=\lambda (w)$ and a vertical edge if $|\lambda (v)-\lambda (w)|=1$.⁵ As a simple example, if Z is bounded, then for each $v\in V^{(n_o+1)}$, there is a vertical edge $[\textbf{o},v]\in \mathscr {E}$.

These edges are not directed and there are no single edge loops. As a set, X is the graph with vertex set $\mathscr {V}$ and edge set $\mathscr {E}$. A vertex could be an endpoint of uncountably many edges in which case the metric space X is not locally compact.

3.1.3 Edge Paths

Two vertices $v\ne w$ in $\mathscr {V}$ are adjacent if $[v,w]\in \mathscr {E}$. An edge path is the concatenation of edges $[v_{i-1},v_i]$ where $(v_i)$ is a (finite or infinite) sequence of pairwise adjacent vertices. So, if $v_0,\dots ,v_n\in \mathscr {V}$ are pairwise adjacent, then

The (possibly infinite) length of an edge path is

$$\begin{aligned} \ell (E):=\sum _i \ell (e_i). \end{aligned}$$

The edge path distance between two points a, b of X is defined via

$$\begin{aligned} l(a,b):=\inf _{\gamma :a\curvearrowright b} \ell (\gamma ) \end{aligned}$$

where the infimum is taken over all paths $\gamma$ in X with endpoints a, b.⁶

Below in §3.3 we demonstrate that each pair of vertices can be joined via an edge path; it follows that X is a rectifiably connected metric space.

Here is another important consequence of the existence of edge paths. Suppose $[v_0,\dots ,v_n]$ is an edge path in X. Then $[v_{i-1},v_i]\in \mathscr {E}$, $|\lambda (v_{i-1})-\lambda (v_i)|\le 1$, and

$$\begin{aligned} 2\textsf{B}_{v_{i-1}}\cap 2\textsf{B}_{v_i}=\textsf{B}[\pi (v_{i-1});2/2^{\lambda (v_{i-1})}] \cap \textsf{B}[\pi (v_i);2/2^{\lambda (v_i)}] \ne \emptyset , \end{aligned}$$

so

$$\begin{aligned} |\pi (v_{i-1})-\pi (v_i)|\le \frac{2}{2^{\lambda (v_{i-1})}}+\frac{2}{2^{\lambda (v_i)}} \le 4\ell \bigl ([v_{i-1},v_i]\bigr ) \end{aligned}$$

and therefore

$$\begin{aligned} |\pi (v_0)-\pi (v_n)|\le \sum _{i=1}^n |\pi (v_{i-1})-\pi (v_i)|\le 4\sum _{i=1}^n\ell \bigl ([v_{i-1},v_i]\bigr ) = 4\,\ell \bigl ([v_0,\dots ,v_n]\bigr ). \end{aligned}$$

In particular, we deduce that $\mathscr {V}\xrightarrow {\pi }Z$ is 4-Lipschitz; i.e.,

$$\begin{aligned} \forall \;v,w\in \mathscr {V}, \quad |\pi (v)-\pi (w)|\le 4\,l(v,w). \end{aligned}$$

(3.1)

3.1.4 Elementary Length Estimates

Here are estimates for some simple edge path distances.

• (3.2a) Adjacent edges. If $e,f\in \mathscr {E}$ are adjacent edges, then $\displaystyle \frac{1}{2}\le \frac{\ell (e)}{\ell (f)}\le 2$.

• (3.2b) Lower bounds for vertices. If $v\ne w$ with $m:=\lambda (v)\wedge \lambda (w)$ and $n:=\lambda (v)\vee \lambda (w)$, then

  $$\begin{aligned} l(v,w)\ge {\left\{ \begin{array}{ll} \dfrac{1}{2^m} \quad &{}\text {if}\, m=n,\\ \dfrac{1}{2^m}-\dfrac{1}{2^n} &{}\text {otherwise}; \end{array}\right. } \end{aligned}$$

  equality holds in the first case if and only if $[v,w]\in \mathscr {E}$ and in the second case if and only if there is a purely vertical edge path $v\curvearrowright w$.

• (3.2c) Non-vertex points. Each point $p\in X\setminus \mathscr {V}$ lies in a unique edge $e_p:=[v_p,w_p]$ that we label so that $l(p,\partial e_p)=l(p,v_p)\le l(p,w_p)$. We say that $a,b\in X\setminus \mathscr {V}$ are separated (by an edge) if and only if $e_a\cap e_b=\emptyset$. Note that when

  $$\begin{aligned} e_a=e=e_b \implies l(a,b)=\ell (e[a,b]), \quad \text {so}\quad [a,b]_l=e[a,b]; \end{aligned}$$

  and when

  $$\begin{aligned} e_a\ne e_b{} & {} \implies l(a,v_a)+l(e_a,e_b)+l(b,v_b) \le l(a,b) \le l(a,w_a)\\{} & {} \qquad \quad +\, l(e_a,e_b)+l(b,w_b). \end{aligned}$$

3.1.5 Pushing Edges

Here we explain how to push an edge, or edge path, so as to decrease its total level (equivalently, to push it away from $\partial X$).⁷ Let $u,v,w\in \mathscr {V}$ with $n:=\lambda (w)=\lambda (v)>\lambda (u)$. Let $z:=\pi (v)$ and pick any parent $v^p=(z_{n-1},n-1)\in V^{(n-1)}$, so $|z_{n-1}-z|<1/2^{n-1}$. Then

$$\begin{aligned} x\in 2\textsf{B}_v \implies |x-z_{n-1}|\le |x-z|+|z-z| < \frac{2}{2^n}+\frac{1}{2^{n-1}}=\frac{2}{2^{n-1}}, \quad \text {so}\, x\in 2\textsf{B}_{v^p}. \end{aligned}$$

It follows that we can push any horizontal edge to a vertical edge

$$\begin{aligned}&{[}v,w]\in \mathscr {E}\implies [v^p,w]\in \mathscr {E} \end{aligned}$$

(3.3a)

and we can push any vertical edge to a horizontal edge

$$\begin{aligned}&{[}u,v]\in \mathscr {E}\implies [u,v^p]\in \mathscr {E}. \end{aligned}$$

(3.3b)

3.2 Boundary of X

Here we describe a bi-Lipschitz embedding $Z\xrightarrow {\varphi }\partial X$. We begin by constructing some simple, but all important, purely vertical edge paths. Then we demonstrate that boundary points can always be joined by a finite length edge path.

3.2.1 Purely vertical paths $\Lambda$

Let $z\in Z$, select any approximating sequence $\underline{z}:=(z_n)_{n\in \mathbb {Z}_o}$, and let $(v_n)_{n\in \mathbb {Z}_o}$ be the associated vertex sequence. Then $z\in \textsf{B}(z_n;1/2^n)\cap \textsf{B}(z_{n+1};1/2^{n+1})$, so there are edges $[v_n,v_{n+1}]\in \mathscr {E}$ and

is a purely vertical edge path in X. For each $m,n\in \mathbb {Z}_o$ with $m<n$, the subpath

has length

$$\begin{aligned} \ell \bigl (\Lambda [v_m,v_n]\bigr )=\sum _{m<i\le n} \frac{1}{2^{i}}=\frac{1}{2^{m}}-\frac{1}{2^{n}}. \end{aligned}$$

It follows that $\Lambda$ is locally rectifiable, even rectifiable when Z is bounded, and, $\Lambda$ has an endpoint in $\partial X$. Indeed, $(v_n)_{n\in \mathbb {Z}_o}$ is l-Cauchy, but not convergent in X, so there exists a point $\zeta \in \partial X$ such that

$$\begin{aligned} l(v_n,\zeta )\rightarrow 0 \quad \text {as }\; n\rightarrow \!+\infty ; \quad \text {evidently,}\quad l(v_n,\zeta )=\ell \bigl (\Lambda [v_n,\zeta )\bigr )=\frac{1}{2^n}. \end{aligned}$$

Moreover, $\zeta$ is independent of the choice of $\underline{z}$, so we can define a map

$$\begin{aligned} Z\xrightarrow {\varphi }\partial X\quad \text {given by}\quad \varphi (z):=\zeta . \end{aligned}$$

We see below (in §3.2.3, (3.4b) and (3.10c)) that $\varphi$ is bi-Lipschitz and that the purely vertical edge path $\Lambda _\zeta :=\Lambda$ (which depends on $\underline{z}$) is both an l-geodesic and a k-geodesic.

• (3.4a) Vertical paths and base points. The base point $o\in Z$ corresponds to the vertex base point $\textbf{o}=(o,n_o)$ and to the boundary base point $\omega :=\varphi (o)\in \partial X$. There are purely vertical edge paths $\Lambda _\omega$ constructed from vertex sequences $(\textbf{o}_n)_{n\in \mathbb {Z}_o}$ where $\textbf{o}_n:=(o_n,n)$ and $(o_n)_{n\in \mathbb {Z}_o}$ is any approximating sequence for o in Z.

• (3.4b) Vertical geodesics. An easy consequence of (3.2b) is that any purely vertical edge path $\Lambda _\zeta$ has the following properties; see also (3.10c).

  • (i) Each compact subpath of $\Lambda _\zeta$ is an l-geodesic segment.

  • (ii) If Z is bounded, $\Lambda _\zeta \cup \{{\zeta }\}$ is an l-geodesic segment in $X\cup \{{\zeta }\}$ with endpoints $\textbf{o},\zeta$.

  • (iii) If Z is unbounded, $\Lambda _\zeta \cup \{{\zeta }\}$ is an l-geodesic ray in $X\cup \{{\zeta }\}$ with initial point $\zeta$.

3.2.2 The functions $\mu ,\nu ,n,m$

Here we define $Z\times Z\xrightarrow {\mu ,m}\dot{\mathbb {Z}}$ and $Z\xrightarrow {\nu ,n}\dot{\mathbb {Z}}$. Let $x,y,z\in Z$; recall that $|z|:=|z-o|$. Put $\xi :=\varphi (x), \eta :=\varphi (y), \zeta :=\varphi (z)\in \partial X$. When $x=y$, we set $\mu (x,y):=\infty$; otherwise, if $x\ne y$, $\mu (x,y)$ is the unique $\mu \in \mathbb {Z}$ with

$$\begin{aligned} \frac{1}{2^{\mu +1}} < |x-y| \le \frac{1}{2^\mu }. \end{aligned}$$

Then $\nu (z):=\mu (z,o)$, so when $z\ne o$, $2^{-(\nu (z)+1)}<|z|\le 2^{-\nu (z)}$. Next,

$$\begin{aligned} n(z):=0\wedge \nu (z) \quad \text {and}\quad m(x,y):=n(x)\vee n(y)=0\wedge \bigl (\nu (x)\vee \nu (y)\bigr ). \end{aligned}$$

We think of $\mu :=\mu (x,y)$ as the level where $\Lambda _\xi$ and $\Lambda _\eta$ get “close” in the sense that there is a horizontal edge $[u_\mu ,w_\mu ]\in \mathscr {E}$ that joins $\Lambda _\xi ,\Lambda _\eta$. Similarly, we think of $\nu (z)$ as the level where $\Lambda _\zeta$ and $\Lambda _\omega$ get “close”. Then $\nu (x)\vee \nu (y)$ tells us which of $\Lambda _\xi ,\Lambda _\eta$ is “closest” to $\Lambda _\omega$, but see (3.5e). The significance of n and m is illustrated in (3.13), (3.15), (3.18).

Below we collect basic information regarding $\mu$ and $\nu$. Here $x,y,z\in Z$; $\mu :=\mu (x,y),\nu :=\nu (z)$; and $(u_n),(w_n)$ are vertex sequences associated with approximating sequences $\underline{x},\underline{y}$.

$$\begin{aligned}&\forall \; n\le \mu , \quad \exists \;[u_n,w_n]\in \mathscr {E} \end{aligned}$$

(3.5a)

$$\begin{aligned}&\exists \;[u_n,w_n]\in \mathscr {E}\implies n\le \mu +3 \end{aligned}$$

(3.5b)

$$\begin{aligned}&u_n=w_n \implies n\le \mu +1 \end{aligned}$$

(3.5c)

$$\begin{aligned}&\nu (z)\ge 0 \iff |z|\le 1 \quad (\text {and here}\, \exists \;[(z_0,0),\textbf{o}_0]\in \mathscr {E}) \end{aligned}$$

(3.5d)

$$\begin{aligned}&|x|\le |y| \implies \nu (x)\ge \nu (y) \implies |x|<2|y| \end{aligned}$$

(3.5e)

$$\begin{aligned}&\frac{|x|}{2}\le |y|\le 2|x|\implies \nu (x)-1\le \nu (y)\le \nu (x)+1 \end{aligned}$$

(3.5f)

$$\begin{aligned}&\frac{|x|}{2}\le |x-y|\le |x|\implies \nu (x)\le \mu (x,y)\le \nu (x)+1 \end{aligned}$$

(3.5g)

3.2.3 Standard Edge Paths $\Gamma$

Here we construct edge path arcs that join boundary points, verify that $\varphi$ is bi-Lipschitz, and explain why $\partial X=\overline{\varphi (Z)}$.

To start, let $x\ne y$ be fixed points in Z. Put $\xi :=\varphi (x),\eta :=\varphi (y)$. Select approximating sequences $\underline{x}:=(x_n)_{n\in \mathbb {Z}_o}, \underline{y}:=(y_n)_{n\in \mathbb {Z}_o}$ with associated vertex sequences $(u_n)_{n\in \mathbb {Z}_o},(w_n)_{n\in \mathbb {Z}_o}$ in $\mathscr {V}$ respectively. Construct vertical geodesics

with terminal points $\xi ,\eta \in \partial X$ respectively. Let $\mu :=\mu (x,y)$ as in §3.2.2.

To obtain an arc, we define

$$\begin{aligned} \Gamma _{\xi \eta }:={\left\{ \begin{array}{ll} \Lambda _\xi [\xi ,u_{\mu +1}]*\Lambda _\eta [w_{\mu +1},\eta ] &{}\text {if}\, u_{\mu +1}=w_{\mu +1} \\ \Lambda _\xi [\xi ,u_\mu ]*[u_\mu ,w_\mu ]*\Lambda _\eta [w_\mu ,\eta ] &{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

(3.6)

We call $\Gamma _{\xi \eta }$ a standard edge path from $\xi$ to $\eta$⁸ it consists of a purely vertical subarc of $\Lambda _\xi$, possibly a single horizontal edge $[u_\mu ,w_\mu ]$, and a purely vertical subarc of $\Lambda _\eta$. Notice that

$$\begin{aligned} \ell (\Gamma _{\xi \eta })\in \left\{ \frac{1}{2^\mu },\frac{2}{2^\mu },\frac{3}{2^\mu }\right\} \end{aligned}$$

depending on whether $u_{\mu +1}=w_{\mu +1}$ or $u_\mu =w_\mu$ or $u_n\ne w_n\;\forall \;n\ge \mu$; thus

$$\begin{aligned} \ell (\Gamma _{\xi \eta })\le \dfrac{3}{2^\mu }<6|x-y|. \end{aligned}$$

Appealing to (3.1) we see that

$$\begin{aligned} |x-y|=\lim _{n\rightarrow \infty } |x_n-y_n| \le 4 \lim _{n\rightarrow \infty } l(u_n,w_n) = 4\,l(\xi ,\eta ) \end{aligned}$$

and therefore

$$\begin{aligned}{} & {} \frac{1}{4}|x-y|\le l(\xi ,\eta ) \le \ell \bigl (\Gamma _{\xi \eta }\bigr ) \le 6|x-y| \end{aligned}$$

(3.7a)

and also

$$\begin{aligned}{} & {} \ell (\Gamma _{\xi \eta })\le 24\, l(\xi ,\eta ). \end{aligned}$$

(3.7b)

Evidently, (3.7a) says that $Z\xrightarrow {\varphi }\partial X$ is a 6-bi-Lipschitz embedding satisfying

$$\begin{aligned} \forall \;x,y\in Z, \quad \frac{1}{4}|x-y|\le l\bigl (\varphi (x),\varphi (y)\bigr ) \le 6|x-y| \end{aligned}$$

and (3.7b) says that $\Gamma _{\xi \eta }$ is a 24-quasiconvex arc. Thus $\varphi$ has a Lipschitz extension $\bar{\varphi }$ to $\bar{Z}$. Moreover, $\partial X\supset \bar{\varphi }(\bar{Z})$, so X is non-complete and hence a quasihyperbolic metric space.

It remains to explain why $\varphi (Z)$ is dense in $\partial X$. Let $\xi \in \partial X$, so there is an l-Cauchy sequence $(x_n)$ in X with $l(x_n,\xi )\rightarrow 0$. Choose paths $\gamma _n:x_n\curvearrowright \xi$ in $X\cup \{{\xi }\}$ with $\ell (\gamma _n)<2l(x_n,\xi )$. Let $v_n$ be the first point along $\gamma _n$ in $\mathscr {V}$, $z_n:=\pi (v_n)$, and $\zeta _n:=\varphi (z_n)\in \varphi (Z)$. Then $v_n$ is an initial point of some edge $e_n\subset \gamma _n$, so

$$\begin{aligned} \ell (\gamma _n)\ge \ell (e_n)\ge \frac{1}{2^{\lambda (v_n)+1}} \end{aligned}$$

and therefore $l(\zeta _n,\xi )\le l(\zeta _n,v_n)+l(v_n,\xi ) \le 2^{-\lambda (v_n)}+\ell (\gamma _n)\rightarrow 0$ as $n\rightarrow \!+\infty$.

Henceforth, we assume Z is complete, so $\partial X=\varphi (Z)$ is 6-bi-Lipschitz equivalent to $Z=\bar{Z}$.

3.2.4 Distance to $\partial X$

Evidently, each $p\in X$ has a nearest boundary point.

• (3.8a) If $p\in \mathscr {V}$, then

  $$\begin{aligned} \delta (p)=\dfrac{1}{2^{\lambda (p)}}=l\bigl (p,\varphi \circ \pi (p)\bigr ). \end{aligned}$$

• (3.8b) If $p\in [v,w]\in \mathscr {E}$ with $\lambda (v)=\lambda (w)+1$, then

  $$\begin{aligned} \delta (p)=l(p,v)+\delta (v)=l(p,v)+\dfrac{1}{2^{\lambda (v)}} = l\bigl (p,\varphi \circ \pi (v)\bigr ). \end{aligned}$$

• (3.8c) If $p\in [v,w]\in \mathscr {E}$ with $\lambda (v)=\lambda (w)$, then

  $$\begin{aligned} \delta (p)=\min \{{l(p,v)+\delta (v),l(p,w)+\delta (w)}\}=\bigl (l(p,v)\wedge l(p,w)\bigr ) +\frac{1}{2^{\lambda (v)}}. \end{aligned}$$

• (3.8d) If $p\in X\setminus \mathscr {V}$, say $p\in e_p\in \mathscr {E}$, then

  $$\begin{aligned} \ell (e_p)={{\,\mathrm{\textsf{dist}}\,}}_l(e_p,\partial X) \le \delta (p) \le \frac{3}{2}\ell (e_p) \end{aligned}$$

  and

  $$\begin{aligned} l(p,v_p)\le \frac{1}{2}\ell (e_p)=\frac{1}{2}{{\,\mathrm{\textsf{dist}}\,}}_l(e_p,\partial X) = \frac{1}{2}\bigl (\delta (v_p)\wedge \delta (w_p)\bigr ). \end{aligned}$$

3.3 Uniformity of X

Any standard edge path $\Gamma _{\zeta \eta }$ joining boundary points $\zeta ,\eta \in \partial X$ is 24-quasiconvex (see (3.7b)) and easily seen to be a double 1-cone arc. We require similar edge paths between vertices points.

Let $v\ne w$ be vertices in $\mathscr {V}$. Choose vertex sequences $\underline{v},\underline{w}$ associated with approximating sequences $\underline{z},\underline{y}$ for $z:=\pi (v),y:=\pi (w)$. Let $\Lambda _\zeta ,\Lambda _\eta ,\Gamma _{\zeta \eta }$ be, respectively, vertical geodesics with terminal points $\zeta :=\varphi (z),\eta :=\varphi (y)\in \partial X$, and a standard edge path $\zeta \curvearrowright \eta$ in $X\cup \{{\zeta ,\eta }\}$.

If $z=y$, then $v,w\in \Lambda _\zeta$ and we let

$$\begin{aligned} \Gamma _{vw}:=\Lambda _\zeta [v,w] \end{aligned}$$

which is an l-geodesic segment and a double 1-cone arc.

Suppose $z\ne y$, so $\zeta \ne \eta$. Let $\mu :=\mu (y,z)$ as in §3.2.2. Assume $n:=\lambda (v)\ge \lambda (w)=:m$. We can use a subarc of $\Gamma _{\zeta \eta }$ when it contains both v, w; this holds if $m\ge \mu +1$, but need not if $m\le \mu$. However, when $m\le \mu$, there is an edge $[v_m,w_m]\in \mathscr {E}$; see (3.5a). With this in mind, we set⁹

$$\begin{aligned} \Gamma _{vw}{:}{=}{\left\{ \begin{array}{ll} \Gamma _{\zeta \eta }[v,w] &{}\text {if}\, m\ge \mu +1 \\ \Lambda _\zeta [v,v_\mu ]*[v_\mu ,w_\mu ]*\Lambda _\eta [w_\mu ,w] &{}\text {if}\, n\ge \mu \ge m \\ {[}v,w_n]*\Lambda [w_n,w] &{}\text {if}\, \mu \ge n. \end{array}\right. } \end{aligned}$$

As in (3.7), when $\pi (v)\ne \pi (w)$, we have

$$\begin{aligned} l(v,w)\le \ell (\Gamma _{vw})<6|\pi (v)-\pi (w)|\le 24\,l(v,w). \end{aligned}$$

In particular, the map $\mathscr {V}\xrightarrow {\pi }Z$ is 6-bi-Lipschitz on each level set $V^{(n)}$. More precisely, for all $v,w\in \mathscr {V}$

$$\begin{aligned} |\pi (v)-\pi (w)|\le 4l(v,w), \quad \text {and if}\, \pi (v)\ne \pi (w), \quad l(v,w)\le 6|\pi (v)-\pi (w)|. \end{aligned}$$

In all cases we deduce that the standard vertex edge path $\Gamma _{vw}:v\curvearrowright w$ is a 24-quasiconvex double 1-cone arc.

We now see that X is a rectifiably connected non-complete locally complete metric space.

Next, let $a\ne b$ be any two points in X. If both are vertices, then $\Gamma _{ab}$ is a 24-uniform arc. The case where exactly one point is a vertex is left for the diligent reader. Assume both $a,b\in X\setminus \mathscr {V}$.

Let $e_a=[v_a,w_a], e_b=[v_b,w_b]$ be the unique edges containing a, b respectively as in (3.2c). If $e_a\cap e_b\ne \emptyset$, then $e_a\cup e_b$ contains a 3-quasiconvex double 1-cone arc $a\curvearrowright b$.

Assume $e_a\cap e_b=\emptyset$. Then $v_a\ne v_b$ and there is a standard vertex edge path $\Gamma _{v_av_b}:v_a\curvearrowright v_b$ Let

$$\begin{aligned} \gamma :=e_a[a,v_a]*\Gamma _{v_av_b}*e_b[v_b,b]:a\curvearrowright b. \end{aligned}$$

Employing (3.2c) we obtain

$$\begin{aligned} l(v_a,v_b)\le l(a,v_a)+l(a,b)+l(b,v_b) \le 2 l(a,b) \end{aligned}$$

and therefore

$$\begin{aligned} \ell (\gamma )=l(a,v_a)+\ell (\Gamma _{v_av_b})+l(b,v_b) \le l(a,b)+24l(v_a,v_b) \le 49 l(a,b). \end{aligned}$$

It remains to check the double cone condition. If $x\in e_a[a,v_a]$, then by (3.8d)

$$\begin{aligned} \ell (\gamma [x,a])=l(x,a)\le \frac{1}{2}\ell (e_a)=\frac{1}{2}{{\,\mathrm{\textsf{dist}}\,}}_l(e_a,\partial X) \le \delta (x) \end{aligned}$$

and similarly for $x\in e_b[v_b,b]$. Assume $x\in \Gamma :=\Gamma _{v_av_b}$. If

$$\begin{aligned} \ell \bigl (\Gamma [x,v_a])\le \frac{1}{2}\delta (v_a), \quad \text {then}\quad \delta (x)\ge \frac{1}{2}\delta (v_a) \end{aligned}$$

and so by (3.8d)

$$\begin{aligned} \ell (\gamma [x,a])=l(a,v_a)+\ell \bigl (\Gamma [x,v_a]\bigr )\le \delta (v_a)\le 2\delta (x). \end{aligned}$$

Similarly for $\ell \bigl (\Gamma [x,v_b]\bigr ) \le \frac{1}{2}\delta (v_b)$. Suppose

$$\begin{aligned} \ell \bigl (\Gamma [x,v_a]\bigr )\ge \frac{1}{2}\delta (v_a) \quad \text {and}\quad \ell \bigl (\Gamma [x,v_b]\bigr )\ge \frac{1}{2}\delta (v_b). \end{aligned}$$

Then again by (3.8d)

$$\begin{aligned} \ell \bigl (\gamma [x,a]\bigr )=\ell \bigl (\Gamma [x,v_a]\bigr )+l(v_a,a) \le 2\ell \bigl (\Gamma [x,v_a]\bigr ) \end{aligned}$$

and similarly $\ell \bigl (\gamma [x,b]\bigr )\le 2\ell \bigl (\Gamma [x,v_b]\bigr )$. Thus, as $\Gamma$ is a double 1-cone arc,

$$\begin{aligned} \ell \bigl (\gamma [x,a]\bigr )\wedge \ell \bigl (\gamma [x,b]\bigr )\le 2 \ell \bigl (\Gamma [x,v_a]\bigr )\wedge \ell \bigl (\Gamma [x,v_b]\bigr ) \le 2\delta (x). \end{aligned}$$

It now follows that in all cases, $\gamma :a\curvearrowright b$ is a 49-quasiconvex double 2-cone arc.

3.4 Quasihyperbolization of X

As X is a rectifiably connected non-complete locally complete metric space, it supports a quasihyperbolic distance $k=k_X$ as described in §2.1.3. Here we discuss the geodesicity of $X_k$; explain why each $\Lambda$ and $\Gamma$ are, respectively, quasihyperbolic geodesics and quasihyperbolic quasi-geodesic lines; establish estimates for certain distances to each $\Lambda$ and $\Gamma$; and verify the hyperbolicity and visualness of $X_k$.

Armed with the information in (3.8a),(3.8b),(3.8c), it is straightforward to compute the quasihyperbolic lengths of edges and we find that

$$\begin{aligned} \text {each vertical edge has quasihyperbolic length}\,\log 2 \end{aligned}$$

and

$$\begin{aligned} \text {each horizontal edge has quasihyperbolic length}\, 2\log \dfrac{3}{2}. \end{aligned}$$

It now follows that $X_k:=(X,k)$ is a geodesic space; that is, for any two points $a,b\in X$, there is a quasihyperbolic geodesic $a\curvearrowright b$ in X. In fact, all quasihyperbolic geodesics between two given vertex points have exactly the same number $N_v\ge 0$ of vertical edges and exactly the same number $N_h\in \{{0,1}\}$ of horizontal edges.

Moreover, there are always quasihyperbolic geodesics joining vertex points that have minimal total level; see §3.1.5. While not essential for this work, these can be described as follows. Let $v,w\in \mathscr {V}$ be distinct vertex points with $\lambda (v)\ge \lambda (w)$. There is a quasihyperbolic geodesic $[v,w]_k$ that has minimal total level with exactly one of the following holding.

• (3.9a) $[v,w]_k$ is a purely vertical geodesic from v down to w.

• (3.9b) $[v,w]_k=[v,u]_k*[u,w]_k$ is the concatenation of two purely vertical geodesics from v down to u and u up to w, for some $u\in \mathscr {V}\setminus \{{v,w}\}$ with $\lambda (u)<\lambda (w)$.

• (3.9c) $[v,w]_k=[v,u]_k*[u,w]$ is the concatenation of a purely vertical geodesic from v down to u (possibly $u=v$) and one horizontal edge $[u,w]\in \mathscr {E}$, for some $u\in \mathscr {V}\setminus \{{w}\}$ with $\lambda (u)=\lambda (w)$.

• (3.9d) $[v,w]_k=[v,u_1]*[u_1,u_2]*[u_2,w]_k$ is the concatenation of three subarcs: two purely vertical geodesics from v down to $u_1$ and $u_2$ up to w, together with one horizontal edge $[u_1,u_2]\in \mathscr {E}$, for some $u_1,u_2\in \mathscr {V}\setminus \{{v,w}\}$ with $\lambda (u_1)=\lambda (u_2)<\lambda (w)$.

This could prove useful in the future. Also, these minimal total level quasihyperbolic geodesics are easily seen to be 24-quasiconvex double 1-cone arcs; these constants vastly improve those given by (2.4b).

3.4.1 Quasihyperbolic geometry of X

Here we collect some information about the quasihyperbolic geometry of X; see also §3.4.2, §3.4.3.

• (3.10a) Simple estimates for k(v, w). Note that for distinct vertices $v\ne w$ in $\mathscr {V}$ we have

  $$\begin{aligned}{} & {} \lambda (v)\ne \lambda (w) \implies k(v,w)\ge \log 2 \\{} & {} \qquad \text {with equality if and only if }\, [v,w]\, \text {is a vertical edge in}\, \mathscr {E}\ \text {and}\\{} & {} \lambda (v)=\lambda (w) \implies k(v,w)\ge 2\log \frac{3}{2} \quad \text {with equality if and only if }\, [v,w]\\{} & {} \qquad \text { is a horizontal edge in}\, \mathscr {E}\ \text {and}\\{} & {} \lambda (v)>\lambda (w) \implies k(v,w)\ge \bigl (\lambda (v)-\lambda (w)\bigr )\log 2 \end{aligned}$$

  with equality if and only if v, w both lie in some purely vertical edge path (in which case $[v,w]_k$ is the subarc of this edge path with endpoints v, w:-).

• (3.10b) Simple estimates for k(a, b). Let $a\ne b$ be distinct points in X with $k(a,b)<\log 2$. Then a, b are not the endpoints of any edge and are not separated by any edge; in fact,

  $$\begin{aligned}{} & {} e_a=e=e_b \implies [a,b]_k=e[a,b] \quad \text {and}\\{} & {} e_a\cap e_b=\{{u}\} \implies [a,b]_k=e_a[a,u]*e_b[u,b]. \end{aligned}$$

• (3.10c) Vertical edge paths are quasihyperbolic geodesics. Let $\Lambda _\zeta$ be any purely vertical edge path in X with terminal point $\zeta \in \partial X$ (as described in §3.2.1).

  • (i) Each compact subpath of $\Lambda _\zeta$ is a quasihyperbolic geodesic segment.

  • (ii) If Z is bounded, $\Lambda _\zeta$ is a quasihyperbolic geodesic ray in X from $\textbf{o}$ to $\zeta$.

  • (iii) If Z is unbounded, $\Lambda _\zeta$ is a quasihyperbolic geodesic line in X with ‘endpoints’ $\zeta ,\infty$.

• (3.10d) Standard edge paths are quasihyperbolic quasi-geodesics. Let $\Gamma :=\Gamma _{\xi \eta }$ be any standard edge path $\xi \curvearrowright \eta$ with endpoints in $\partial X$ (as defined in §3.2.3). Then $\Gamma$ is a quasihyperbolic quasi-geodesic line in X satisfying

  $$\begin{aligned} \forall \;a,b\in \Gamma , \quad \ell _k\bigl (\Gamma [a,b]\bigr )\le 240\,k(a,b). \end{aligned}$$

Proof of (3.10d)

Let $a,b\in \Gamma$. The asserted inequality is clear if a, b both lie in one of $\Lambda _\xi [\xi ,u_\mu ]$ or $[u_\mu ,w_\mu ]$ or $\Lambda _\eta [w_\mu ,\eta ]$ (where $\mu =\mu (x,y)$ and $\Lambda _\xi ,\Lambda _\eta$ are constructed using $(u_n),(w_n)$).

From (3.10b) it is clear that when $k(a,b)<\log 2$, $\Gamma [a,b]=[a,b]_k$ is a quasihyperbolic geodesic. Suppose $k(a,b)\ge \log 2$.

Move along $[a,b]_k$ from a to b (respectively, from b to a) and let u (respectively, w) be the first vertex encountered. Assume $u\in \Lambda _\xi , w\in \Lambda _\eta$ with $n:=\lambda (u)\ge \lambda (w)=:m\ge \mu$. By (3.2b),

$$\begin{aligned} l(u,w)\ge \frac{1}{2^{m+1}} \quad \text {so}\quad l(u,\xi )=\frac{1}{2^n}\le 2\,l(u,w) \quad \text {and}\quad l(w,\eta )=\frac{1}{2^m}\le 2\,l(u,w) \end{aligned}$$

whence

$$\begin{aligned} l(\xi ,\eta )\le l(\xi ,u)+l(u,w)+l(w,\eta ) \le 5\, l(u,w) \end{aligned}$$

and therefore by (3.7b)

$$\begin{aligned} \ell \bigl (\Gamma [u,w]\bigr )\le \ell (\Gamma ) \le 24\,l(\xi ,\eta ) \le 120\,l(u,w). \end{aligned}$$

Now we examine the three cases depending on the exact definition of $\Gamma$ as given in §3.2.3; we find that

$$\begin{aligned} \ell (\Gamma ) \in \left\{ \frac{1}{2^\mu },\frac{2}{2^\mu },\frac{3}{2^\mu }\right\} . \end{aligned}$$

In each case we readily compute $\ell _k\bigl (\Gamma [u,w]\bigr )$ and deduce that

$$\begin{aligned} k(u,w)\ge \log \Bigl (1+\frac{l(u,w)}{\delta (u)\wedge \delta (w)}\Bigr ) \ge \frac{1}{120} \log \Bigl (1+2^n\ell (\Gamma )\Bigr ) \ge \frac{1}{240}\ell _k\bigl (\Gamma [u,w]\bigr ). \end{aligned}$$

It now follows that

$$\begin{aligned} \ell _k\bigl (\Gamma [a,b]\bigr )&= \ell _k\bigl (\Gamma [a,u]\bigr ) + \ell _k\bigl (\Gamma [u,w]\bigr ) + \ell _k\bigl (\Gamma [w,b]\bigr ) \\&\le k(a,u)+240\,k(u,w)+k(w,b) \le 240\,k(a,b). \end{aligned}$$

$\square$

3.4.2 Distances to $\Lambda _\zeta$

Here we estimate the edge path and quasihyperbolic distances from the vertex basepoint $\textbf{o}$ to any purely vertical edge path $\Lambda _\zeta$; see (3.11) and (3.13). When Z is bounded every $\Lambda _\zeta$ is an l-geodesic $\textbf{o}\curvearrowright \zeta$ (see (3.4b)(ii)), so throughout this subsubsection we assume Z is not bounded and let $(\textbf{o}_n)$ be any vertex sequence as in (3.4a).

Let $z\in Z$, put $\zeta :=\varphi (z)$, and let $\Lambda :=\Lambda _\zeta$ be constructed as in §3.2.1 using a vertex sequence $(v_n)$ obtained from any approximating sequence $\underline{z}=(z_n)$ for z.

We require estimates for $l(\textbf{o},\Lambda )={{\,\mathrm{\textsf{dist}}\,}}_l(\textbf{o},\Lambda )$. Evidently, $l(\textbf{o},\Lambda )=0$ if and only if $\textbf{o}\in \Lambda$ which holds if and only if $v_0=\textbf{o}$ or equivalently if and only if $z_0=o$, which in turn implies that $|z|<1$. If $z=o$, then $z_0=o$ and $l(\textbf{o},\Lambda )=0$.

We assume $l(\textbf{o},\Lambda )\ne 0$, so $\textbf{o}\not \in \Lambda$ and $z_0\ne o\ne z$. We demonstrate that

$$\begin{aligned}&l(\textbf{o},\Lambda )\ge (1/2)\vee (|z|/16){} & {} \text {always,} \end{aligned}$$

(3.11a)

$$\begin{aligned}&l(\textbf{o},\Lambda )\le 1{} & {} \text {if}\, |z|\le 1, \end{aligned}$$

(3.11b)

$$\begin{aligned}&l(\textbf{o},\Lambda )\le 4\,|z|{} & {} \text {if}\, |z|\ge 1, \end{aligned}$$

(3.11c)

$$\begin{aligned}&l(\textbf{o},\Lambda )\ge |z|/8{} & {} \text {if}\, |z|>8. \end{aligned}$$

(3.11d)

Since each edge from $\textbf{o}$ has length at least 1/2, (3.11a) holds if $|z|\le 8$, and also when $|z|>8$ by (3.11d). When $|z|\le 1$, $\nu :=\nu (z)=\mu (z,o)\ge 0$, so by (3.5a), there is an edge $[v_0,\textbf{o}]\in \mathscr {E}$ and thus

$$\begin{aligned} l(\textbf{o},\Lambda )\le l(\textbf{o},v_0)= \ell \bigl ([v_0,\textbf{o}]\bigr )=1 \end{aligned}$$

which gives (3.11b). When $|z|\ge 1$, $\nu \le 0$, (3.5a) says there is an edge $[v_\nu ,\textbf{o}_\nu ]\in \mathscr {E}$, and now

$$\begin{aligned} l(\textbf{o},\Lambda )\le l(\textbf{o},v_\nu )\le l(\textbf{o},\textbf{o}_\nu )+l(\textbf{o}_\nu ,v_\nu ) = \Bigr (\frac{1}{2^\nu }-1\Bigr )+\frac{1}{2^\nu }< \frac{4}{2^{\nu +1}} < 4\,|z| \end{aligned}$$

so (3.11c) holds.

Suppose $|z|>8$. Then $1/2^\nu \ge |z|>8$, so $\nu \le -2$. Let $v\in \Lambda \cap \mathscr {V}$. If $n:=\lambda (v)\le \nu +1$, then

$$\begin{aligned} l(\textbf{o},v) \ge \frac{1}{2^n}-1 \ge \frac{1}{2^{\nu +1}}-1 \ge \frac{1}{2}\,|z|-1 \ge \frac{3}{8}\,|z|. \end{aligned}$$

On the other hand, if $n\ge \nu +2$, then

$$\begin{aligned} |z_n-z|<\frac{1}{2^n}\le \frac{1}{2}\cdot \frac{1}{2^{\nu +1}} < \frac{1}{2}\,|z| \end{aligned}$$

so $|z_n|\ge |z|-|z-z_n|>\frac{1}{2}|z|$ and therefore by (3.1)

$$\begin{aligned} l(\textbf{o},v)\ge \frac{1}{4}|\pi (\textbf{o})-\pi (v)|=\frac{1}{4}|z_n|>\frac{1}{8}\,|z|. \end{aligned}$$

As every path $\textbf{o}\curvearrowright \Lambda$ must pass through some $v\in \Lambda \cap \mathscr {V}$, (3.11d) follows.

The following estimate is used in the proofs of both (3.13) and (3.15).

Lemma 3.12

Let $\eta :=\varphi (y),\zeta :=\varphi (z)\in \partial X$ be given. Suppose $\nu (y)\le \nu (z)\le 0$. Let $w\in \mathscr {V}\cap \bigl (\Lambda _\eta \cup \Lambda _\zeta )$ and let u be the $\ell$-midpoint of some $[\textbf{o},w]_k$. Then

$$\begin{aligned} k(u,v_{\nu (z)})\simeq 0. \end{aligned}$$

Proof

We demonstrate that $k(u,v_{\nu (z)})\le C$ for some absolute constant C. We assume $[\textbf{o},w]_k$ is a minimal total level quasihyperbolic geodesic. Then $[\textbf{o},w]_k$ is a 24-quasiconvex double 1-cone arc,¹⁰ so $\delta (u)\ge \ell \bigl ([\textbf{o},w]_k\bigr )/2$.

By (3.5e), $|z|<2|y|$, so (3.11a) yields

$$\begin{aligned} \ell \bigl ([\textbf{o},w]_k\bigr )\ge l(\textbf{o},\Lambda _\eta )\wedge l(\textbf{o},\Lambda _\zeta ) \ge \frac{|y|}{16} \wedge \frac{|z|}{16} \ge \frac{|z|}{32}. \end{aligned}$$

Since $\nu =\nu (z)=\mu (z,o)$ (3.5a) asserts there is an edge $[v_\nu ,\textbf{o}_\nu ]\in \mathscr {E}$, so

$$\begin{aligned} l(v_\nu ,\textbf{o})\le l(v_\nu ,\textbf{o}_\nu )+l(\textbf{o}_\nu ,\textbf{o}) = \frac{1}{2^\nu }+\Bigl (\frac{1}{2^\nu }-1\Bigr )< \frac{4}{2^{\nu +1}} < 4|z| \le 128\, \ell \bigl ([\textbf{o},w]_k\bigr ). \end{aligned}$$

Thus

$$\begin{aligned} l(u,v_\nu )\le l(u,\textbf{o})+l(v_\nu ,\textbf{o}) \le 129\,\ell \bigl ([\textbf{o},w]_k\bigr ) \end{aligned}$$

and therefore by (2.4a)

$$\begin{aligned} k(u,v_\nu )\le 4A^2\log \Bigl (1+\frac{l(u,v_\nu )}{\delta (u)}\Bigr ) \le 4A^2\log 259=:C. \end{aligned}$$

Evidently, $A=49$ is an absolute constant, so C is too. $\square$

Now we verify the rough estimate

$$\begin{aligned} \forall \;\zeta =\varphi (z)\in \partial X, \quad k(\textbf{o},\Lambda _\zeta ) {{\,\mathrm{\simeq }\,}}-n(z)\log 2. \end{aligned}$$

(3.13)

If $\textbf{o}\in \Lambda :=\Lambda _\zeta$, $z_0=o$, so $|z|<1, \nu (z)\ge 0, n(z)=0$, and $k(\textbf{o},\Lambda )=-n(z)\log 2$. Assume $\textbf{o}\not \in \Lambda$.

When $\nu :=\nu (z)\ge 0$ (equivalently, $|z|\le 1$), there is an edge $[\textbf{o},v_0]\in \mathscr {E}$. Since $v_0\in \Lambda$ and $n(z)=0$,

$$\begin{aligned} -n(z)\log 2=0\le k(\textbf{o},\Lambda ) \le k(\textbf{o},v_0)= 2\log 3/2=:C_0. \end{aligned}$$

Suppose $\nu <0$. Here $n:=n(z)=\nu =\mu (z,o)$, so by (3.5a) there exists an edge $[\textbf{o}_n,v_n]\in \mathscr {E}$ and

$$\begin{aligned} k(\textbf{o},\Lambda ) \le k(\textbf{o},v_n) \le k(\textbf{o},\textbf{o}_n) +k(\textbf{o}_n,v_n) = -n\cdot \log 2+C_0. \end{aligned}$$

Establishing a lower bound requires more effort. Fix $w\in \Lambda \cap \mathscr {V}$. Let u be the $\ell$-midpoint of the quasihyperbolic segment $[\textbf{o},w]_k$. According to Lemma 3.12, $k(u,v_n)\le C_1$ for an absolute constant $C_1$. So,

$$\begin{aligned} k(\textbf{o},w)\ge k(\textbf{o},u)\ge k(\textbf{o},\textbf{o}_n) - k(\textbf{o}_n,v_n) - k(v_n,u) \ge -n\cdot \log 2-C_0-C_1. \end{aligned}$$

As every path $\textbf{o}\curvearrowright \Lambda$ must pass through some $w\in \Lambda \cap \mathscr {V}$, it now follows that

$$\begin{aligned} -n\cdot \log 2 -C_0-C_1 \le k(\textbf{o},\Lambda ) \le -n\cdot \log 2+C_0 \end{aligned}$$

which establishes (3.13).

A careful reading of the above argument reveals that in all cases we have

$$\begin{aligned} k(\textbf{o},\Lambda _\zeta ) \simeq k(\textbf{o},v_{n(z)}) \quad \text {where}\, \zeta :=\varphi (z)\in \partial X. \end{aligned}$$

(3.14)

3.4.3 Distance to $\Gamma _{\zeta \eta }$

Here we establish our first estimate (see also (3.18)) for the quasihyperbolic distance from the vertex basepoint $\textbf{o}$ to any $\Gamma _{\zeta \eta }$:

$$\begin{aligned} \forall \;\eta :=\varphi (y),\zeta :=\varphi (z)\in \partial X, \quad k(\textbf{o},\Gamma _{\zeta \eta }) \simeq {\left\{ \begin{array}{ll} -m\cdot \log 2 &{}\text {if}\, \mu \le m\\ (\mu -2m)\log 2 &{}\text {if}\, \mu \ge m \end{array}\right. } \end{aligned}$$

(3.15)

where $\mu :=\mu (y,z)$ and $m:=m(y,z)$ are defined as in §3.2.2.

Proof of (3.15)

Let $\Gamma :=\Gamma _{\zeta \eta }$ as defined in (3.6). We assume $\nu (z)\ge \nu (y)$, so $m=\nu (z)\wedge 0=n(z)$. Since $m\le \nu (z)=\mu (z,o)$, (3.5a) asserts that there exist an edge $[\textbf{o}_m,v_m]\in \mathscr {E}$.

Suppose $\mu <m$; this is the easy case. Here $v_{m}\in \Gamma$, so

$$\begin{aligned} k(\textbf{o},\Gamma )\le k(\textbf{o},v_{m})\le k(\textbf{o},\textbf{o}_m)+k(\textbf{o}_m,v_m))=-m\cdot \log 2+2\log (3/2). \end{aligned}$$

We’re done if $m=0$, so assume $m<0$. Let $w\in \Gamma \cap \mathscr {V}$. By (3.13), if $w\in \Lambda _\zeta$, then

$$\begin{aligned} k(\textbf{o},w)\ge k(\textbf{o},\Lambda _\zeta )\simeq -n(z)\log 2 = -m\cdot \log 2 \end{aligned}$$

and if $w\in \Lambda _\eta$, then, as $n(y)\le n(z)=m$,

$$\begin{aligned} k(\textbf{o},w)\ge k(\textbf{o},\Lambda _\eta )\simeq -n(y)\log 2 \ge -m\cdot \log 2. \end{aligned}$$

Since every path $\textbf{o}\curvearrowright \Gamma$ must pass through some $w\in \Gamma \cap \mathscr {V}$, (3.15) now follows.

Suppose $\mu \ge m$; this case requires more effort. Since $v_{\mu +1}\in \Gamma$,¹¹

$$\begin{aligned} k(\textbf{o},\Gamma )&\le k(\textbf{o},v_{\mu +1}) \le k(\textbf{o},\textbf{o}_m)+k(\textbf{o}_m,v_m)+k(v_m,v_{\mu +1}) \\&=-m\cdot \log +2\log (3/2)+(\mu +1-m)\log 2 \\&=(\mu -2m)\log 2+2\log 3/2+\log 2. \end{aligned}$$

It remains to show that $k(\textbf{o},\Gamma )\ge (\mu -2m)\log 2-C$ for some absolute constant C.

Let $w\in \Gamma \cap \mathscr {V}$; so $\lambda (w)\ge \mu$. If $m=0$, then $\lambda (w)\ge \mu \ge m=0=\lambda (\textbf{o})$, so by (3.10a),

$$\begin{aligned} k(\textbf{o},w)\ge \mu \log 2=(\mu -2m)\log 2. \end{aligned}$$

Assume $m<0$; then $m=\nu (z)<0$. Let u be the $\ell$-midpoint of the quasihyperbolic segment $[\textbf{o},w]_k$. According to Lemma 3.12, $k(u,v_m)\simeq 0$ with an absolute constant. Thus

$$\begin{aligned} k(\textbf{o},u)\ge k(\textbf{o},v_m) - k(u,v_m) \simeq k(\textbf{o},v_m) \ge -m\cdot \log 2 \end{aligned}$$

and

$$\begin{aligned} k(u,w)\ge k(v_m,w)-k(v_m,u) \simeq k(v_m,w) \ge (\mu -m)\log 2 \end{aligned}$$

whence

$$\begin{aligned} k(\textbf{o},w)=k(\textbf{o},u)+k(u,w)\ge (\mu -2m)\log 2 - C. \end{aligned}$$

As every path $\textbf{o}\curvearrowright \Lambda$ must pass through some $w\in \Lambda \cap \mathscr {V}$, it now follows that

$$\begin{aligned} k(\textbf{o},\Lambda )\ge (\mu -2m)\log 2 - C \end{aligned}$$

where C is some absolute constant. $\square$

A careful reading of the above argument reveals that in all cases we have

$$\begin{aligned} k(\textbf{o},\Gamma _{\zeta \eta }) \simeq k(\textbf{o},w) \quad \text {where}\, w\in \Gamma _{\zeta \eta }\,\text { is}\quad w:={\left\{ \begin{array}{ll} v_m &{}\text {when}\, \mu < m\\ v_{\mu +1} &{}\text {when}\, \mu \ge m \end{array}\right. } \end{aligned}$$

(3.16)

and we are still assuming $\eta :=\varphi (y),\zeta :=\varphi (z), \mu :=\mu (y,z), m:=m(y,z)$ and $\nu (z)\ge \nu (y)$.

3.4.4 Hyperbolicity and Visualness of $X_k$

Since $X_k$ is geodesic, we can appeal to Theorem 2.5 to assert that $X_k$ is Gromov hyperbolic; also, the Gromov constant $\theta =\theta (A)$ depends only on the uniformity constant $A:=49$.

Moreover, there is a natural bijection $\partial _G X\xrightarrow {\iota }\hat{\partial } X$. To simplify notation, we write $\iota (\zeta )=\zeta$; that is, we employ $\zeta \in \hat{\partial } X$ to denote a point in $\partial _G X$ understanding this as described in the last part of Theorem 2.5. In particular, when Z is unbounded, there are three points $\infty$ in each of $\hat{Z},\hat{\partial } X,\partial _G X$ that correspond to each other via the maps $\varphi$ and $\iota$.

If we think of $\partial _G X$ as equivalence classes of quasihyperbolic quasi-geodesic rays, then the endpoint $\zeta =\varphi (z)\in \partial X$ of a purely vertical $\Lambda _\zeta$ corresponds to the point in $\partial _G X$ given by the equivalence class of $\Lambda _\zeta [v_0,\zeta )$, and when Z is unbounded, the “other endpoint” $\infty \in \dot{\partial }X$ of $\Lambda _\zeta$ corresponds to the point $\infty$ in $\partial _G X$ given by the equivalence class of $\Lambda _\zeta [v_0,\infty )$. See (3.10c).

Now we demonstrate that $X_k$ is visual with an absolute constant K. It suffices to establish the visual property for each vertex $v\in \mathscr {V}$. We exhibit a point $\xi \in \partial _G X$ with $k(v,\textbf{o})\le \bigl (v\mid \xi \bigr )_\textbf{o}+K$. If Z bounded, each vertex lies on a quasihyperbolic geodesic ray emanating from $\textbf{o}$, so we may assume that Z is unbounded.

Let $z:=\pi (v),\zeta :=\varphi (z),\Lambda :=\Lambda _\zeta ,v':=v_{n(z)}$, and define

$$\begin{aligned} \Upsilon :=[\textbf{o},v']_k*\Lambda [v',\xi ) \quad \text {where}\quad \xi :={\left\{ \begin{array}{ll} \zeta &{}\text {if}\, \lambda (v)\ge n(z), \\ \infty &{}\text {if}\, \lambda (v)<n(z); \end{array}\right. } \end{aligned}$$

so, $v\in \Lambda [v',\xi )\subset \Upsilon$. Below we prove that $\Upsilon$ is a quasihyperbolic rough-quasi-geodesic ray (with absolute constants). Given this, we note that for all $u\in \Upsilon (v,\xi )$,

$$\begin{aligned} k(v,\textbf{o})\le \bigl (v\mid u\bigr )_\textbf{o}+K=\frac{1}{2}\bigl (k(v,\textbf{o})+k(u,\textbf{o})-k(v,u)\bigr )+K \end{aligned}$$

if and only if

$$\begin{aligned} \bigl (\textbf{o}\mid u\bigr )_v=\frac{1}{2}\bigl (k(v,\textbf{o})+k(u,v)-k(u,\textbf{o})\bigr )\le K. \end{aligned}$$

But, as $X_k$ is geodesic and hyperbolic and $v\in \Upsilon [\textbf{o},u]$, geodesic stability tells us that

$$\begin{aligned} \bigl (\textbf{o}\mid u\bigr )_v {{\,\mathrm{\simeq }\,}}k\bigl (v,[\textbf{o},u]_k\bigr ) {{\,\mathrm{\simeq }\,}}k\bigl (v,\Upsilon [\textbf{o},u]\bigr )=0. \end{aligned}$$

Employing [21, Lemma 5.11] we now deduce that

$$\begin{aligned} \bigl (v\mid \xi \bigr )_\textbf{o}+\theta \ge \limsup _{\Upsilon \ni u\rightarrow \xi }\,\bigl (v\mid u\bigr )_\textbf{o}\ge k(v,\textbf{o})-K. \end{aligned}$$

To see that $\Upsilon$ is a quasihyperbolic rough-quasi-geodesic, we verify the quasihyperbolic rough-chordarc property for $\Upsilon$. As $\Upsilon$ is the concatenation of two k-geodesics, we need only find constants L, C such that for all $a\in [\textbf{o},v']_k$ and $b\in \Lambda [v',\xi )$,

$$\begin{aligned} k(a,v')+k(v',b)=\ell _k\bigl (\Upsilon [a,b]\bigr )\le L\,k(a,b)+C. \end{aligned}$$

Suppose $n(z)=0$. Then $v'=(z_0,0)$, $\exists \;[\textbf{o},v']\in \mathscr {E}$, and $k(a,v')\le C_0:=2\log (3/2)$, so

$$\begin{aligned} k(a,v')+k(v',b)\le 2k(a,v')+k(a,b)\le k(a,b)+2C_0. \end{aligned}$$

Suppose $n(z)<0$. From (3.14) we have an absolute $C_1$ such that

$$\begin{aligned} k(\textbf{o},a)+k(a,v'){} & {} =k(\textbf{o},v')\le k(\textbf{o},\Lambda )+C_1 \le k(\textbf{o},a)+k(a,\Lambda )+C_1\\{} & {} \le k(\textbf{o},a)+k(a,b)+C_1 \end{aligned}$$

so $k(a,v')\le k(a,b)+C_1$ and therefore

$$\begin{aligned} k(a,v')+k(v',b)\le 2k(a,v')+k(a,b)\le 3k(a,b)+2C_1. \end{aligned}$$

3.5 QuasiMöbius Equivalence

Here we corroborate that the canonical gauge on $\partial _G X$ is naturally quasimöbius equivalent to the conformal gauge on $\hat{Z}$. Recall that the latter is quasimöbius equivalent to the conformal gauge on $\bar{Z}=Z$.¹²

Since $X_k=(X,k)$ is our geodesic Gromov hyperbolic space, we write $k_\varepsilon =k_{\varepsilon ,\textbf{o}}$ for a standard visual distance on $\partial _G X$; here $\varepsilon \in (0,\varepsilon _0]$ is a fixed visual parameter and $\textbf{o}$ is our fixed vertex base point. As in §3.4.4, we employ Theorem 2.5 to identify $\hat{Z},\hat{\partial } X,\partial _G X$ via the maps $\varphi$ and $\iota$.

Standard considerations (e.g., see [21, Lemma 5.11]) reveal that for all $\xi ,\eta \in \partial _G X$

$$\begin{aligned} k_\varepsilon (\xi ,\eta ) {{\,\mathrm{\approx }\,}}\exp \bigl (-\varepsilon (\xi \vert \eta )_\textbf{o}\bigr ) {{\,\mathrm{\approx }\,}}\exp \bigl (-\varepsilon \, k(\textbf{o},\Gamma _{\xi \eta })\bigr ) \end{aligned}$$

where $(\xi \vert \eta )_\textbf{o}$ is the usual Gromov product, all constants are absolute, and we employ the crucial fact that $\Gamma _{\xi \eta }$ is a quasihyperbolic quasi-geodesic line as verified in (3.10d).

If Z is bounded, X is too, and $X_k$ is $\log 3/2$-roughly starlike with respect to $\textbf{o}$. Here we can mimic the argument for the last part of [2, Theorem 3.6]; see pp. 24–25 therein.

We assume that Z is unbounded, so X is also unbounded. It would be convenient to be able to use the results in [6] to reduce the unbounded case to the bounded case; e.g., as done in [14].¹³ The lack of local compactness in our setting requires an alternate approach.

By using continuity of cross ratios, it suffices to consider distinct points $z,y,x,t\in Z$, say, with $\varphi$-images $\zeta ,\eta ,\xi ,\vartheta \in \partial X=\partial _G X$. We have absolute cross ratios

$$\begin{aligned} r:=|x,z,t,y|:=\frac{|x-z||t-y|}{|x-t||z-y|} \quad \text {and}\quad r_\varepsilon =r_{\varepsilon ,\textbf{o}}:=|\xi ,\zeta ,\vartheta ,\eta |_\varepsilon :=\frac{k_\varepsilon (\xi ,\zeta )k_\varepsilon (\vartheta ,\eta )}{k_\varepsilon (\xi ,\vartheta )k_\varepsilon (\zeta ,\eta )}. \end{aligned}$$

We demonstrate that

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon \quad \text {with absolute constants}. \end{aligned}$$

(3.17)

By approximating each $k_\varepsilon$ distance using the appropriate $k(\textbf{o},\Gamma )$ distance we easily see that

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (k(\textbf{o},\Gamma _{\xi \zeta }) + k(\textbf{o},\Gamma _{\vartheta \eta }) - k(\textbf{o},\Gamma _{\xi \vartheta }) - k(\textbf{o},\Gamma _{\zeta \eta })\bigr )\Bigr ). \end{aligned}$$

Our simple brute force approach is straightforward, but tedious and involves multiple cases and subcases. The earnest reader should check the nearly trivial case where $z=o$ and $t=\infty$.

While (3.15) provides precise, albeit rough, estimates for $k(\textbf{o},\Gamma )$, the following cruder estimates are more useful here. Let $a,b\in Z$, $\zeta :=\varphi (a),\eta :=\varphi (b)$, and $\Gamma :=\Gamma _{\zeta \eta }$. Suppose $0<|a|\le |b|$. Then:

where $\mu :=\mu (a,b), \nu :=\nu (a)\ge \nu (b), m:=m(a,b)=\nu \wedge 0$ are defined as in §3.2.2.

Below we repeatedly, and implicitly, appeal to the estimates

$$\begin{aligned}{} & {} |a-b|\le \frac{1}{2}\bigl (|a|\vee |b|\bigr ) \implies |a|{{\,\mathrm{\approx }\,}}|b| \quad \text {and}\\{} & {} |a-b|\ge \frac{1}{2}\bigl (|a|\wedge |b|\bigr ) \implies |a-b|{{\,\mathrm{\approx }\,}}|a|\vee |b|. \end{aligned}$$

For example, in (3.18c), (3.18d) we get $|a-b|{{\,\mathrm{\approx }\,}}|b|$, $|a|{{\,\mathrm{\approx }\,}}|b|$; also in (3.18a), (3.18b) if $|b|\ge 2$.

Proof of (3.17)

Below we establish (3.17) for all $x,y,t\in Z$ in the case where $|z|\le 1$. Assume $|z|>1$. Select $p\in Z^{(0)}$ with $|z-p|<1$. By replacing $\textbf{o}$ with the new vertex base point $\textbf{p}:=(p,0)$ (so e.g. everywhere $|x|, k(\textbf{o},\Lambda ), k(\textbf{o},\Gamma ), k_{\varepsilon ,\textbf{o}}$ are replaced by $|x-p|, k(\textbf{p},\Lambda ), k(\textbf{p},\Gamma ), k_{\varepsilon ,\textbf{p}}$ respectively) we can apply the same arguments—those given below as well as those for (3.13), (3.15), (3.18), etc.—to see that for all $x,y,t\in Z$,

$$\begin{aligned} r_{\varepsilon ,\textbf{p}}:=|\xi ,\zeta ,\vartheta ,\eta |_{\varepsilon ,\textbf{p}}:=\frac{k_{\varepsilon ,\textbf{p}}(\xi ,\zeta )k_{\varepsilon ,\textbf{p}}(\vartheta ,\eta )}{k_{\varepsilon ,\textbf{p}}(\xi ,\vartheta )k_{\varepsilon ,\textbf{p}}(\zeta ,\eta )} {{\,\mathrm{\approx }\,}}r^\varepsilon \quad \text {with absolute constants}. \end{aligned}$$

An appeal to Fact 2.2 gives us $r_{\varepsilon ,\textbf{o}}{{\,\mathrm{\approx }\,}}r_{\varepsilon ,\textbf{p}}$ (with constant 16), so (3.17) holds in all cases.

Let $x,y,z,t\in Z$. Assume $|z|\le 1$. The 24 permutations of z, y, x, t yield at most six values for their absolute cross ratios, three numbers and their reciprocals. Because of this, we may assume that $|x|\le |y|$ and $|z|\le |t|$. With these considerations, it suffices to examine the following cases and their subcases.

$$\begin{aligned}&|x|,|y|,|t|\le 2. \end{aligned}$$

(3.19a)

$$\begin{aligned}&|x|\ge 2, \;\text {and}\, |t|\,\text { satisfying}\, |t|\le 1\, \text {or}\, 1\le |t|\le 2\, \text {or}\, |t|\ge 2. \end{aligned}$$

(3.19b)

$$\begin{aligned}&|t|\ge 2\ge |x|, \;\text {and}\, |y|\,\text { satisfying}\, |y|\le 1\, \text {or}\, 1\le |y|\le 2\, \text {or}\, |y|\ge 2. \end{aligned}$$

(3.19c)

$$\begin{aligned}&|y|\ge 2>|x|\vee |t|. \end{aligned}$$

(3.19d)

Again, we assume $|x|\le |y|, |z|\le |t|$, and $|z|\le 1$.

In case (3.19a), (3.10a) says that for each $\Gamma \in \{{\Gamma _{\xi \zeta },\Gamma _{\vartheta \eta },\Gamma _{\xi \vartheta },\Gamma _{\zeta \eta }}\}$, $k(\textbf{o},\Gamma ){{\,\mathrm{\simeq }\,}}\mu \cdot \log 2$ for the appropriate $\mu$. As $2^{-\mu (a,b)}{{\,\mathrm{\approx }\,}}|a-b|$, $r_\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon$ readily follows.

For case (3.19b), $|y|\ge |x|\ge 2$; so $|x-z|{{\,\mathrm{\approx }\,}}|x|$ and $|y-z|{{\,\mathrm{\approx }\,}}|y|$. Here (3.18b) says that

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \zeta }){{\,\mathrm{\simeq }\,}}0 {{\,\mathrm{\simeq }\,}}k(\textbf{o},\Gamma _{\eta \zeta }). \end{aligned}$$

In (3.19b) with $|t|\le 1$, (3.18b) again says that

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}0 {{\,\mathrm{\simeq }\,}}k(\textbf{o},\Gamma _{\eta \vartheta }) \quad \text {but also}\quad |x-t|{{\,\mathrm{\approx }\,}}|x| \quad \text {and}\quad |y-t|{{\,\mathrm{\approx }\,}}|y| \end{aligned}$$

and so $r_\varepsilon {{\,\mathrm{\approx }\,}}1{{\,\mathrm{\approx }\,}}r$.

In (3.19b) with $1\le |t|\le 2\le |x|\le |y|$, we consider four subsubcases

• (3.19b.i) $|x-t|\vee |y-t|\ge 1$,

• (3.19b.ii) $|x-t|\le 1\le |y-t|$,

• (3.19b.iii) $|y-t|\le 1\le |x-t|$,

• (3.19b.iv) $|x-t|\wedge |y-t|\le 1$.

For (3.19b.i), $|x-t|{{\,\mathrm{\approx }\,}}|x|,|y-t|{{\,\mathrm{\approx }\,}}|y|$ and $k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}0 {{\,\mathrm{\simeq }\,}}k(\textbf{o},\Gamma _{\eta \vartheta })$, so again $r{{\,\mathrm{\approx }\,}}1{{\,\mathrm{\approx }\,}}r_\varepsilon$. For (3.19b.iv), $2\le |x|\le |x-t|+|t|\le 3$ and similarly $|y|{{\,\mathrm{\approx }\,}}1$, so $r{{\,\mathrm{\approx }\,}}|y-t|/|x-t|$. Also, by (3.10a)

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}\mu (x,t)\log 2 \quad \text {and}\quad k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}\mu (y,t)\log 2 \end{aligned}$$

so

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (0 + \mu (y,t) - \mu (x,t) - 0\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|y-t|}{|x-t|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

The subsubcases (3.19b.ii), (3.19b.iii) are similar.

In (3.19b) with $|t|\ge 2$, we consider three subsubcases

• (3.19b.v) $2\le |t|\le |x|\le |y|$,    $(\text {here}\, \nu (t)\ge \nu (x)\ge \nu (y))$

• (3.19b.vi) $2\le |x|\le |t|\le |y|$,    $(\text {here}\, \nu (x)\ge \nu (t)\ge \nu (y))$

• (3.19b.vii) $|t|\ge |y|$.                $(\, \nu (x)\ge \nu (y)\ge \nu (t))$

For (3.19b.v): If $|t|\le 2\bigl (|x-t|\wedge |y-t|\bigr )$, $|x-t|{{\,\mathrm{\approx }\,}}|x|$ and $|y-t|{{\,\mathrm{\approx }\,}}|y|$, so $r{{\,\mathrm{\approx }\,}}1$; then by (3.18c)

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}-\nu (t)\log 2 {{\,\mathrm{\simeq }\,}}k(\textbf{o},\Gamma _{\eta \vartheta }) \quad \text {so also}\quad r_\varepsilon {{\,\mathrm{\approx }\,}}1. \end{aligned}$$

If $|t|\ge 2\bigl (|x-t|\vee |y-t|)$, $|x|{{\,\mathrm{\approx }\,}}|t|{{\,\mathrm{\approx }\,}}|y|$, so $r{{\,\mathrm{\approx }\,}}|y-t|/|x-t|$; also, by (3.18d)

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}\bigl (\mu (x,t)-2\nu (t)\bigr )\log 2 \quad \text {and}\quad k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}\bigl (\mu (y,t)-2\nu (t)\bigr )\log 2 \end{aligned}$$

so

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (0 + \mu (y,t) - \mu (x,t) - 0\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|y-t|}{|x-t|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

The cases where $2|x-t|\le |t|\le 2|y-t|$ or $2|y-t|\le |t|\le 2|x-t|$ are similar.

For (3.19b.vi): If $|x|\le 2|x-t|$ and $|t|\le 2|y-t|$, $|x-t|{{\,\mathrm{\approx }\,}}|t|$ and $|y-t|{{\,\mathrm{\approx }\,}}|y|$, so $r{{\,\mathrm{\approx }\,}}|x|/|t|$; but,

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}-\nu (x)\log 2 \quad \text {and}\quad k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}-\nu (t)\log 2 \end{aligned}$$

by (3.18c), and therefore

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (-\nu (t) + \nu (x)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x|}{|t|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

If $|x|\le 2|x-t|$ and $|t|\ge 2|y-t|$, $|x-t|{{\,\mathrm{\approx }\,}}|t|$ and $|y|{{\,\mathrm{\approx }\,}}|t|$, so $r{{\,\mathrm{\approx }\,}}|x||y-t|/|t|^2$; also,

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}-\nu (x)\log 2 \quad \text {and}\quad k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}\bigl (\mu (y,t)-2\nu (t)\bigr )\log 2 \end{aligned}$$

by ((3.18)c,d), and so

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (y,t)-2\nu (t) +\nu (x)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|y-t||x|}{|t|^2} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

The cases $|x|\ge 2|x-t|, |t|\le 2|y-t|$ or $|x|\ge 2|x-t|,|t|\ge 2|y-t|$, are similar.

For (3.19b.vii): We argue as above; e.g., if $|x|\le 2|x-t|$ and $|y|\ge 2|y-t|$, $|x-t|{{\,\mathrm{\approx }\,}}|t|$ and $|y|{{\,\mathrm{\approx }\,}}|t|$, so $r{{\,\mathrm{\approx }\,}}|x||y-t|/|t|^2$, but again by ((3.18)c,d),

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}-\nu (x)\log 2 \quad \text {and}\quad k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}\bigl (\mu (y,t)-2\nu (y)\bigr )\log 2 \end{aligned}$$

so

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (y,t)-2\nu (y) +\nu (x)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|y-t||x|}{|y|^2} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

For case (3.19c), $|t|\ge 2\ge |x|$, so by (3.10a) $k(\textbf{o},\Gamma _{\xi \zeta }){{\,\mathrm{\simeq }\,}}\mu (x,z)\log 2$; also, $k(\textbf{o},\Gamma _{\xi \vartheta })$ depends on whether $|x-t|\le 1$ or $|x-t|\ge 1$; and when $|y|\le 2$, similar estimates hold for $k(\textbf{o},\Gamma _{\eta \zeta }),k(\textbf{o},\Gamma _{\eta \vartheta })$. If $|y|\le 1$, $|x-t|{{\,\mathrm{\approx }\,}}|t|{{\,\mathrm{\approx }\,}}|y-t|$ so $r{{\,\mathrm{\approx }\,}}|x-z|/|y-z|$; also by (3.18b)

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}0{{\,\mathrm{\simeq }\,}}k(\textbf{o},\Gamma _{\eta \vartheta }) \end{aligned}$$

whence

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (x,z)-\mu (y,z)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x-z|}{|y-z|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

In (3.19c) with $1\le |y|\le 2\le |t|$, $|y|{{\,\mathrm{\approx }\,}}1$, we consider four subsubcases

• (3.19c.i) $|x-t|\wedge |y-t|\ge 1$,

• (3.19c.ii) $|x-t|\le 1\le |y-t|$,

• (3.19c.iii) $|y-t|\le 1\le |x-t|$,

• (3.19c.iv) $|x-t|\vee |y-t|\le 1$.

For (3.19c.i), $|x-t|{{\,\mathrm{\approx }\,}}|t|{{\,\mathrm{\approx }\,}}|y-t|$, so $r{{\,\mathrm{\approx }\,}}|x-z|/|y-z|$; also, $k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}0 {{\,\mathrm{\simeq }\,}}k(\textbf{o},\Gamma _{\eta \vartheta })$, so again $r_\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon$. For (3.19c.iii), $|x-t|{{\,\mathrm{\approx }\,}}|t|{{\,\mathrm{\approx }\,}}|y|{{\,\mathrm{\approx }\,}}1$, so $r{{\,\mathrm{\approx }\,}}|x-z||y-t|/|y-z|$. Also, by (3.18b) and (3.10a)

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}0 \quad \text {and}\quad k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}\mu (y,t)\log 2 \end{aligned}$$

so

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (x,z)+\mu (y,t)-\mu (y,z)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x-z||y-t|}{|y-z|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

The subsubcase (3.19c.ii) is similar and (3.19c.iv) is easy.

Next we examine (3.19c) with $|y|\ge 2$ (still with $|x|\le 2\le |t|$). Now $|y-z|{{\,\mathrm{\approx }\,}}|y|$ and (3.10a) (with $a=z,b=y$) tells us that $k(\textbf{o},\Gamma _{\eta \zeta }){{\,\mathrm{\simeq }\,}}0$. Here there are eight subsubcases to inspect depending on whether $|x-t|\le 1$ or $|x-t|\ge 1$, $|y|\ge |t|$ or $|y|\le |t|$, $|y|\wedge |t|\le 2|y-t|$ or $|y|\wedge |t|\ge 2|y-t|$.

Assume $|x-t|\le 1$. Then $|x|{{\,\mathrm{\approx }\,}}|t|{{\,\mathrm{\approx }\,}}1$ and by (3.10a), $k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}\mu (x,t)\log 2$. Suppose $|y|\le |t|$. If $|y|\le 2|y-t|$, then $|y-t|{{\,\mathrm{\approx }\,}}|t|{{\,\mathrm{\approx }\,}}1$ so $r{{\,\mathrm{\approx }\,}}|x-z|/|x-t||y|$, and by (3.18c), $k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}-\nu (y)\log 2$, whence

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (x,z)-\nu (y)-\mu (x,t)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x-z|}{|x-t||y|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

If $|y|\ge 2|y-t|$, then $|y|{{\,\mathrm{\approx }\,}}|t|{{\,\mathrm{\approx }\,}}1$ and by (3.18d), $k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}\bigl (\mu (y,t)-2\nu (y)\bigr )\log 2$. Therefore,

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (x,z)+\mu (y,t)-2\nu (y)-\mu (x,t)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x-z||y-t|}{|x-t||y|^2} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon \end{aligned}$$

because $|y|^2{{\,\mathrm{\approx }\,}}|y|{{\,\mathrm{\approx }\,}}|y-z|$. Similar arguments apply when $|y|\ge |t|$.

Assume $|x-t|\ge 1$. Then $|x-t|{{\,\mathrm{\approx }\,}}|t|$ and by (3.18b), $k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}0$. Suppose $|y|\ge |t|$. If $|t|\le 2|y-t|$, then $|y-t|{{\,\mathrm{\approx }\,}}|y|$ so $r{{\,\mathrm{\approx }\,}}|x-z|/|t|$, and by (3.18c), $k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}-\nu (t)\log 2$, whence

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (x,z)-\nu (t)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x-z|}{|t|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

If $|t|\ge 2|y-t|$, then $|y|{{\,\mathrm{\approx }\,}}|t|$ and by (3.18d), $k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}\bigl (\mu (y,t)-2\nu (t)\bigr )\log 2$. Therefore,

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (x,z)+\mu (y,t)-2\nu (t)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x-z||y-t|}{|t|^2} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon \end{aligned}$$

because $|t|^2{{\,\mathrm{\approx }\,}}|t||y|{{\,\mathrm{\approx }\,}}|x-t||y-z|$. Similar arguments apply when $|y|\le |t|$.

Finally, we look at case (3.19d) where $|y|\ge |x|\vee |t|$. Here $|y-z|{{\,\mathrm{\approx }\,}}|y|$ and (3.10a), (3.18b) reveal that

$$\begin{aligned} k(\textbf{o},\Gamma _{\xi \zeta }){{\,\mathrm{\simeq }\,}}\mu (x,z)\log 2,\quad k(\textbf{o},\Gamma _{\xi \vartheta }){{\,\mathrm{\simeq }\,}}\mu (x,t)\log 2, \quad \text {and}\quad k(\textbf{o},\Gamma _{\eta \zeta }){{\,\mathrm{\simeq }\,}}0. \end{aligned}$$

If $|y-t|\le 1$, $|y|{{\,\mathrm{\approx }\,}}1$ and by (3.10a) (with $a=t,b=y$), $k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}\mu (y,t)\log 2$, so

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (x,z)+\mu (y,t)-\mu (x,t)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x-z||y-t|}{|x-t|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon . \end{aligned}$$

If $|y-t|\ge 1$, $|y-t|{{\,\mathrm{\approx }\,}}|y|$ and by (3.18b) (with $a=t,b=y$), $k(\textbf{o},\Gamma _{\eta \vartheta }){{\,\mathrm{\simeq }\,}}0$, so

$$\begin{aligned} r_\varepsilon {{\,\mathrm{\approx }\,}}\exp \Bigl (-\varepsilon \bigl (\mu (x,z)-\mu (x,t)\bigr )\log 2\Bigr ) {{\,\mathrm{\approx }\,}}\Biggl (\frac{|x-z|}{|x-t|} \Biggr )^\varepsilon {{\,\mathrm{\approx }\,}}r^\varepsilon \end{aligned}$$

because $|y-z|{{\,\mathrm{\approx }\,}}|y|{{\,\mathrm{\approx }\,}}|y-t|$. $\square$