††footnotetext:  File: Visibility-6-6-23.tex, printed: 2024-8-28, 3.23

# Visible quasihyperbolic geodesics

Vasudevarao Allu  Vasudevarao Allu, Discipline of Mathematics, School of Basic
Sciences, Indian Institute of Technology Bhubaneswar, Argul, Bhubaneswar,
PIN-752050, Odisha (State), India<EMAIL_ADDRESS>and  Abhishek Pandey
Abhishek Pandey, Discipline of Mathematics, School of Basic Sciences, Indian
Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha
(State), India<EMAIL_ADDRESS>
###### Abstract.

In this article, we initiate the study of visibility property in the context
of quasihyperbolic metric. Let $\Omega$ be a bounded domain in
$\mathbb{R}^{n}$ and $\partial_{Euc}\Omega$ be its Euclidean boundary. We say
that the pair $\\{p,q\\}\subset\partial_{Euc}\Omega$ has visible
quasihyperbolic geodesic if there exist open neighborhoods $U_{p}$ of $p$ and
$U_{q}$ of $q$ and a compact set $K_{\\{p,q\\}}\subset\Omega$ such that for
any quasihyperbolic geodesic which joins points in $U_{p}$ and $U_{q}$
intersects with $K$. If a domain has visible quasihyperbolic geodesic for
every pair of points, we say that it is a visibility domain. The major part of
this paper is devoted to provide a rich collection of visibility domains. In
this direction we provide a general visibility criteria for a domain to be a
visibility domain, and using this we obtain the visibility of Uniform domains,
John domains and domains satisfying quasihyperbolic boundary conditions. We
also study the visibility of bounded hyperbolic domains with respect to
hyperbolic and quasihyperbolic metric. Further, we explore the relation
between Gromov hyperbolicity and visibility. As an application of visibility
we prove a sufficient condition for the continuous extension of
quasihyperbolic isometries and quasi-isometries. Further, motivated by the
work of Bharali and Zimmer [Trans. Amer. Math. Soc. (2022)] and using the end
compactification we define the visibility of unbounded domains and study its
relationship with Gromov hyperbolicity. Finally, we provide some useful
examples of visibility domains which are neither John domain nor domains
satisfying quasihyperbolic boundary conditions.

###### Key words and phrases:

Gromov hyperbolic spaces, John domains, quasihyperbolic metric,
quasihyperbolic geodesics, quasiconformal mappings, visibility

###### 2020 Mathematics Subject Classification:

Primary 30F45, 30L10, 30L99, 30C65. Secondary 51F99, 53C22.

###### Contents

  1. 1 Introduction and results
    1. 1.1 Visibility
    2. 1.2 Visibility of domains
    3. 1.3 Gromov hyperbolicity and visibility
    4. 1.4 Continuous extension of quasi-isometries and isometries
    5. 1.5 The end compactification, visibility and Gromov hyperbolicity
    6. 1.6 Outline of the paper
    7. 1.7 Basic Notations
  2. 2 Preliminaries
    1. 2.1 Metric geometry
    2. 2.2 Hyperbolic geometry
    3. 2.3 Quasihyperbolic metric
    4. 2.4 Gromov hyperbolic space
    5. 2.5 Gromov compactification
    6. 2.6 Gromov product
    7. 2.7 Quasihyperbolic Domains
    8. 2.8 Gromov hyperbolicity in quasihyperbolic geometry
  3. 3 Visibility of some classical domains
    1. 3.1 General visibility criteria
    2. 3.2 H-visibility domain and QH-visibility domain
  4. 4 Gromov hyperbolicity and visibility
  5. 5 Visibility domains via conformal and quasiconformal mappings
  6. 6 Continuous extension of quasihyperbolic isometries and quasi-isometries
  7. 7 The end compactification, visibility and Gromov hyperbolicity
  8. 8 Some Examples
  9. 9 Appendix
  10. 10 Concluding remarks and topics for further exploration
  11. 11 Acknowledgements

##  1\. Introduction and results

The quasihyperbolic metric in $\mathbb{R}^{n}$ is a generalization of
hyperbolic metric, which was introduced by Gehring and Palka [24, Section 2]
where they have studied the characterization of domains $D$ in one point
compactification of $\mathbb{R}^{n}$ which are quasiconformally equivalent to
the unit ball in $\mathbb{R}^{n}$. Characterizing domains which are
quasiconformaly equivalent to the unit ball is an important problem in the
higher-dimensional quasiconformal theory. It is clear that such domains are
quasiconformally homogeneous. Gehring and Palka [24] considered
quasiconformally homogeneous domains to get new characterization for domains
quasiconformally equivalent to the unit ball. Gehring and Palka have proved
that the maximal dilatation of such quasiconformal mappings can be estimated
in terms of quasihyperbolic metric (see [24, Section 3]). Using these
estimates they have obtained very useful results related to quasiconformally
homogeneous domains (see [24, Section 4]). Therefore, the importance of the
quasihyperbolic metric is quite clear as it is well-behaved with the
quasiconformal mappings.

The present paper is devoted to the study of quasihyperbolic geometry and we
consider the following natural question related to quasihyperbolic geometry

######  Problem 1.1.

How a domain $\Omega\subsetneq\mathbb{R}^{n}$ is curved when it is equipped
with the quasihyperbolic metric?

Let $\Omega$ be a proper subdomain in $\mathbb{R}^{2}$, and $\rho$ be a
$C^{2}$ smooth metric. The Gaussian curvature of this metric at
$z_{0}\in\Omega$ is defined as

(1.2)
$\mathcal{K}_{\rho}(z_{0})=-\frac{\Delta\log\,\rho(z_{0})}{(\rho(z_{0}))^{2}}$

Quasihyperbolic metric arises from the density
$d(z,\partial_{Euc}\Omega)^{-1}$, due to the fact that this density need not
be differentiable, the Gaussian curvature is not defined. For instance, the
Riemannian metric tensor associated to the quasihyperbolic metric in the unit
ball is not differntiable at the origin. More generally, if there exists a
point $x\in\Omega$ for which there are atleast two boundary points which gives
the distance between $x$ and $\partial_{Euc}\Omega$, then density
$d(z,\partial_{Euc}\Omega)^{-1}$ is not differentiable at $x$. This happens
due to the existance of medial axis of domain $\Omega$. As this density is not
differentiable in the entire domain, quasihyperbolic metric is outside the
standard framework of the Riemannian metrics. Thus, to answer Problem 1.1 we
first need some suitable notion of curvature. Heins [34] has considered this
problem for a more general class of metrics and defined the notion of lower
and upper Gaussian curvature for more general metric (which need not to be
$C^{2}$). Using the integral formula for Laplacian, Heins in [34], defined the
upper Gaussian curvature $\overline{\mathcal{K}}_{\rho}$ and lower Gaussian
curvature $\underline{\mathcal{K}}_{\rho}$ of the conformal metric
$\rho(z)|dz|$. Upper and lower Gaussian curvature for quasihyperbolic metric
always exists but it may be $-\infty$. In the context of quasihyperbolic
metric, Martin and Osgood [39] have worked with this notion of curvature in
the sense of Heins, for planar domains. Further, Hästö [32] has studied these
curvatures and applied it to characterize the quasihyperbolic isometries.

If we consider quasihyperbolic metric on the upper-half space in
$\mathbb{R}^{n}$, it coincides with the hyperbolic metric, and hence it is of
constant negative curvature $-1$. Moreover, Martin and Osgood [39] have proved
that quasihyperbolic metric in planar domains have nonpositive generalized
Gaussian curvature. Thus, one may curious about: when can we say that a domain
$\Omega$ in $\mathbb{R}^{n}$ is negatively or in general nonpositively curved
(in some sense) when equipped with the quasihyperbolic metric? To say that a
domain $\Omega$ in $\mathbb{R}^{n}$ equipped with the quasihyperbolic metric
is negatively or nonpositively curved (in some sense), we need some notion.
Also since the quasihyperbolic metric is outside the farme work of Riemannian
metric, we want to capture curvature in quasihyperbolic geometry without with
tensor-free approaches to negative or nonpositive curvature. This can be done
by studying a suitable notion of negative or nonpositive curvature for metric
spaces which coincides with some notion of curvature in Riemannian manifolds
when considering it as a metric space. Given a metric space there are several
notions of it being negatively or nonpositively curved such as:

### Bausemann convexity:

In 1898, Hadamard made the a fundamental observation in his pioneering paper
[31] that the distance function from a fixed point in a surface $S$ smoothly
embedded in $\mathbb{R}^{3}$ and equipped with a Riemannian metric of non-
positive curvature, to a point that moves along a global geodesic in this
surface is convex. This observation made the basis of the definition of metric
spaces of non-positive curvature in the sense of Busemann. In 1948, Busemann
[19] introduced the notion Busemann convexity and defined the locally Busemann
space. A geodesic metric space is said to be nonpositively curved in the sense
of Busemann if it is locally Busemann space. Such spaces are also called
locally convex metric spaces. This notion of nonpositive curvature coincides
with the notion of nonpositive sectional curvatur of a Riemannian manifold.
More precisely, a Riemannian manifold is locally Busemann if, and only if, it
is of nonpositive sectional curvature. For much more on the theory of
nonpositively curved spaces in the sense of Busemann we refer to [45].

### CAT(k) conditions:

In 1957, Alexandrov [2] introduced several equivalent definitions for a metric
space to have curvature bounded above from $k$ for any real number $k$.
Nowadays, these definitions are called CAT($k$) conditions, a term introduced
by Gromov. All these conditions involve comparing the space to a well
understood model space. A geodesic metric space $(X,d)$ is CAT($k$) if every
geodesic triangle satisfies the CAT($k$) condition. A geodesic metric space
$(X,d)$ has curvature $\leq k$ if for every $x\in X$ there is a ball which is
CAT($k$), $X$ is said to be nonpositively curved if it has curvature $\leq 0$.
In such a case, we say that metric space is nonpositively curved in the sense
of Alexandrov. These conditions are of great importance. For instance, suppose
$k\leq 0$. If $X$ is a Riemannian manifold then the sectional curvature $K\leq
k$ if, and only if, it is locally CAT($k$) (see [3, Chapter 1, 3.7] or [16,
Part II.1. Appendix]). For global CAT($k$) condition, we have, a Riemannian
manifold with sectional curvature $K\leq k$ is CAT($k$) if, and only if, it is
simply connected. For the theory of CAT($k$) spaces we refer to the book by
Bridson and Haefliger [16] and [1].

######  Remark 1.1.

Note that if a metric space is nonpositively curved in the sense of Alexandrov
then it is nonpositively curved in the sense of Busemann but not conversly
(see [17]). Thus, the Busemann notion of nonpositive curvature is a weaker
notion.

### Gromov hyperbolicity:

We will discuss this concept later in detail but at this stage we just give a
brief introduction to it. Gromov hyperbolicity is a property of a general
metric space to be negatively curved in the sense of coarse geometry. In 1987,
Gromov [30] introduced this concept in the setting of geometric group theory
precisely, for Cayley graphs of finitely generated groups. A geodesic metric
space is said to be Gromov hyperbolic if it is $\delta\geq 0$ hyperbolic. That
is, there is a uniform $\delta$ such that all the geodesic triangles are
$\delta$-thin. That is, each of its sides is contained in the
$\delta$-neighborhood of the union of the other two sides. It is known that
every CAT($k$) space, $k<0$ is Gromov hyperbolic for $\delta=(\log
3)/\sqrt{-k}$, and hence every simply connected Riemannian manifold of
sectional curvature at most $k$ is Gromov hyperbolic. For more details on
this, we refer to [16].

There is an interesting article by Stephen M. Buckley [17] which beautifully
relates all the above mentioned concepts of negatively or nonpositively curved
spaces. At this stage, we would like to mention that Gromov hyperbolicity is
one of the notion of negative curvature for a metric space and the negative
curvature property of a domain in $\mathbb{R}^{n}$ equipped with
quasihyperbolic metric in the Gromov sense (i.e. $(\Omega,k_{\Omega})$ is
Gromov hyperbolic) has been extensively studied.

Visibility was first introduced by Eberlein and O’ Neill [21], where they
introduced a general construction of compactification of Hadamrd manifolds by
attaching an abstract boundary or boundary at infinity, which is called ideal
boundary. Using the notion of ideal boundary Eberlein and O’ Neill defined the
term called visibility manifold for complete Riemannian manifold with
nonpositive sectional curvature. It then truns out that visibility is a notion
of strict negative curvature in the Riemannian sense.

Hence a suitable notion of visibility in the quasihyperbolic geometry can be
consider as a weak notion of negative curvature, and motivated by this we
initiate the study of visibility property in the context of quasihyperbolic
metric. Recently, visibility property in the context of Kobayashi distance has
been systematically studied, see [9, 8, 15, 20] and references therein.

The aim of this paper is to study the concept of visibility in the context of
quasihyperbolic metric and compare it with the Gromov hyperbolicity. To define
visibility in the context of quasihyperbolic metric requires some explanation,
but before describing the notion of visibility and results of this paper in
details, we shall mention the following motivating example which will give us
the idea of visibility.

### Compactification of $\mathbb{H}^{2}$ by attaching the boundary at infinity

Consider the Poincaré upper-half plane model of hyperbolic space
$\mathbb{H}^{2}$. The geodesics in $\mathbb{H}^{2}$ are the semi-circles
orthogonal to the real axis and the vertical straight lines. Moreover, given
any two points in $\mathbb{H}^{2}$ there exists a unique geodesic passing
through them. Fix $x_{0}\in\mathbb{H}^{2}$ and let $\mathcal{G}_{x_{0}}$ be
the set of geodesic rays $\gamma:[0,\infty)\rightarrow\mathbb{H}^{2}$ starting
from $x_{0}$. Let $\gamma,\sigma\in\mathcal{G}_{x_{0}}$. We say
$\gamma\sim\sigma$ if $\sup_{t\geq
0}\rho_{\mathbb{H}^{2}}(\gamma(t),\sigma(t))<\infty$. Denote
$\partial_{I}\mathbb{H}^{2}=\mathcal{G}_{x_{0}}/\sim$ by the boundary at
infinity and we call members of it, points at infinity. This space has the
following properties

  * (i)

Each geodesic ray lands at a point on the Euclidean boundary.

  * (ii)

Two equivalent geodesic rays lands at the same Euclidean boundary point.

This gives a compactification $\mathbb{H}^{2}\cup\partial_{I}\mathbb{H}^{2}$
of $(\mathbb{H}^{2},\rho_{\mathbb{H}^{2}})$, which is homoemorphic to the
closed unit ball in the complex plane. It is easy to see that for each
$x\in\mathbb{H}^{2}$ and $\xi\in\partial_{I}\mathbb{H}^{2}$, there exists a
geodesic ray joining $x$ and $\xi$. Moreover, any two distinct points in
$\partial_{I}\mathbb{H}^{2}$ can be joined by a geodesic line (this is called
visibility).

The analogous compactification of arbitrary Hadamard manifolds was first given
by Eberlein and O’Neill [21] by attaching the boundary at infinity, which we
call the ideal boundary and this compactification for Hadamard manifold is
homeomorphic to a closed $n$-ball. The notion of ideal boundary is a central
concept in the theory of negative or nonpositively curved metric spaces and
this notion comes after defining a suitable notion of negative or nonpositive
curvature. This notion also has a generalization in CAT($0$) spaces and Gromov
hyperbolic spaces respectively. In Gromov hyperbolic spaces the boundary at
infinity is called Gromov boundary. We now mention the precise definition of
visibility.

###  1.1. Visibility

A Hadamard manifold $H$ is a complete, simply-connected Riemannian manifold
with non-positive sectional curvature $K\leq 0$. The universal cover of any
complete Riemannian manifold $M$ with non-positive sectional curvatire $K\leq
0$ is a Hadamard manifold. Further, a complete manifold $M$ of dimension
$n\geq 2$ with non-positive sectional curvature are precisely the quotient
manifolds $M=H/D$, where $D$ is group of isometries of $H$ which acts properly
discontinuously on $H$.

In 1973, Eberlein and Neill [21] introduced the concept of visibility for
complete Riemannian manifolds having non-positive sectional curvature as a
generalization of strictly negative curvature. Their definition can be read as
follows.

######  Definition 1.1.

$M=H/D$ is a visibility manifold if, for any point $x\neq y$ in the ideal
boundary $H(\infty)$ there is a geodesic line joining $x$ and $y$.

It is well-known (see [11, Lemma 9.10], [21]) that if $H$ is complete, simply-
connected Riemannian manifold with sectional curvature $K\leq c<0$, then
$M=H/D$ is a visibility manifold. In particular, every complete, simply-
connected Riemannian manifold with constant negative curvature is a visibility
manifold. In fact visibility is the natural replacement for the strict
negative curvature.

The visibility condition can be defined in many different equivalent ways (see
[4, pp. 54]). One such is the following (see or [21, Proposition 4.4]).

######  Definition 1.2.

$M=H/D$ is a visibility manifold if, and only if, for any $z,w\in H(\infty)$,
$z\neq w$ there exists a compact set $K\subset H$ such that for any sequence
$x_{k},y_{k}\in H$, with $x_{k}\to z$ and $y_{k}\to w$ the geodesic
$\gamma_{k}:[0,T]\to H$ joining $p_{k}$ and $q_{k}$ intersects with $K$ for
sufficiently large $k$.

######  Remark 1.2.

To study visibility property in the context of quasihyperbolic geometry, the
natural way is to consider Definition 1.1, but for this we need the notion of
ideal boundary of the space $(\Omega,k_{\Omega})$. For the notion of ideal
boundary we need a suitable notion of negative or nonpositive curvature. One
such notion is CAT($0$) space and visibility propetry has also been extended
to CAT($0$) spaces (see [16, Chapetr II.9]). Thus, we can study visibility in
the sense of Eberlein and O’Neill (i.e. connecting ideal boundary points by
geodesic line) if we know when $(\Omega,k_{\Omega})$ is CAT($0$). But this is
going to be restrictive because of the following two facts:

  * (i)

By Cartan-Hadamard theorem we know that a complete metric space is CAT($0$) if
and only if it is locally CAT($0$) and simply connected and,

  * (ii)

the quasihyperbolic metric induces the same topology and $(\Omega,k_{\Omega})$
is a complete metric space.

Thus, in view of the above two facts, to study visibility in the sense of
Eberlein and O’Neill (see Definition 1.1) we can only consider simply
connected domains in $\mathbb{R}^{n}$. Nevertheless, motivated by the work of
Bharali-Zimmer [9] and Bracci et. al.[15], we define visibility in the context
of quasihyperbolic metric as follows.

######  Definition 1.3.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ and
$\partial_{Euc}\Omega$ be its Euclidean boundary. Let
$p,q\in\partial_{Euc}\Omega$, $p\neq q$. We say that the pair $\\{p,q\\}$ has
visible quasihyperbolic geodesic if there exist neighborhoods $U,V$ of $p,q$
respectively such that $\overline{U}\cap\overline{V}=\phi$ and a compact set
$K_{\\{p,q\\}}\subset\Omega$ such that for any quasihyperbolic geodesic
$\gamma:[0,T]\to\Omega$ with $\gamma(0)\in U$ and $\gamma(T)\in V$, we have

$\gamma([0,T])\cap K\neq\phi.$

######  Definition 1.4.

We say that a bounded domain $\Omega$ in $\mathbb{R}^{n}$ has visibility
property if any pair of points $\\{p,q\\}\subset\partial_{Euc}\Omega$, $p\neq
q$ has visible quasihyperbolic geodesic.

######  Definition 1.5.

A bounded domain $\Omega$ in $\mathbb{R}^{n}$ is said to be visibility domain
if it has visibility property.

######  Remark 1.3.

Note that the definition of visibility domain is equivalent to the following.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Then $\Omega$ is said to
be visibility domain if, and only if, for any $p,q\in\partial_{Euc}\Omega$,
$p\neq q$ there exists a compact set $K\subset\Omega$ such that for any
sequence $p_{k},q_{k}\in\Omega$, with $p_{k}\to p$ and $q_{k}\to q$ there
exists positive integer $k_{0}$ such that any quasihyperbolic geodesic
$\gamma_{k}:[0,T]\to\Omega$ joining $p_{k}$ and $q_{k}$ (i.e., with
$\gamma_{k}(0)=p_{k}$ and $\gamma_{k}(T)=q_{k}$), we have

$\gamma_{k}([0,T])\cap K\neq\phi,\,\,\mbox{ for all }k\geq k_{0}.$

Geometrically speaking, a bounded domain in $\mathbb{R}^{n}$ is a visibility
domain if the quasihyperbolic geodesics bend inside the domain when connecting
points close the the Euclidean boundary $\partial_{Euc}\Omega$. We are now
going to introduce our main results.

At this stage, we must mention that since motivated by the work on visibility
in the context of Kobayashi distance, we are initiating the study on
visibility property with respect to quasihyperbolic metric therefore, we
follow some of the ideas from [9, 15, 20, 15] to prove some of our main
results.

###  1.2. Visibility of domains

We first consider the following natural question.

######  Problem 1.3.

Which bounded domains $\Omega\subset\mathbb{R}^{n}$ are visibility domains?

We refer to Problem 1.3 as a visibility problem. To ensure the significance of
the visibility problem, we need to answer the following two questions
affirmatively.

  * A.

[Problem I] Is there a rich collection of domains that are visibility domains?

  * B.

[Problem II] Is there a reasonably good collection of domains that are
visibility domain but not Gromov hyperbolic? We want to answer this problem
affirmatively so that one can distinguish between these two notions of
negative curvature.

The major part of this paper is devoted to Problem I and Problem II. We first
consider Problem I with focus on the classical Uniform domain and John domain.
It is well-known that, in the Poincaré disk model of the hyperbolic space,
hyperbolic geodesic between two points $x,y\in\mathbb{D}$, denote it by
$\gamma_{hyp}[x,y]$, have the following properties:

  * (i)

$l(\gamma_{hyp}[x,y])\leq C|x-y|$, and

  * (ii)

$\min\\{l(\gamma_{hyp}[x,z]),l(\gamma_{hyp}[z,y])\\}\leq C\,d(z,\partial D)$,

for all $z\in\gamma$, where $C=\pi/2$. The second condition is known as the
cone arc condition by which we defined the John domains (see Definition 2),
which was first introduced by John [36] in the context of elasticity theory.
If the curve satisfies both conditions mentioned above it is called double
cone arc by which we define the uniform domains introduced by Martio and
Sarvas [40] (see Definition 2). We prove the following interesting result.

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded $c$-John domain. If every
quasihyperbolic geodesic in $\Omega$ is a $b=b(c)$-cone arc then $\Omega$ is a
visibility domain.

This result is quite interesting because characterizing John domains whose
quasihyperbolic geodesics are cone arcs is a classical problem in
quasihyperbolic geometry. Proposition 1.1 says that visibility is a necessary
condition. As we know that every quasihyperbolic geodesic in Gromov hyperbolic
John domain is a cone arc [60, Corollary 1.2], in view of Theorem 1.1, we have
the following corollary.

######  Corollary 1.4.

Every Gromov hyperbolic John domain with center $x_{0}$ is a visibility
domain.

Nevertheless, we want to remove the condition of cone arc on quasihyperbolic
geodesics from Theorem 1.1. A Careful observation shows that John domain
satisfies a certain growth condition for quasihyperbolic metric. Thus, this
motivates us to check the visibility of domains which have certain growth
condition with respect to quasihyperbolic metric. Domain satisfing some growth
conditions have great importance in geometric function theory. It is well-
known that every $K$-quasiconformal map is locally Hölder continuous. To
conclude the global Hölder continuity we need some extrs geometric assumptions
on the domians. Becker and Pommerenke [7] have proved that If
$f:\mathbb{D}\to\Omega\subset\mathbb{C}$ is a conformal mapping, then $f$ is
globally $\beta$ Hölder continuous, $0<\beta\leq 1$ , if and only if, the
hyperbolic metric $h_{\Omega}$ in $\Omega$ satisfies a logarithmic growth
condition

$h_{\Omega}(f(0),z)\leq\frac{1}{\beta}log\left(\frac{\delta_{\Omega}(f(0))}{\delta_{\Omega}(z)}\right)+C_{0}.$

Gehring and Martio [27] have further extended this result for multiply
connected domains and domains in higher dimensions. For a $K$-quasiconformal
map $f:\Omega\to\Omega^{\prime}$ Gehring and Martio have proved the global
Hölder continuity of $f$ providing $\Omega$ is sufficiently nice (see [27, p.
204]) and $\Omega^{\prime}$ satisfies the logarithmic growth condition with
respect to quasihyperbolic metric.

$k_{\Omega^{\prime}}(x_{0},x)\leq\frac{1}{\beta}log\left(\frac{\delta_{\Omega^{\prime}}(x_{0})}{\delta_{\Omega^{\prime}}(x)}\right)+C_{0}.$

Thus, it is quite evident that studying visibility of such domains is of
interest. Indeed, we prove the following general result which we call as
general visibility criteria.

######  Theorem 1.2.

[General Visibility criteria] Let $\Omega\subset\mathbb{R}^{n}$ be a bounded
domain. Fix a base point $x_{0}\in\Omega$ and $\phi:(0,\infty)\to(0,\infty)$
be an strictly increasing function with $\phi(t)\to\infty$ as $t\to\infty$
such that

(1.5) $\int_{0}^{\infty}\frac{dt}{\phi^{-1}(t)}<\infty$

If $\Omega$ satisfies the following growth condition

(1.6)
$k_{\Omega}(x_{0},x)\leq\phi\left(\frac{\delta_{\Omega}(x_{0})}{\delta_{\Omega}(x)}\right),$

then $\Omega$ is a visibility domain.

With this general visibility criteria we obtain a very rich collection of
visibility domains (see Sect. 8). One of the main consequences of this is the
following theorem

######  Theorem 1.3.

Every bounded John domain is a visibility domain.

######  Remark 1.4.

Following Gotoh [29], it is often convenient to write the conditions in
Theorem 1.2 in the following form

$\delta_{\Omega}(x)\leq\delta_{\Omega}(x_{0})\psi(k_{\Omega}(x_{0},x))$

where we assume $\psi:[0,\infty)\to(0,\infty)$ is non-increasing, and that

$\int_{0}^{\infty}\psi(t)dt<\infty.$

###  1.3. Gromov hyperbolicity and visibility

We now focus on Problem II. Since every Gromov hyperbolic space has visibility
property when considering the Gromov boundary in the Gromov topology (see [16,
Part III, H, Lemma 3.2]). Therefore to answer Problem II 1.3, we study how
visibility of a bounded domain $\Omega$ is related to Gromov hyperbolicity of
$(\Omega,k_{\Omega})$. It is clear that if $(\Omega,k_{\Omega})$ is Gromov
hyperbolic and Gromov boundary is homeomorphic to the Euclidean boundary of
$\Omega$, then $\Omega$ is a visibility domain. Thus, we need to consider the
folllowing problem.

######  Problem 1.7.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ and $(\Omega,k_{\Omega})$
be Gromov hyperbolic. For which such domain the following is true:
The identity map from $(\Omega,k_{D})$ to $(\Omega,d_{Euc})$ extends to a
homeomorphism from $\overline{\Omega}^{G}$ to $\overline{\Omega}$?

Problem 1.7 is an interesting problem in the theory of Gromov hyperbolicity in
quasihyperbolic geometry. In general, the identity map may not even extend to
a continuous map from $\overline{\Omega}^{G}$ to $\overline{\Omega}$, and even
if it does the extension may not be injective. Moreover, in view of [12,
Theorem 1.11], for a bounded domain $\Omega$ in $\mathbb{R}^{n}$, the
following are equivalent

  * (i)

$\Omega$ is uniform

  * (ii)

$(\Omega,k_{\Omega})$ is Gromov hyperbolic and the identity map
$id:(\Omega,k_{D})\to(\Omega,d_{Euc})$ extends as a homeomorphism
$\Phi:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc}$ such that $\Phi$ is
quasisymmetrric homeomorphism from $\partial_{G}\Omega$ onto
$\partial_{Euc}\Omega$.

We prove the following result which is the reminiscent of the result of [15,
Theorem 3.3] in the context of Kobayashi metric.

######  Theorem 1.4.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain. Assume that
$(\Omega,k_{D})$ is Gromov hyperbolic. Then the following are equiavlent

  * (1)

$\Omega$ is a visibility domain

  * (2)

the identity map

$id:(\Omega,k_{\Omega})\to(\Omega,d_{Euc})$

extends as a continuous surjective map

$\Phi:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc}.$

######  Remark 1.5.

Theorem 1.4 together with Theorem 1.3 gives us that for every bounded John
domain, identity map extends as a continuous surjective map from the Gromov
boundary onto the Euclidean boundary.

We also study under what conditions when such extension is homeomorphism?
Further we provide a complete solution to Problem 1.7. Infact, we prove that
the such domains are only visibility domains which has no geodesic loops in
$\overline{\Omega}^{Euc}$.

######  Theorem 1.5.

The extension $\Phi$ in Theorem A is homeomorphism if, and only if, $\Omega$
has no geodesic loops in $\overline{\Omega}^{Euc}$.

Thus, in view of Theorem 1.4 we have another proof of the following important
corollary.

######  Corollary 1.8.

Every bounded uniform domain is a visibility domain. Moreover, it has no
geodesic loops in $\overline{\Omega}^{Euc}$.

Theorem 1.4 and Theorem 1.5 are reminiscent of the result of Bracci et. al.
[15, Theorem 3.3] in the context of visibility with respect to the Kobayashi
distance. Further, we study visibility domain using the boundary extension of
conformal and quasiconformal mappings, and as an application of Theorem 1.4
and Theorem 1.5 we prove the following result.

######  Theorem 1.6.

Let $\Omega\in\mathbb{R}^{n}$, $n\geq 2$ be a bounded domain which is
quasiconformally equivalent to unit ball $\mathbb{B}^{n}$. If $\Omega$ is
locally connected along the boundary, then $\Omega$ is a visibility domain
which has no geodesic loops in $\overline{\Omega}^{Euc}$.

For $n=2$, we have the following simple characterization of visibility in the
case of simply connected domains.

######  Theorem 1.7.

Let $\Omega\in\mathbb{C}$ be simply connected. Then the following are
equivalent

  * (i)

$\Omega$ is a visibility domain

  * (ii)

$\partial_{Euc}\Omega$ is locally connected.

Moreover, if in addition, $\Omega$ is a visibility domain then $\Omega$ has no
geodesic loop in $\overline{\Omega}^{Euc}$ if, and only if, $\Omega$ is
locally connected along the boundary, or simply, $\Omega$ is a Jordan domain.

######  Remark 1.6.

By a result of Wilder [59, pp. 66], that every Jordan domain is locally
connected along the boundary and by [49, Theorem 17.22] we have that a domain
which is quasiconformally equivalent to a ball is Jordan domain if, and only
if, it is locally connected along the boundary.

###  1.4. Continuous extension of quasi-isometries and isometries

The main motivation to study continuous extension of quasi-isometry is the
fact that every quasiconformal map is a rough quasi-isometry with respect to
quasihyperbolic metric [25, Theorem 3], and continuous extension of
quasiconformal map is always of some interest. For more details on the
continuous extension of quasiconformal mappings we refer to [49]. We first
define a domain which we call as quasihyperbolically well behaved which is
motivated by the no geodesic loop property (see Section 6). We prove the
following two results related to isometries and $(1,\mu)$-quasihyperbolic
quasi-isometry.

######  Theorem 1.8.

Suppose $\Omega\subset\mathbb{R}^{n}$ is quasihyperbolically well behaved
domain and $\Omega^{\prime}\subset\mathbb{R}^{n}$ is a visibility domain. Then
every quasihyperbolic isometry $f:(D,k_{D})\to(\Omega,k_{\Omega})$ extends to
a continuous map
$F:\overline{\Omega}^{Euc}\to\overline{\Omega^{\prime}}^{Euc}$.

######  Theorem 1.9.

Suppose $\Omega\subset\mathbb{R}^{n}$ is quasihyperbolically well behaved
domain and $\Omega^{\prime}\subset\mathbb{R}^{n}$ is a visibility domain. Then
every continuous $(1,\mu)$-quasihyperbolic quasi-isometric embedding
$f:(\Omega,k_{\Omega})\to(\Omega^{\prime},k_{\Omega^{\prime}})$ extends to a
continuous map $F:\overline{\Omega}^{Euc}\to\overline{\Omega^{\prime}}^{Euc}$.

Further as an application of Theorem 1.4 and Theorem 1.5, we prove that every
quasiconformal map has a homeomorphic extension to the Euclidean boundary
under the condition that domains are Gromov hyperbolic and visibility domains
with no geodesic loops.

######  Theorem 1.10.

Let $\Omega$ and $D$ be bounded domains in $\mathbb{R}^{n}$. If $\Omega$ and
$D$ are both Gromov hyperbolic and visibility domain with no geodesic loops,
then any quasi-isometry $f:(\Omega,k_{\Omega})\to(D,k_{D})$ which is
homeomorphism extends to a homeomorphism
$F:\overline{\Omega}^{Euc}\to\overline{D}^{Euc}$. In particular, every
quasiconformal map $f:\Omega\to D$ has a homeomorphic extension to the
Euclidean boundary.

###  1.5. The end compactification, visibility and Gromov hyperbolicity

Further, we study the visibility property of quasihyperbolic metric for
unbounded domains. Motivated by the work of Bharali and Zimmer [10], we use
the end compactification to define the visibility of unbounded domains with
respect to quasihyperbolic metric. For the details of the end compactifiction
see Sect. 7.

For a domain $\Omega\subset\mathbb{R}^{n}$, let $\overline{\Omega}^{End}$
denote the end point compactification of $\overline{\Omega}$. Write
$\partial\overline{\Omega}^{End}=\overline{\Omega}^{End}\setminus\Omega$. With
these notations we define visibility property as follows:

######  Definition 1.6.

Let $\Omega\subset\mathbb{R}^{n}$ be a domain (not necessarily bounded). We
say $\Omega$ is a visibility domain with respect to quasihyperbolic metric (or
simply a visibility domain) if $\Omega$ has the following property: if
$z,w\in\partial\overline{\Omega}^{End}$ are distinct points and $V_{z}$ and
$V_{w}$ are $\overline{\Omega}^{End}$-open neighborhoods of $z$ and $w$
respectively, whose closures in $\overline{\Omega}^{End}$ are disjoint, then
there exists a compact set $K\subset\Omega$ such that: if
$\gamma:[0,T]\to\Omega$ is any quasihyperbolic geodesic with $\gamma(0)\in
V_{z}$ and $\gamma(T)\in V_{w}$, then

$\gamma([0,T])\cap K\neq\phi.$

We then study the relationship between Gromov hyperbolicity and unbounded
visibility and prove the following variant of Theorem 1.4.

######  Theorem 1.11.

Let $\Omega\subset\mathbb{R}^{n}$ be a domain. Suppose $(\Omega,k_{\Omega})$
is Gromov hypoerbolic and $\Omega$ is GCEC-compatible domain. Then the
following are equivalent

  * (i)

$\Omega$ is a visibility domain

  * (ii)

the identity map $id_{\Omega}:\Omega\to\Omega$ extends to a continuous
surjective map from $\overline{\Omega}^{G}$ onto $\overline{\Omega}^{End}$

See Definition 7.1 for GCEC compatible domain.

###  1.6. Outline of the paper

The organization of the paper is as follows. In Sect.2, we recall some basic
facts and tools which are going to used in proving our main results. In
Sect.3, we prove the general visibility criteria and using this we prove
visibility of some classical domain such as John domains, domains satisfying
quasihyperbolic boundary conditions. In Sect.4, we prove Theorem 1.4 and
Theorem 1.5. In sect.5, we prove Theorem 1.6 and Theorem 1.7. In Sect.6, we
study continuous extension of quasihyperbolic isometries and quasiisometries
and prove Theorem 1.8 and Theorem 1.9. In section 7, we study the visibility
of unbounded domains using end compactification. In Sect.8, we provide some
examples of visibility domains which are neither John domains nor the domains
which satisfies quasihyperbolic boundary conditions.

###  1.7. Basic Notations

We end the introduction by fixing some basic notations which are used in this
paper.

  * (1)

For $x\in\mathbb{R}^{n}$, $||x||$ will denote the standard Euclidean norm and,
for $x,y\in\mathbb{R}^{n}$, $d_{Euc}(x,y)=||x-y||$ will denote the Euclidean
distance.

  * (2)

For a bounded domain $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$,
$\partial_{Euc}\Omega$ and $\overline{\Omega}^{Euc}$ denote the Euclidean
boundary and Euclidean closure of $\Omega$ respectively.

  * (3)

$\overline{\Omega}^{End}$ denote the end compactification of
$\overline{\Omega}^{Euc}$.

  * (4)

$E\left(\overline{\Omega}^{Euc}\right)$ denote the space of end of
$\overline{\Omega}^{Euc}$

  * (5)

$\overline{\Omega}^{End}=\overline{\Omega}^{Euc}\cup
E\left(\overline{\Omega}^{Euc}\right)$.

  * (6)

$\partial\overline{\Omega}^{End}=\overline{\Omega}^{End}\setminus\Omega=\partial_{Euc}\Omega\cup
E\left(\overline{\Omega}^{Euc}\right)$ denote the boundary of end
compactification.

##  2\. Preliminaries

###  2.1. Metric geometry

Let $(X,d)$ be a metric space. A curve is a continuous function
$\gamma:[a,b]\rightarrow X$. Let $\mathcal{P}$ denote set of all partitions
$a=t_{0}<t_{1}<t_{2}<\cdots<t_{n}=b$ of the interval $[a,b]$. The length of
the curve $\gamma$ in the metric space $(X,d)$ is

$l_{d}(\gamma)=\sup_{\mathcal{P}}\sum_{k=0}^{n-1}d(\gamma(t_{k}),\gamma(t_{k+1})).$

A curve is said to be rectifiable if $l_{d}(\gamma)<\infty$. A metric space
$X$ is said to be rectifiably connected if every pair of points $x,y\in X$ can
be joined by a rectifiable curve. For a rectifiable curve $\gamma$ we define
arc length $s:[a,b]\rightarrow\left[0,l_{d}(\gamma)\right]$ by
$s(t)=l_{d}(\gamma|_{[a,t]})$. The arc length function is of bounded
variation. For any rectifiable curve $\gamma:[a,b]\to X$, there is a unique
map $\gamma_{s}:[0,l_{d}(\gamma)]\to X$ such that $\gamma=\gamma_{s}\circ s$,
and such a curve $\gamma_{s}$ is called the arclength parametrization of
$\gamma$.

For $x,y\in X$, the inner length metric $\lambda_{X}(x,y)$ is defined by

$\lambda_{X}(x,y)=\inf\\{l_{d}(\gamma):\gamma\mbox{ is a rectifiable arc
joining }x\mbox{ and }y\\}.$

Let $\rho:X\rightarrow[0,\infty]$ be a Borel function. The $\rho$-length of a
rectifiable curve $\gamma$ is

$l_{\rho}(\gamma)=\int_{\gamma}\rho\,ds=\int_{a}^{b}\rho(\gamma(t))\,ds(t)=\int_{0}^{l_{d}(\gamma)}\rho\circ\gamma_{s}(t)\,dt.$

If $X$ is rectifiably connected, then the $\rho$-distance between two points
$x,y\in X$ is

$d_{\rho}(x,y)=\inf\\{l_{\rho}(\gamma):\gamma\mbox{ is a rectifiable curve
joining }~~x,y\mbox{ in }X\\}.$

where infimum is taken over all rectifiable curves joining $x$ and $y$ in $X$.
In general, $d_{\rho}$ need not be a metric since $d_{\rho}(x,y)$ could be
zero or infinite. In order to ensure that $(X,d_{\rho})$ is a metric space, we
must require that $0<d_{\rho}(x,y)<\infty$. To differentiate this, we call
such $\rho$ a metric-density or simply, density on $X$. However, if $\rho$ is
positive and continuous then $d_{\rho}$ is a metric and, we say $(X,d_{\rho})$
is a conformal deformation of $(X,d)$ by the conformal factor $\rho$.

### Maps, Paths and geodesics

######  Definition 2.1.

A curve $\gamma:[a,b]\to X$ is said to be geodesic if it is an isometry i.e.,
for all $t,t^{\prime}\in[a,b]$,

$d(\gamma(t),\gamma(t^{\prime}))=|t-t^{\prime}|.$

An isometry $\gamma:[0,\infty)\to X$ is called a geodesic ray, and an isometry
$\gamma:(-\infty,\infty)\to X$ is called a geodesic line. A metric space $X$
is said to be:

  * (1)

geodesic if every pair of points $x,y\in X$ can be joined by a geodesic,

  * (2)

proper if every closed ball is compact in $X$

  * (3)

intrinsic or path or length space if for all $x,y\in X$, we have

$d(x,y)=\lambda_{X}(x,y).$

To see the connection between these spaces we have the following well-known
theorem of Hoepf and Rinow (see, [16, I, Proposition 3.7]).

Theorem A. Let $(X,d)$ be an intrinsic space. Assume $(X,d)$ is complete and
locally compact. Then

  * (1)

$(X,d)$ is proper.

  * (2)

$(X,d)$ is geodesic.

######  Remark 2.1.

It is worth to mention that Martin [38, Theorem 2.8] proved that if $\rho$ is
a continuous density and it defines a complete metric space $(X,d_{\rho})$,
then $(X,d_{\rho})$ is a geodesic metric space.

######  Definition 2.2.

Let $\lambda\geq 1$ and $\mu\geq 0$ and $(X,d)$ and $(Y,d^{\prime})$ be metric
spaces. We say that a map (not necessarily continuous) $f:X\to Y$ is a
$(\lambda,\mu)$\- rough quasi-isometry or just quasi-isometry if

(2.1) $\lambda^{-1}d(x,y)-\mu\leq d^{\prime}(f(x),f(y))\leq\lambda
d(x,y)+\mu,$

for all $x,y\in X$, and for every $y\in Y$, we have

$d^{\prime}(y,f(X))\leq\mu.$

In the case where $f:I\to Y$ is a map of an real interval $I$, we say that
such a map is $(\lambda,\mu)$-quasi-isometric path. Further, $f$ is said to be
quasi-isometric embedding if only (2.1) is satisfiesd.

######  Remark 2.2.

An arc $\gamma$ joining $x$ and $y$ in a metric space $X$ is said to be
$\lambda$-quasigeodesic, $\lambda\geq 1$, if

$l(\gamma[u,v])\leq\lambda d(u,v)$

for all $u,v\in\gamma$. In such a case the arc length parametrization
$\gamma_{s}:[0,l(\gamma)]\to\gamma$ satisfies the inequality

$\lambda^{-1}|t-t^{\prime}|\leq
d(\gamma_{s}(t),\gamma_{s}(t^{\prime}))\leq\lambda|t-t^{\prime}|.$

Thus $\gamma$ is $(\lambda,0)$-quasi-isometry.

###  2.2. Hyperbolic geometry

######  Definition 2.3.

Let $\rho:\mathbb{D}\rightarrow(0,\infty)$ defined by

$\rho(z)=\frac{1}{1-|z|^{2}}\,.$

Clearly, $\rho$ is continuous. Thus, the hyperbolic length of $\gamma$ is
defined as

(2.2)
$l_{h}(\gamma)=\int_{\gamma}{\rho}\,ds=\int_{\gamma}\frac{|dz|}{1-|z|^{2}}.$

######  Definition 2.4.

Let $z_{1},z_{2}\in\mathbb{D}$. Then the hyperbolic distance $h_{D}$ of
$z_{1},z_{2}$ is defined by

(2.3)
$h_{\mathbb{D}}(z_{1},z_{2})=\inf_{\gamma}\int_{\gamma}\frac{|dz|}{1-|z|^{2}}$

where the infimum is taken over all piecewise smooth curves $\gamma$
connecting $z_{1}$ and $z_{2}$.

For every hyperbolic plane domain $\Omega$ (i.e., $\mathbb{C}\setminus\Omega$
contains atleast two points) there is a unique maximal (or unique complete)
metric on $\Omega$ that has constant Gaussian curvature $-1$. For more details
we refer to [6]. We call such metric as the hyperbolic metric on $\Omega$, and
we denote it by $h_{\Omega}$.

### Hyperbolic Geodesic.

A curve $\gamma$ in $\Omega$ joining $x$ and $y\in\Omega$ is said to be
hyperbolic geodesic if

$l_{h}(\gamma)=h_{\Omega}(x,y)$

###  2.3. Quasihyperbolic metric

Let $\Omega$ be proper subdomain in $\mathbb{R}^{n}$ and let
$d_{Euc},\partial_{Euc}\Omega$ and $\overline{\Omega}^{Euc}$ denote the
Euclidean metric on $\Omega$, Euclidean boundary and Euclidean closure of
$\Omega$ respectively. Define $k:\Omega\to(0,\infty)$ by

$k(z)=\frac{1}{\delta_{\Omega}(z)}=\frac{1}{d(z,\partial_{Euc}\Omega)}.$

Define the quasihyperbolic length of a rectifiable curve $\gamma$ in $\Omega$
by

$l_{k}(\gamma)=\int_{\gamma}\rho(z)\,ds=\int_{\gamma}\frac{ds}{d(z,\partial_{Euc}\Omega)}.$

Let $k_{D}$ denote the quasihyperbolic metric in $\Omega$ which is defined by

(2.4) $k_{\Omega}(x,y)=\inf_{\gamma}l_{k}(\gamma)$

where the infimum is taken over all rectifiable curves $\gamma$ in $\Omega$
joining $x$ to $y$.

Quasihyperbolic metric satisfies the following properties.

  * (i)

$k_{\Omega}=h_{\Omega}$ when $\Omega$ is a half space in $\mathbb{R}^{n}$,
where $h$ is the hyperbolic metric

  * (ii)

$k_{\Omega}\leq h_{\Omega}\leq 2k_{\Omega}$, when $\Omega$ is a ball in
$\mathbb{R}^{n}$.

  * (iii)

$k_{\Omega}/2\leq h_{\Omega}\leq 2k_{\Omega}$, when $\Omega$ planar simply
connected domain.

  * (iv)

$c_{\Omega}k_{\Omega}\leq h_{\Omega}\leq 2k_{\Omega}$, when $\Omega$ is a
modulated domain, where $c_{\Omega}$ is constant depending only on the domain
$\Omega$.

  * (v)

$h_{\Omega}\leq 2k_{\Omega}$ for every hyperbolic domain $\Omega$.

For further geometric properties of quasihyperbolic metric, we refer to [25,
58].

In view of [12, Proposition 2.8], it is well-known that $(\Omega,k_{\Omega})$
is a complete, proper and geodesic metric space.

### Properties of quasihyperbolic metric

We recall some basic estimates for the quasihyperbolic metric which have been
first introduced by Gehring and Palka [24, 2.1] in $\mathbb{R}^{n}$. Let
$\Omega\subsetneq\mathbb{R}^{n}$ be a domain and $x,y\in\Omega$ and let
$\gamma$ be a rectifiable curve in $\Omega$ joining $x$ and $y$. Then we have
the following:

(2.5) $k_{\Omega}(x,y)\geq
log\left(1+\frac{\lambda_{\Omega}(x,y)}{\min\\{\Delta(x),\Delta(y)\\}}\right)\geq
log\left(1+\frac{|x-y|}{\min\\{\Delta(x),\Delta(y)\\}}\right)\geq\left|log\frac{\Delta(y)}{\Delta(x)}\right|$

and

(2.6) $l_{k}(\gamma)\geq
log\left(1+\frac{l(\gamma)}{\min\\{\Delta(x),\Delta(y)\\}}\right).$

The inequality 2.5 implies that

(2.7) $\lim_{y\to\partial_{Euc}\Omega}k_{\Omega}(x,y)=\infty$

for $x\in\Omega$. Using (2.7), we have that $(\Omega,k_{\Omega})$ is a
complete metric space. Moreover, it induces the usual topology [24, Corollary
2.2]. The following lemma is a simple fact which we will frequently use in our
proofs.

######  Lemma 2.8.

[24, Corollary 2.3] Let $\Omega$ be a proper subdomain in $\mathbb{R}^{n}$.
Then for each set $F\subset\Omega$,

$k_{\Omega}(F)=\sup_{x,y\in F}k_{\Omega}(x,y)<\infty$

if, and only if, $\overline{F}\subset\Omega$.

The following observation immediately follows from (2.5).

######  Observation 1.

Let $\Omega$ be a bounded domain. If sequences $\\{z_{k}\\}$,
$\\{w_{k}\\}\subset\Omega$ converge to two different points on
$\partial_{Euc}\Omega$, then

$\lim_{k\to\infty}k_{\Omega}(z_{k},w_{k})=\infty.$

### Quasi-hyperbolic geodesic

A rectifiable curve $\gamma$ from $x$ to $y$ in $\Omega$ is said to be
quasihyperbolic geodesic or simply, geodesic, if
$k_{\Omega}(x,y)=l_{k}(\gamma)$. Obviously, each subarc of a quasi-hyperbolic
geodesic is again a geodesic. Gehring and Osgood [25, Lemma 1] have proved the
existence of quasihyperbolic geodesics between any two points of the domain
$\Omega$. For important results on quasihyperbolic geodesics in domain in
$\mathbb{R}^{n}$, we refer to Martin [38]. Moreover, Martin [38, Corollary
4.8] has proved that quasihyperbolic geodesics in $\mathbb{R}^{n}$ are
$C^{1,1}$ smooth i.e., the arc length parametrization has Lipschitz continuous
derivatives.

######  Remark 2.3.

It is often convenient to parametrize an arc by quasihyperbolic length. We say
that $g:[0,r]\to\Omega$ is a quasihyperbolic parametrization if
$l_{k}(g[0,t])=t$ for all $t\in[0,r]$. Then $r=l_{k}(g)$ and

$|g^{\prime}(t)|=\delta_{\Omega}(g(t))$

almost everywhere. Every rectifiable arc $\gamma\subset\Omega$ has a
quasihyperbolic parametrization $g:[0,r]\to\gamma$, and $g$ satisfies the
Lipschitz condition

$|g(s)-g(t)|\leq M|s-t|$

where $M=\max\\{\delta(x):x\in\gamma\\}$.

If $\gamma$ is a quasihyperbolic geodesic, then
$g:[0,r]\to(\Omega,k_{\Omega})$ is an isometry and we say that $g$ is a
quasihyperbolic geodesic path from $g(0)$ to $g(r)$. Then in view of
Martin[38, 4.8], $g$ is $C^{1}$.

Throughout this paper, we denote, in short, quasihyperbolic geodesic as qh-
geodesic whenever required.

###  2.4. Gromov hyperbolic space

Let $(X,d)$ be a geodesic metric space, a geodesic triangle is the union of
geodesics $\gamma_{i}:[a_{i},b_{i}]\to X$, $i=1,2,3$ such that $a_{i}<b_{i}$
for every $i=1,2,3$ and
$\gamma_{1}(b_{1})=\gamma_{2}(a_{1}),\gamma_{2}(b_{2})=\gamma_{3}(a_{3}),\gamma_{3}(b_{2})=\gamma_{1}(a_{1})$.
The geodesics $\gamma_{1},\gamma_{2}$ and $\gamma_{3}$ are called sides of the
geodesic triangle.

######  Definition 2.5.

Let $\delta\geq 0$. A geodesic metric space $(X,d)$ is said to be
$\delta$-hyperbolic if every geodesic triangle is $\delta$-thin. That is, each
side of the geodesic triangle is contained in the $\delta$-neighborhood of the
union of the other two sides.

######  Definition 2.6.

A geodesic metric space $(X,d)$ is said to be Gromov hyperbolic if it is
$\delta$-hyperbolic for some $\delta\geq 0$.

### Examples

Clearly, any bounded geodesic metric space is Gromov hyperbolic. Clearly
$\mathbb{R}$ is Gromov hyperbolic with $\delta=0$. For $n\geq 2$,
$\mathbb{R}^{n}$ is not Gromov hyperbolic. The hyperbolic space
$\mathbb{H}^{n}$ is Gromov hyperbolic with $\delta=\log 3$. Moreover, every
CAT($k$) space, $k<0$ is Gromov hyperbolic for $\delta=(\log\,3)/\sqrt{-k}$.
In particular, every complete simply connected Riemannian manifold with
sectional curvature bounded from above by $-k$, with $k>0$, is Gromov
hyperbolic. One can construct more examples by considering the Gromov
hyperbolicity of Riemann surfaces, for this we refer to [48]. In particular,
some classical surfaces such as punctured unit disk and the annuli with
Poincaré metric are Gromov hyperbolic (for the bound on the hyperbolicity
constant see [48, Theorem 5.1].

### Geodesics and quasigeodesics in Gromov hyperbolic spaces

One of the important propetry of Gromov hyperbolicity is that it is invariant
under quasi-isometries. The proof of this result is based on the stability of
geodesics in $\delta$-hyperbolic spaces. Geodesics in Gromov hyperbolic spaces
exhibit some stability and are generally well-behaved. which we will discuss
next. In particular, quasi-geodesics and geodesics sharing the same endpoints
stay close to each other.

######  Definition 2.7.

Let $X$ and $Y$ be two non-empty subsets of a metric space $(M,d)$. We define
their Hausdorff distance $d^{\mathcal{H}}(X,Y)$ by

$d^{\mathcal{H}}(X,Y)=\max\left\\{\sup_{x\in X}d(x,Y),\,\sup_{y\in
Y}d(X,y)\right\\},$

where $d(a,B)=\inf_{b\in B}d(a,b)$ denotes the distance from a point $a\in M$
to a subset $B\subset M$.

######  Theorem 2.1.

(Shadowing lemma/ Geodesic stability) Suppose $(X,d)$ be a Gromov hyperbolic
space with $\delta\geq 0$. For any $\lambda\geq 1,\mu\geq 0$ there exists a
constant $M=M(\delta,\lambda,\mu)$ such that: If $\gamma:[a_{1},b_{1}]\to X$
is a $(\lambda,\mu)$-quasi-geodesic and $\sigma:[a_{2},b_{2}]\to X$ is a
geodesic with $\gamma(a_{1})=\sigma(a_{2})$, $\gamma(a_{2})=\sigma(a_{2})$,
then

$d^{\mathcal{H}}(\gamma([a_{1},b_{1}]),\sigma([a_{2},b_{2}]))\leq M,$

Theorem 2.1 has the following important consequence.

######  Corollary 2.9.

A geodesic metric space $X$ is hyperbolic if and only if, for every
$\lambda\geq 1$ and every $\mu\geq 0$, there exists a constant $M$ such that
every $(\lambda,\epsilon)$-quasi-geodesic triangle in $X$ is $M$-thin. (If $X$
is $\delta$-hyperbolic, then $M$ depends only on $\delta,\lambda$ and
$\epsilon$.)

Further, Theorem 2.1 gives the invariance of Gromov hyperbolicity under quasi-
isometry.

######  Theorem 2.2.

Let $X$ and $Y$ be geodesic metric spaces and let $f:X\rightarrow Y$ be a
quasi-isometric embedding. If $Y$ is Gromov hyperbolic then $X$ is Gromov
hyperbolic. (i.e., If $Y$ is $\delta$-hyperbolic and $f:X\rightarrow Y$ is a
$(\lambda,\mu)$-quasi-isometric embedding, then $X^{\prime}$ is
$\delta^{\prime}$-hyperbolic, where $\delta^{\prime}$ depends only on
$\delta,\lambda$ and $\mu$). In addition, if $f(X)=Y$ then $X$ is Gromov
hyperbolic if, and only if, $Y$ is Gromov hyperbolic

######  Remark 2.4.

Theorem 2.2 can be used to construct non-examples of Gromov hyperbolic spaces.
For instance, consider the two-dimensional jungle-gym (a $\mathbb{Z}^{2}$
covering of a Riemann surface of genus $\geq 2$) equipped with a metric of
constant negative curvature. This is quasi-isometric to $\mathbb{R}^{2}$ and
hence not Gromov hyperbolic. Infact, geodesic triangles are nearly Euclidean.

###  2.5. Gromov compactification

For the rest of this subsection we assume that $(X,d)$ is a proper geodesic
Gromov hyperbolic metric space. Let $p\in X$ and $\mathcal{G}_{p}$ be the
space of all geodesic rays $\gamma:[0,\infty)\to X$ with $\gamma(0)=p$,
endowed with the topology of uniform convergence on compact subsets of
$[0,\infty)$. Define an equivalence relation $\sim$ on $\mathcal{G}_{p}$ as
follows: Let $\gamma,\sigma\in\mathcal{G}_{p}$ be two geodesic rays. Then

$\gamma\sim\sigma\iff\sup_{t\geq 0}d(\gamma(t),\sigma(t))<\infty.$

######  Definition 2.8.

  * (1)

The Gromov boundary of $X$ is denoted by $\partial_{G}X$ and is defined as the
quotient space $\mathcal{G}_{p}/\sim$ endowed with the quotient topology.

  * (2)

The Gromov closure of $X$ is denoted by $\overline{X}^{G}$ and is defined as
$\overline{X}^{G}=X\cap\partial_{G}X$.

######  Remark 2.5.

The definition of the Gromov boundary does not depend on the base point.
Indeed for given $p,q\in X$, there is a natural map
$J:\mathcal{G}_{p}/\sim\,\to\mathcal{G}_{q}/\sim$. Let
$[\gamma]\in\mathcal{G}_{p}$, where $\gamma:[0,\infty)\to X$ is a geodesic ray
with $\gamma(0)=p$. For each $n\in\mathbb{N}$, consider a sequence of
geodesics $\gamma_{n}:[0,R_{n}]\to X$ with $\gamma_{n}(0)=q$ and
$\gamma_{n}(R_{n})=\gamma(n)$. Then by Arźela-Ascoli’s theorem, up to a
subsequence, we can assume that $\gamma_{n}$ converges uniformly on every
compact subset to a geodesic ray $\sigma:[0,\infty)\to X$ with $\gamma(0)=q$.
Now define, $J([\gamma])=[\sigma]$. Since $(X,d)$ is Gromov hyperbolic, $J$ is
well defined and onto. Infact $J$ is a homeomorphism (see [16, Chapter
III.H.3, Lemma 3.3, Lemma 3.6 and Proposition 3.7]).

### Gromov topology

The set $\overline{X}^{G}$ admits a natural topology and with this topology
$\overline{X}^{G}$ is a compactification of $X$, known as Gromov
compactification of $X$ (see for instance [16, Chapter III.H.3, Proposition
3.7]).

To understand this topology we need some additional notation: A generalized
ray is either a geodesic ray or a geodesic. For a geodesic ray
$\gamma\in\mathcal{G}_{p}$ define $\mbox{End}(\gamma)=\\{\sigma:[0,\infty)\to
X:\sigma\mbox{ is a geodesic ray with }\gamma(0)=p\mbox{ and
}\sigma\sim\gamma\\}=[\gamma]$, and for a geodesic $\gamma:[0,R]\to X$ with
$\gamma(0)=p$, where $R>0$, define $\mbox{ End}(\gamma)=\gamma(R)$. Then a
sequence $\\{x_{n}\\}_{n\in\mathbb{N}}$ in $\overline{X}^{G}$ converges to
$x\in\overline{X}^{G}$ (denoted by $x_{n}\stackrel{{\scriptstyle
Gromov}}{{\longrightarrow}}x$), if, and only if, for every generalized ray
$\gamma_{n}$ with $\gamma_{n}(0)=p$ and $\mbox{ End}(\gamma_{n})=x_{n}$ its
every subsequence has a subsequence which converges uniformly on every compact
subset to a generalized ray $\gamma$ with $\gamma(0)=x_{0}$ and $\mbox{
End}(\gamma)=x$.

###  2.6. Gromov product

There is an alternate description of Gromov hyperbolic space via Gromov
prodfuct which is useful in the following sense

  * (i)

This notion allows us to define Gromov hyperbolicity for arbitrary metric
space (not necessarily geodesic).

  * (ii)

This notion allows us to define Gromov boundary for arbitrary metric space
(not necessairly Gromov hyperbolic)

  * (iii)

This notion allows us to define a canonical gauge of metrics on Gromov
boundary, which are known as visual metrics.

######  Definition 2.9.

Fix a base point $o\in X$ and let $x,y\in X$. The Gromov product of $x$ and
$y$ is defined as

$(x|y)_{o}=\frac{1}{2}(d(x,o)+d(o,y)-d(x,y)).$

We deine the Gromov hyperbolicity of arbitrary metric space which coincides
with the thin triangles definition in the case of geodesic metric space.

######  Definition 2.10.

Let $\delta\geq 0$. A metric space $(X,d)$ is said to be $\delta$-hyperbolic
if there exists a $\delta\geq 0$ such that $\forall\,x,y,z,p\in X$, the
following inequality holds:

$(x|z)_{p}\,\geq\,\min\\{(x|y)_{p},(y|z)_{p}\\}-\delta.$

######  Definition 2.11.

  * (1)

We say that a sequence $\\{x_{n}\\}\subset X$ is a Gromov sequence if it goes
to infinity in Gromov sense i.e., if

$\lim_{m,n\to\infty}(x_{n}|x_{m})_{o}=+\infty.$

  * (2)

Let $\Gamma^{G}$ denote the set of Gromov sequence and
$\\{x_{n}\\},\\{y_{n}\\}\in\Gamma^{G}$. If $X$ is Gromov hyperbolic then the
following relation

$x_{n}\sim_{s}y_{n}\iff\lim_{n\to\infty}(x_{n}|y_{n})_{o}=\infty$

is an equivalence relation on $\Gamma^{G}$. This definition is also irrelevant
of chosen point $o$, since $|(x|y)_{p}-(x|y)_{q}|\leq d(p,q)$.

  * (3)

We define the set

$\partial_{G}^{S}X:=\Gamma^{G}/\sim_{s}.$

as the Gromov boundary.

The set $\partial_{G}^{S}X$ admits a canonical bijection with $\partial_{G}X$,
that is,

$\partial_{G}^{S}X\to\partial_{G}X\,\,\,\,\,\mbox{ by
}\,\,\,\,\,[\gamma]\to[(\gamma(n))_{n\in\mathbb{N}}]_{s}.$

The following proposition says that two sequences converge to the same points
of the Gromov boundary if, and only if, they are equivalent.

######  Proposition 2.10.

Let $x_{n}$, $y_{n}$ be two sequences such that

$x_{n}\to\xi\in\partial_{G}X,\,\,\,\,\,\,y_{n}\to\eta\in\partial_{G}X.$

Then $\xi=\eta$ if, and only if, $(x_{n}|y_{n})_{o}\to\infty$.

Moreover the Gromov boundary $\partial_{G}X$ has a canonical quasiconformal
structure in the following sense.

######  Theorem 2.3.

Let $X$ and $Y$ be proper geodesic Gromov hyperbolic spaces. If $f:X\to Y$ is
quasi-isometry, then $f$ extends to a homeomorphism
$\overline{f}:\partial_{G}X\to\partial_{G}Y$. Moreover, the boundary map is
quasi-conformal.

### Visibility in Gromov hyperbolic spaces

In view of the Gromov topology of $\overline{X}^{G}$ we have the following
observation for geodesic rays and geodesic lines.

######  Remark 2.6.

For any geodesic ray $\gamma:[0,\infty)\to X$,

$\lim_{t\to\infty}\gamma(t)=\mbox{ End }(\gamma)=[\gamma]\in\partial_{G}X$

and for any geodesic line $\gamma:(-\infty,\infty)\to X$, both the following
limits exist and

$\lim_{t\to-\infty}\gamma(t)\neq\lim_{t\to\infty}\gamma(t).$

For a geodesic line $\gamma$, we denote
$\gamma(-\infty)=\lim_{t\to-\infty}\gamma(t)$ and
$\gamma(\infty)=\lim_{t\to\infty}\gamma(t)$.

Following theorem tells us that any point in the Gromov hyperbolic space can
be joined to any Gromov boundary point via geodesic ray.

######  Theorem 2.4.

[16, Chapter III.H.3, Lemma 3.1] For each $p\in X$ and $\xi\in\partial_{G}X$
there exists a geodesic ray $\gamma:[0,\infty)\to X$ such that $\mbox{ End
}(\gamma)=\xi$.

Gromov hyperbolic space has the following visibility property.

######  Theorem 2.5.

[16, Chapter III.H.3, Lemma 3.2] Let $(X,d)$ be a proper, geodesic and Gromov
hyperbolic metric space, then for each distinct points
$\xi,\eta\in\partial_{G}X$,
there exists a geodesic line $\gamma$ with $\gamma(-\infty)=\xi$ and
$\gamma(\infty)=\eta$.

Theorem 2.5 can be interpret in the following equivalent way. For the proof
see Section 9.

######  Theorem 2.6.

Let $(X,d)$ be a proper, geodesic and Gromov hyperbolic metric space. If
$\xi,\eta\in\partial_{G}X$ and $V_{\xi}$ and $V_{\eta}$ are neighborhoods in
$\overline{X}^{G}$ of $\xi$ and $\eta$ respectively so that
$\overline{V_{\xi}}\cap\overline{V_{\eta}}=\phi$ then there exists a compact
set $K\subset X$ such that: if $\gamma:[0,T]\to X$ is a geodesics with
$\gamma(0)\in V_{\xi}$ and $\gamma(T)\in V_{\eta}$, then $\gamma([0,T])\cup
K\neq\phi$.

One can observe that the Theorem 2.6 is equivalent to the following.

######  Remark 2.7.

Let $(X,d)$ be a proper, geodesic and Gromov hyperbolic metric space. If
$\xi,\eta\in\partial_{G}X$, $\xi\neq\eta$ and $z_{k}$, $w_{k}$ be any sequence
with $z_{k}\stackrel{{\scriptstyle Gromov}}{{\longrightarrow}}\xi$ and
$w_{k}\stackrel{{\scriptstyle Gromov}}{{\longrightarrow}}\eta$, then there
exists a compact set $K\subset X$ with the following property: if
$\gamma_{k}:[0,T_{k}]\to X$ be a sequence of geodesics with
$\gamma_{k}(0)=z_{k}$ and $\gamma_{k}(T_{k})=w_{k}$, then
$\gamma_{k}([0,T_{k}])\cap K\neq\phi$.

###  2.7. Quasihyperbolic Domains

In this section we recall some classical domains for which we will study the
visibility property.

### Uniform Domains

Let $C\geq 1$. A domain $\Omega\subset\mathbb{R}^{n}$ is said to be
$C$-uniform domain if for each pair of points $x,y\in\Omega$, there exists a
rectifiable arc $\gamma$ in $\Omega$ joining $x$ and $y$ such that

  * (i)

$\min\\{l(\gamma[x,z]),l(\gamma[z,y])\\}\leq C\,d(z,\partial D)$, for all
$z\in\gamma$, and

  * (ii)

$l(\gamma)\leq C|x-y|$,

where $l(\gamma)$ denotes the arc length of $\gamma$ in $(D,|.\,|)$, where
$|.\,|$ is metric induced by norm, and $\gamma[x,z]$ denotes the part of
$\gamma$ between $x$ and $z$. Also we say that curve $\gamma$ is a double
$C$-cone arc.

### Inner uniform domains

Let $C\geq 1$. A domain $\Omega\subset\mathbb{R}^{n}$ is said to be $C$-inner
uniform in the norm metric if for each pair of points $x,y\in\Omega$, there
exists a rectifiable arc $\gamma$ in $D$ joining $x$ to $y$ such that

  * (i)

$\min\\{l(\gamma[x,z]),l(\gamma[z,y])\\}\leq C\,d(z,\partial D)$, for all
$z\in\gamma$,

  * (ii)

$l(\gamma)\leq C\lambda_{D}(x,y)$.

### John Domain

Let $C\geq 1$. A domain $\Omega\subset\mathbb{R}^{n}$ is said to be $C$-John
domain with center $x_{0}\in\Omega$ if every point $x\in\Omega$ can be joined
to $x_{0}$ by a rectifiable curve $\gamma$ in $\Omega$ which satisfies

(2.11) $l(\gamma[x,y])\leq C\delta_{\Omega}(y)\,\,\,\mbox{ for all
}y\in\gamma$

and we say that such a curve $\gamma$ is a $C$-cone arc. Note that from above
definition it follows that $\Omega$ is bounded. The following definition works
well for unbounded domains.

######  Definition 2.12.

Let $C\geq 1$. A domain $\Omega\subset\mathbb{R}^{n}$ is said to be $C$-John
domain if for each pair of points $x,y\in\Omega$, there exists a rectifiable
arc $\gamma$ in $\Omega$ joining $x$ and $y$ such that

$\min\\{l(\gamma[x,z]),l(\gamma[z,y])\\}\leq C\delta_{\Omega}(z),\mbox{ for
all }z\in\gamma.$

######  Remark 2.8.

By [60, Lemma 2.2], we have that if $\Omega$ is $C$-John domain with center
$x_{0}\in\Omega$, the $\Omega$ is $C$-John. Moreover if $\Omega$ is $C$-John
then $\Omega$ is $4C^{2}$-John with center $x_{0}\in\Omega$, where $x_{0}$ is
such that $\delta_{\Omega}(x_{0})=\max\delta_{\Omega}(x)$. Since we are
considering bounded domains only so we will use both the definitions according
to our convinience.

######  Remark 2.9.

We mention some of the remarks which gives the connection between the
aforesaid domains.

  * (i)

$c$-uniform implies inner $c$-uniform implies $c$-John

  * (ii)

$\mathbb{R}^{2}\setminus\\{(x,0):x\geq 0\\}$ is inner uniform but not unifrom.

  * (iii)

$\mathbb{R}^{2}\setminus\\{(n,0):n\in\mathbb{N}\\}$ is a John domain but not
inner uniform.

### Domains satisfying quasihyperbolic boundary conditions (QHBC)

Following [27] We say that a domain $\Omega\subset\mathbb{R}^{n}$ satisfies a
quasihyperbolic boundary condition if for some $x_{0}$ there exists $\beta>0$
and $C_{0}=C_{0}(x_{0})$ such that

(2.12)
$k_{\Omega}(x_{0},x)\leq\frac{1}{\beta}\log\dfrac{\delta_{\Omega}(x_{0})}{\delta_{\Omega}(x)}+C_{0}\,\,\,$

for all $x\in\Omega$. It easily follows that if $\Omega$ satisfies (2.12) for
some $x_{0}\in\Omega$, then $\Omega$ satisfies (2.12) for any $x_{1}\in\Omega$
with same $\beta$ and possibly a different $C=C_{0}+k_{\Omega}(x_{0},x_{1})$.
Also, notice that, in view of (2.5), it is necessary that $\beta\leq 1$ . In
short, we say that $\Omega$ is $\beta$-QHBC domain. Every $\beta$-QHBC domain
$\Omega$ is bounded (see [27, Lemma 3.9]). Infact, $\mbox{diam}(\Omega)\leq
2\beta e^{C_{0}/\beta}$.

######  Remark 2.10.

In view of [27, Lemma 3.11], we have that every $C$-John domain with center
$x_{0}\in\Omega$ is a $\beta$-QHBC domain.

###  2.8. Gromov hyperbolicity in quasihyperbolic geometry

######  Definition 2.13.

Let $\delta\geq 0$. A domain $\Omega\subset\mathbb{R}^{n}$ is said to be
$\delta$-hyperbolic if $(\Omega,k_{\Omega})$ is $\delta$-hyperbolic.

######  Definition 2.14.

A domain $\Omega\subset\mathbb{R}^{n}$ is said to be Gromov hyperbolic if
$(\Omega,k_{\Omega})$ is $\delta$-hyperbolic for some $\delta\geq 0$.

It is known that following domains in $\mathbb{R}^{n}$ are Gromov hyperbolic

  * (1)

All $c$\- uniform and inner $c$\- uniform domains are Gromov
$\delta=\delta(c)$ hyperbolic;

  * (2)

bounded convex domains in $\mathbb{R}^{n}$;

  * (3)

Quasiconformal image of Gromov hyperbolic domains. In particular, proper
simply connected domains in $\mathbb{C}$ are Gromov hyperbolic.

$\mathbb{R}^{2}\setminus\\{(n,0):n\in\mathbb{N}\\}$ is a John domain which is
not Gromov hyperbolic.

### Gehring-Hayman condition

Let $\Omega\subset\mathbb{R}^{n}$ be a proper subdomain. $\Omega$ satisfies a
Gehring-Hayman condition if there is a constant $C_{gh}\geq 1$ such that for
each pair of points $x,y\in\Omega$ and for each quasi-hyperbolic geodesic
joining $x$ and $y$, it holds that-

$l(\gamma[x,y])\leq C_{gh}l(\sigma[x,y]),$

where $\sigma$ is any other curve joining $x$ and $y$ in $\Omega$. In other
words, it says that quasihyperbolic geodesics are essentially the shortest
curves in $\Omega$.

### Ball separation condition

Let $\Omega\subset\mathbb{R}^{n}$ be a proper subdomain. $\Omega$ satisfies a
ball separation condition if there exists a constant $C_{bs}\geq 1$ such that
for each pair of points $x$ and $y$, for each quasihyperbolic geodesic
$\gamma$ joining $x$ and $y$, for every $z\in\gamma[x,y]$, and for every curve
$\sigma$ joining $x$ and $y$ it holds that-

$B(z,C_{bs}d(z,\partial\Omega))\cap\sigma[x,y]\neq\phi$

Bonk et. al. [12, Proposition 7.12, 7.14] have proved that Gehring-Hayman
condition and Ball separation condition are necessary for
$(\Omega,k_{\Omega})$ to be Gromov hyperbolic. Later, in 2003, Balogh and
Buckley [5, Theorem 0.1] proved that both the conditions together are
sufficient also. Thus, combining these two we have the following result.

######  Theorem 2.7.

[12, Proposition 7.12, 7.14], [5, Theorem 0.1] Let
$\Omega\subset\mathbb{R}^{n}$ be a proper subdomain. Then $(\Omega,k)$ is
Gromov hyperbolic if, and only if, $\Omega$ satisfies both a Gehring-Hayman
condition and a ball separation condition.

##  3\. Visibility of some classical domains

In this section we study the visibility of some classical domains introduced
in subsection 2.7. We first provide the proof of Theorem 1.1.

Proof of Theorem 1.1. Suppose $\Omega$ is not a visibility domain i.e. there
does not exist any compact set with the desired property. Assume there exists
$p,q\in\partial_{Euc}\Omega$, $p\neq q$, sequences
$\\{x_{k}\\},\\{y_{k}\\}\subset\Omega$ with $x_{k}\rightarrow p$ and
$y_{k}\rightarrow q$, and a sequence of quasihyperbolic geodesics
$\gamma_{k}:[a_{k},b_{k}]\rightarrow\Omega$ with $\gamma_{k}(a_{k})=x_{k}$ and
$\gamma_{k}(b_{k})=y_{k}$, such that

(3.1) $\max_{{}t\in[a_{k},{b_{k}]}}\delta_{\Omega}(\gamma_{k}(t))\rightarrow
0~,\mbox{ as }k\rightarrow\infty$

Then by suitable parametrization, we can assume $a_{k}\leq 0\leq b_{k}$ for
all $k\in\mathbb{N}$ and

$\delta_{\Omega}(\gamma_{k}(0))=\max_{t\in[a_{k},{b_{k}]}}\delta_{\Omega}(\gamma_{k}(t))\rightarrow
0~,\mbox{ as }k\rightarrow\infty$

Next we will prove the following claim which will give the contradiction

Claim: $\delta_{\Omega}(\gamma_{k}(0))\not\rightarrow 0\mbox{ as }k\to\infty$
Proof: Since $p,q\in\partial_{Euc}\Omega,p\neq q$, therefore there exists
$R>0$ and $k_{0}\in\mathbb{N}$ such that the following three holds

  * (i)

$|p-q|>R$,

  * (ii)

$|x_{k}-p|\leq R/3$, for all $k\geq k_{0}$ and,

  * (iii)

$|y_{k}-p|\geq R$, for for all $k\geq k_{0}$.

Let $\theta_{k}$ be the first point on $\gamma_{k}$ with $|\theta_{k}-p|=R$.
Consider $\sigma_{k}=\gamma_{k}[x_{k},\theta_{k}]$ and let
$z_{k}\in\sigma_{k}$ be a point with $|z_{k}-p|=2R/3$. Since $\sigma_{k}$ is
also quasihyperbolic geodesic in $\Omega$ joining $x_{k}$ and $\theta_{k}$ and
$\Omega$ is a $C$\- John domain in which every quasihyperbolic geodesic is a
double $B=B(C)$-cone arc, therefore we have

$\displaystyle B\delta_{\Omega}(z_{k})$ $\displaystyle\geq$
$\displaystyle\min\\{l(\gamma_{k}[x_{k},z_{k}]),l(\gamma_{k}[z_{k},\theta_{k}])\\}$
$\displaystyle\geq$ $\displaystyle\min\\{|z_{k}-x_{k}|,|z_{k}-\theta_{k}|\\}$

Now,

$\displaystyle|z_{k}-\theta_{k}|$ $\displaystyle=$
$\displaystyle|z_{k}-p+p-\theta_{k}|$ $\displaystyle\geq$
$\displaystyle|\theta_{k}-p|-|z_{k}-p|=R-\frac{2R}{3}=\frac{R}{3}$
$\displaystyle\geq$ $\displaystyle\frac{R}{3}$

and

$\displaystyle|x_{k}-z_{k}|$ $\displaystyle=$ $\displaystyle|x_{k}-p+p-z_{k}|$
$\displaystyle\geq$
$\displaystyle|x_{k}-p|-|z_{k}-p|=\frac{2R}{3}-\frac{R}{3}=\frac{R}{3}$
$\displaystyle\geq$ $\displaystyle\frac{R}{3}$

Therefore, in view of (3) we get

(3.3) $\delta_{\Omega}(z_{k})\geq\frac{R}{3B}$

Since $\delta_{\Omega}(\gamma_{k}(0))\geq\delta_{\Omega}(z_{k})$, therefore by
(3.3) it follows that

$\delta_{\Omega}(\gamma_{k}(0))\not\rightarrow 0\,\,\,\mbox{ as }k\to\infty.$

This completes the proof of the claim. Hence $\Omega$ is a visibility domain.
∎

### Visibility of Uniform domain

Recall the following result of Gehring and Osgood [25]

######  Theorem 3.1.

Each quasihyperbolic geodesic in a $c$-uniform domain
$\Omega\subset\mathbb{R}^{n}$ is a double $b$-cone arc, where the constant $b$
depends only on $c$.

As we know that every bounded uniform domain is John domain, Theorem 3.1 gives
us the following corollary

######  Corollary 3.4.

Every bounded uniform domain is a visibility domain.

Also since every bounded convex domain is uniform domain (see [52]), we have
the following corollary

######  Corollary 3.5.

Every bounded convex domain in $\mathbb{R}^{n}$ is a visibility domain.

###  3.1. General visibility criteria

This subsection is devoted to the proof of Theorem 1.2. To do so we need the
following lemma and it easily follows from (1.5).

######  Lemma 3.6.

Let $\phi$ as in Theorem 1.2. Then for every $\epsilon>0$ there exists
constants $-\infty<a^{*}<b^{*}<+\infty$ such that

$\int_{-\infty}^{a^{*}}\frac{1}{\phi^{-1}\left(\frac{|t|}{2}\right)}dt<\epsilon,$
$\int_{b^{*}}^{+\infty}\frac{1}{\phi^{-1}\left(\frac{t}{2}\right)}dt<\epsilon.$

Proof of Theorem 1.2. Suppose $\Omega$ is not a visibility domain i.e. there
does not exist any compact set with the desired property. Then there are
$p,q\in\partial_{Euc}\Omega$, $p\neq q$, sequences
$\\{x_{n}\\},\\{y_{n}\\}\subset\Omega$ with $x_{n}\rightarrow p$ and
$y_{n}\rightarrow q$ as $n\to\infty$, and a sequence of quasihyperbolic
geodesics $\gamma_{n}:[a_{n},b_{n}]\to\Omega$ joining $x_{k}$ and $y_{k}$ such
that

(3.7) $\max_{t\in[a_{n},b_{n}]}\delta_{\Omega}(\gamma_{n}(t))\rightarrow
0~,\mbox{ as }n\rightarrow\infty$

By a suitable quasihyperbolic parameterization
$g_{n}:[a_{n},b_{n}]\rightarrow\Omega$ of $\gamma_{n}$ with
$g_{n}(a_{n})=x_{n}$, $g_{n}(b_{n})=y_{n}$ and
$|b_{n}-a_{n}|=l_{k}(g_{n}[a_{n},b_{n}])$, we can assume that, for all
$n,a_{n}\leq 0\leq b_{n}$ and

(3.8)
$\max_{t\in[a_{n},b_{n}]}\delta_{\Omega}(g_{n}(t))=\delta_{\Omega}(g_{n}(0))$

Furthermore, by Remark 2.3 we have that each $g_{n}$ satisfies the following
Lipschitz condition

$|g_{n}(s)-g_{n}(t)|\leq M_{n}|s-t|,\,\,\,s,t\in[a_{n},b_{n}],$

where, $M_{n}=\max\\{\delta_{\Omega}(x):x\in\gamma_{n}\\}$. In view of (3.7),
we have that $M_{n}\to 0$. This gives us that each $g_{n}$ is $M$-Lipschitz
with respect to the Euclidean distance, where $M=\sup_{n\geq 1}\\{M_{n}\\}$
i.e., for all $n$, we have

$|g_{n}(s)-g_{n}(t)|\leq M|s-t|,\,\,\,s,t\in[a_{n},b_{n}].$

Therefore, by Arzela-Ascoli theorem, up to subsequence we can assume:

  * (i)

$a_{k}\rightarrow a\in[-\infty,0]$ and $b_{k}\rightarrow b\in[0,\infty]$

  * (ii)

$g_{k}$ converges uniformly on every compact subset of $(a,b)$ to a continuous
map $g:(a,b)\rightarrow\overline{\Omega}^{Euc}$.

  * (iii)

$g_{k}(a_{k})\rightarrow p$ and $g_{k}(b_{k})\rightarrow q$

We made the following observation

######  Observation 2.

We conclude from the fact that

$|g_{n}(a_{n})-g_{n}(b_{n})|\leq M|a_{n}-b_{n}|,\,\,\,\mbox{ for all }n$

that $a<b$. By Remark 2.3, we have that
$g_{n}:[a_{n},b_{n}]\to(\Omega,k_{\Omega})$ is an isometry. This gives us that

$|b-a|=\lim_{k\to\infty}|b_{k}-a_{k}|=\lim_{k\to\infty}k_{\Omega}(g_{k}(b_{k}),g_{k}(a_{k}))=\infty\,\,\,(\mbox{
by Observation \ref{observation}}).$

Therefore, it is clear that both $a$ and $b$ together can’t be finite.

######  Claim 1.

$g:(a,b)\rightarrow\overline{\Omega}^{Euc}$ is a constant map.

Proof of Claim 1. In view of (3.7), we have that $\delta_{\Omega}(g_{k}(t))$
converges uniformly to $0$. Then for $s<t\in(a,b)$, we have

$\displaystyle||g(s)-g(t)||$ $\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}||g_{k}(s)-g_{k}(t)||$
$\displaystyle\leq$
$\displaystyle\lim_{k\rightarrow\infty}\int_{s}^{t}||g_{k}^{\prime}(t)||dt=\lim_{k\rightarrow\infty}\int_{s}^{t}\delta_{\Omega}(g_{k}(t))=0$

This yields that $g$ is a constant map. ∎

Further using the property of the domain $\Omega$, we will establish a
contradiction by proving the following

######  Claim 2.

$g:(a,b)\rightarrow\overline{\Omega}^{Euc}$ is not a constant map.

Proof of Claim 2. Since $\Omega$ satisfies

$k_{\Omega}(x_{0},x)\leq\phi\left(\frac{\delta_{\Omega}(x_{0})}{\delta_{\Omega}(x)}\right)\mbox{
for all }x\in\Omega,$

and $g_{k}$ is an isometry, we have

$\displaystyle|t|=k_{\Omega}(g_{k}(0),g_{k}(t))$ $\displaystyle\leq$
$\displaystyle k_{\Omega}(g_{k}(0),x_{0}))+k_{\Omega}(x_{0},g_{k}(t))$
$\displaystyle\leq$
$\displaystyle\phi\left(\frac{\delta_{\Omega}(x_{0})}{\delta_{\Omega}(g_{k}(0))}\right)+\phi\left(\frac{\delta_{\Omega}(x_{0})}{\delta_{\Omega}(g_{k}(t))}\right)$
$\displaystyle\leq$ $\displaystyle
2\phi\left(\frac{\delta_{\Omega}(x_{0})}{\delta_{\Omega}(g_{k}(t))}\right)\,\,\,(\mbox{by
}\eqref{***})$

In view of Observation 2, we have only following two cases to consider:

  * Case 1:

Both $a$ and $b$ are infinite.

  * Case 2:

Either only $a$ or only $b$ is finite. In other words, Either only $a$ or only
$b$ is infinite.

Let us first consider the Case 1 and assume that $a=-\infty$ and $b=\infty$.
The fact that $b_{n}\to\infty$ and $a_{n}\to-\infty$ and the function $\phi$
is strictly increasing with $\lim\limits_{t\to\infty}\phi(t)=\infty$ imply
that there is a positive integer $n_{0}$ and positive constants $A$ and $B$
such that we have the following

  * (i)

whenever $t\in(B,b_{n}]$ and $n\geq n_{0}$, we have

$\frac{|t|}{2}\in\mbox{{range}}(\phi),\mbox{ and hence,
}\delta_{\Omega}(g_{k}(t))\leq\frac{\delta_{\Omega}(x_{0})}{\phi^{-1}\left(\frac{|t|}{2}\right)}$

  * (ii)

whenever $t\in(a_{n},-A]$ and $n\geq n_{0}$, we have

$\frac{|t|}{2}\in\mbox{{range}}(\phi),\mbox{ and hence,
}\delta_{\Omega}(g_{k}(t))\leq\frac{\delta_{\Omega}(x_{0})}{\phi^{-1}\left(\frac{|t|}{2}\right)}$

In view of Lemma 3.6, let $\epsilon=||p-q||/2$, then we can choose
$a^{*}\in(-\infty,-A)$ and $b^{*}\in(B,\infty)$ such that

(3.9)
$||p-q||>\int_{b^{*}}^{+\infty}\frac{\delta_{\Omega}(x_{0})}{\phi^{-1}\left(\frac{t}{2}\right)}\,dt+\int_{-\infty}^{a^{*}}\frac{\delta_{\Omega}(x_{0})}{\phi^{-1}\left(\frac{|t|}{2}\right)}\,dt$

Therefore, we have

$\displaystyle||g(b^{*})-g(a^{*})||$ $\displaystyle=$
$\displaystyle\lim_{k\to\infty}||g_{k}(b^{*})-g_{k}(a^{*})||$
$\displaystyle\geq$
$\displaystyle\limsup_{k\rightarrow\infty}(||g_{k}(b_{k})-g_{k}(a_{k})||-||g_{k}(b_{k})-g_{k}(b^{*})||-||g_{k}(a_{k})-\gamma_{k}(a^{*})||)$
$\displaystyle\geq$
$\displaystyle||p-q||-\limsup_{k\rightarrow\infty}\int_{b^{*}}^{b_{k}}||g_{k}^{\prime}(t)||\,dt-\limsup_{k\rightarrow\infty}\int_{a_{k}}^{a^{*}}||g_{k}^{\prime}(t)||\,dt$
$\displaystyle\geq$
$\displaystyle||p-q||-\limsup_{k\to\infty}\int_{b^{*}}^{b_{k}}\delta_{\Omega}(g_{k}(t))\,dt-\limsup_{k\to\infty}\int_{a_{k}}^{a^{*}}\delta_{\Omega}(g_{k}(t))\,dt$
$\displaystyle\geq$
$\displaystyle||p-q||-\limsup_{k\rightarrow\infty}\int_{b^{*}}^{b_{k}}\frac{\delta_{\Omega}(x_{0})}{\phi^{-1}\left(\frac{t}{2}\right)}\,dt-\limsup_{k\rightarrow\infty}\int_{a_{k}}^{a^{*}}\frac{\delta_{\Omega}(x_{0})}{\phi^{-1}\left(\frac{|t|}{2}\right)}\,dt$
$\displaystyle\geq$
$\displaystyle||p-q||-\int_{b^{*}}^{+\infty}\frac{\delta_{\Omega}(x_{0})}{\phi^{-1}\left(\frac{t}{2}\right)}\,dt-\int_{-\infty}^{a^{*}}\frac{\delta_{\Omega}(x_{0})}{\phi^{-1}\left(\frac{|t|}{2}\right)}\,dt>0\,\,\,(\mbox{by
}\eqref{2}).$

Thus, $g$ is non-constant. Now, let us consider the Case 2 and assume that $a$
is finite, and hence then $b$ is infinite. Proceed as in the proof of the Case
1, we get that $g$ is non-constant. ∎

Combining both the cases the final conclusion is that $g$ is non-constant,
which contradicts Claim 1. This gives the existence of compact set $K$. Hence
$\Omega$ is a visibility domain. This completes the proof of Theorem 1.2. ∎

Our general visibility criteria gives rise to a very rich collection of
visibility domains, which we will mention below.

### Visibility of $\beta$\- QHBC domains

From general visibility criteria it follows that every $\beta$-QHBC is a
visibility domain by taking $\phi(t)=1/\beta\log t+C_{0}$.

### Visibility of John domain

We want to show that the cone arc condition on quasihyperbolic geodesics in
Theorem 1.1 is not required. By [29, p. 152] we have that every bounded
$C$-John domain satisfies the following quasihyperbolic boundary conditions

$k_{\Omega}(x,x_{0})\leq
C\log\dfrac{\delta_{\Omega}(x_{0})}{\delta_{\Omega}(x)}+\beta,\,\,\,\mbox{ for
all }x\in\Omega,$

where $C\geq 1$ and $\beta=C+C\log(1+C)$. Thus, taking $\phi(t)=C\log t+\beta$
in general visibility criteria it follows that every bounded John domain is a
visibility domain.

###  3.2. H-visibility domain and QH-visibility domain

The way we have defined the notion of visibility with respect to
quasihyperbolic metric, in the same way we can define the visibility of
hyperbolic domain with respect to hyperbolic metric.

######  Definition 3.1.

Let $\Omega$ be a bounded hyperbolic domain in $\mathbb{C}$. Then $\Omega$ is
said to be $H$-visibility domain if for every pair of points
$p,q\in\partial_{Euc}\Omega$, $p\neq q$ there exists neighborhoods $U,V$ of
$p,q$ respectively such that $\overline{U}\cap\overline{V}=\phi$ and a compact
set $K_{\\{p,q\\}}\subset\Omega$ such that for any hyperbolic geodesic
$\gamma:[0,T]\to\Omega$ with $\gamma(0)\in U$ and $\gamma(T)\in V$, we have

$\gamma([0,T])\cap K\neq\phi.$

### Conventions

We will follow the following conventions, whenever necessary. For a bounded
hyperbolic domain $\Omega$ we say

  * (1)

$\Omega$ is H-visibility domain if it satisfies Definition 3.1

  * (2)

$\Omega$ is QH-visibility domain if it satisfies Definition 1.3.

In 2020, Buckley and Herron [18, Theorem B] proved a marvolous result about
Gromov hyperbolicity which says that $\Omega$ is Gromov hyperbolic with
respect to hyperbolic metric if, and only if, it is Gromov hyperbolic with
respect to quasihyperbolic metric. Therefore we can simply say $\Omega$ is
Gromov hyperbolic. Thus, in view of this, one can ask the natural question
that do H-visibility and QH-visibility equivalent for a bounded hyperbolic
domain in $\mathbb{C}$? Recall the following very interesting result of
Buckley and Herron [18, Theorem A],

######  Theorem 3.2.

For each $A\geq 1$ there are explicit constants $H$ and $K$ that depends only
on $A$ such that for any hyperbolic $A$-quasigeodesic $\gamma^{h}$ and any
quasihyperbolic $A$-quasigeodesic $\gamma^{k}$ both with ends point $a$ and
$b$,

$l_{k}(\gamma^{h})\leq K\,k(a,b)\,\,\,\mbox{ and }\,\,\,l_{h}(\gamma^{k})\leq
H\,h(a,b).$

Using Theorem 3.2 it is easy to establish the following result concerning
simultaneous visibility of hyperbolic domain with respect to hyperbolic and
quasihyperbolic metric.

######  Theorem 3.3.

Let $\Omega$ be a bounded hyperbolic domain. If $\Omega$ is Gromov hyperbolic,
then the following are equivalent

  * (1)

$\Omega$ is H-visibility domain

  * (2)

$\Omega$ is a QH- visibility domain.

Proof. Let $\Omega$ be a bounded hyperbolic domain which is Gromov hyperbolic.
Suppose $\Omega$ is H-visibility domain. Let $p,q\in\partial_{Euc}\Omega$,
$p\neq q$ and $x_{k},y_{k}$ be any sequence in $\Omega$ such that $x_{k}\to p$
and $y_{k}\to q$. Let $\gamma_{k}$ be a sequence of quasihyperbolic geodesics
joining $x_{k}$ and $y_{k}$. Then by Theorem 3.2, we have

$l_{h}(\gamma_{k})\leq H\,h(x_{k},y_{k}).$

That is $\gamma_{k}$ is hyperbolic $H$-quasigeodesic. Let $\sigma_{k}$ be any
sequence of hyperbolic geodesics joining $x_{k}$ and $y_{k}$. Since $\Omega$
is a visibility domain, for sufficiently large $k$, $\sigma_{k}$ intersects
with a common compact set $K$. Since $\Omega$ is Gromov hyperbolic, there
exists some $R$ depending on $H$ and Gromov hyperbolic constant such that the
Hausdoreff distance between $\sigma_{i}$ for any fixed $i$ and $\gamma_{k}$
for all $k$, is less than $R$. Therefore, $\gamma_{k}$ also intersects with
the same compact set $K$ for sufficiently large $k$, and hence $\Omega$ is QH-
visibility domain. Conversely, suppose $\Omega$ is QH-visibility domain,
following the same argument as above we conclude that $\Omega$ is H-visibility
domain. ∎

##  4\. Gromov hyperbolicity and visibility

In this section we prove Theorem 1.4 and Theorem 1.5. To proceed further, need
some preperation. We begin with the following definition.

######  Definition 4.1.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ and
$\gamma:[0,\infty)\to\Omega$ be a qh-geodesic ray. We denote the cluster set
of $\gamma$ at $\infty$ by $\Gamma^{\infty}_{\gamma}$ and is defined as

$\Gamma^{\infty}_{\gamma}=\left\\{x\in\overline{\Omega}^{Euc}:\mbox{there
exists a sequence }t_{k}\to\infty\mbox{ such that
}\lim_{k\to\infty}\gamma(t_{k})=x\right\\}$

similarly, for qh-geodesic line we can define the cluster set at $-\infty$ as

$\Gamma^{-\infty}_{\gamma}=\left\\{x\in\overline{\Omega}^{Euc}:\mbox{there
exists a sequence }t_{k}\to-\infty\mbox{ such that
}\lim_{k\to-\infty}\gamma(t_{k})=x\right\\}$

In the following Lemma we show that for every sequence $\\{t_{k}\\}$ with
$t_{k}\to\infty$, $\lim_{k\rightarrow\infty}\gamma(t_{k})$ always belongs to
the boundary, and hence always exists.

######  Lemma 4.1.

Let $\Omega$ be a bouded domain in $\mathbb{R}^{n}$. Then
$\Gamma^{\infty}_{\gamma}\subset\partial_{Euc}\Omega$.

Proof. Let $x\in\Gamma^{\infty}_{\gamma}$ i.e., there exists a sequence
$t_{k}\to\infty$ such that $\lim_{t\to\infty}\gamma(t_{k})=x$. Since $\gamma$
is a qh-geodesic ray i.e.,, $\gamma:[0,\infty)\to(\Omega,k_{D})$ is an
isometry, therefore we have
$|t_{k}|=k_{\Omega}(\gamma(0),\gamma(t_{k}))\to\infty$. In view of [24,
Corollary 2.3], we have that
$\lim_{k\to\infty}\gamma(t_{k})\in\partial_{Euc}\Omega$. Thus,
$x\in\partial_{Euc}\Omega$. ∎

In the next lemma, we show that in the case of visibility domain, for every
sequence $\\{t_{k}\\}$ with $t_{k}\to\infty$, the limit
$\lim_{k\rightarrow\infty}\gamma(t_{k})$ is same. We give this as special name
in the following definition.

######  Definition 4.2.

We say that a qh-geodesic ray $\gamma:[0,\infty)\to\Omega$ lands at a point
$p\in\partial_{Euc}\Omega$ if $\Gamma^{\infty}_{\gamma}=\\{p\\}$.

######  Lemma 4.2.

Let $\Omega$ be a visibility domain. Then any qh-geodesic ray $\gamma$ lands
at a point $p\in~\partial_{Euc}\Omega$.

Proof. Suppose there are two distinct points $p,q\in\Gamma^{\infty}_{\gamma}$.
i.e there exists sequences $s_{k}\to\infty$ and $t_{k}\to\infty$ such that
$\lim_{k\to\infty}\gamma(s_{k})=p$ and $\lim_{k\to\infty}\gamma(t_{k})=q$.
Upto subsequence we can assume $s_{k}<t_{k}<s_{k+1}$ for all $k$. Consider the
qh-geodesics $\gamma_{k}=\gamma|_{[}s_{k},t_{k}]$, then by the visibility
property, there exists a compact set $K\subset\Omega$ such that for
sufficiently large $k$, there exists $s^{\prime}_{k}\in(s_{k},t_{k})$ such
that $\gamma(s^{\prime}_{k})\in K$. But then $\Gamma^{\infty}_{\gamma}\cap
K\neq\phi$, which is a contradiction. ∎

Next we show that in visibility domain, equivalent qh-geodesic rays lands at
the same point.

######  Lemma 4.3.

Let $\Omega$ be visibility domain in $\mathbb{R}^{n}$. Fix $x_{0}\in\Omega$
and let $\gamma$ and $\sigma$ be any two qh-geodesic rays with
$\gamma(0)=x_{0}=\sigma(0)$ such that $\gamma\sim\sigma$. Then $\gamma$ and
$\sigma$ lands at a same point in $\partial_{Euc}\Omega$.

Proof. Let $\Omega$ be a visibility domain. In view of lemma 4.2, by
contradiction suppose that $\gamma$ and $\sigma$ lands at two distinct point
$p,q$, $p\neq q$ in $\partial_{Euc}\Omega$. There exists a sequence
$\\{s_{k}\\}$ with $s_{k}\to\infty$ such that
$\lim_{k\to\infty}\gamma(s_{k})=p$ and $\lim_{k\to\infty}\sigma(s_{k})=q$.
Write $z_{k}=\gamma(s_{k})$ and $w_{k}=\sigma(s_{k})$, it follows that
$z_{k}\to p$ and $w_{k}\to q$. In view of Observation 1, we have that
$\lim_{k\to\infty}k_{\Omega}(z_{k},w_{k})=\infty$, which is a contradiction to
the fact that $\gamma\sim\sigma$. ∎

######  Lemma 4.4.

Suppose $\Omega$ is a visibility domain. Fix any point
$p\in\partial_{Euc}\Omega$ and $x_{0}\in\Omega$, then there exists a qh-
geodesic ray $\gamma$ issuing from $x_{0}$ which lands at $p$.

Proof. Since $p\in\partial_{Euc}\Omega$, therefore there is a sequence
$\\{x_{k}\\}\subset\Omega$ which converges to point $p$. Let $\gamma_{k}$ be
any sequence of qh-geodesics joining $x_{0}$ and $x_{k}$, then as
$(\Omega,k_{\Omega})$ is a proper metric space, by Arzelá-Ascoli theorem there
exists a subsequence of $\\{\gamma_{k}\\}$ which converges uniformly on
compacta to a qh-geodesic ray $\gamma$. In view of Lemma 4.2, we have that
$\gamma$ lands at a point $q\in~\partial_{Euc}\Omega$. We have to show that
$p=q$. If not, let $p\neq q$, so there exists sequences $s_{k}\to\infty$ and
$t_{k}\to\infty$ such that $\lim_{k\to\infty}\gamma_{k}(s_{k})=p$ and
$\lim_{k\to\infty}\gamma_{k}(t_{k})=q$. Upto a subsequence we can assume
$s_{k}<t_{k}$. Then by visibility property, there exists a comapct set
$K\subset\Omega$ such that $\gamma_{k}|_{[s_{k},t_{k}]}$ intersects with $K$
for sufficiently large $k$, say $\gamma_{k}(t^{\prime}_{k})\in K$ for some
$t^{\prime}_{k}\in[s_{k},t_{k}]$. Therefore,

$t^{\prime}_{k}=k_{\Omega}(x_{0}=\gamma_{k}(0),\gamma_{k}(t^{\prime}_{k}))\leq\max_{x\in
K}k_{\Omega}(x_{x},x).$

which gives that $t^{\prime}_{k}$ does not tend to $\infty$, a contradiction.
Hence, $p=q$. This completes the proof. ∎

######  Lemma 4.5.

Let $\Omega$ be a bounded domain. Suppose $\Omega$ is a visibility domain. Let
$T_{k}\in(0,\infty]$, if $\gamma_{k}:[0,T_{k})\to\Omega$ be a sequence of qh-
geodesics converges locally uniformly on compacta to a geodesic ray
$\gamma:[0,\infty)\to\Omega$, then

$\lim\limits_{t\to\infty}\gamma(t)=\lim\limits_{k\to\infty}\lim\limits_{t\to
T_{k}}\gamma_{k}(t)$

Proof. It is clear that if $T_{k}=\infty$, then $\lim\limits_{t\to
T_{k}}\gamma_{k}(t)$ always belong to the Euclidean boundary of $\Omega$, and
hence $\lim\limits_{k\to\infty}\lim\limits_{t\to
T_{k}}\gamma_{k}(t)\in\partial_{Euc}\Omega$. If $T_{k}\in\mathbb{R}$, then by
Lemma 4.2, $\gamma$ lands at a point, say, $\xi$. Let
$\lim\limits_{t\to\infty}\gamma(t)=\xi\in\partial_{Euc}\Omega$,
$\lim\limits_{t\to T_{k}}\gamma_{k}(t)=\xi_{k}\in\Omega$, and

$\lim\limits_{k\to\infty}\lim\limits_{t\to
T_{k}}\gamma_{k}(t)=\xi_{\infty}\in\partial_{Euc}\Omega.$

Assume by contradiction $\xi\neq\xi_{\infty}$. Since
$\lim\limits_{k\to\infty}\gamma_{k}(t)=\gamma(t)$ for every $t$, we can choose
a sequence $\alpha_{k}\to\infty$ such that
$\lim\limits_{k\to\infty}\gamma_{k}(\alpha_{k})=\xi$. We can also choose a
sequence $\beta_{k}\to\infty$ such that
$\lim\limits_{k\to\infty}\gamma_{k}(\beta_{k})=\xi_{\infty}$. Upto a
subsequence we can assume that $\alpha_{k}<\beta_{k}$. Then by visibility
property, there exists a comapct set $K\subset\Omega$ such that
$\gamma_{k}|_{[\alpha_{k},\beta_{k}]}$ intersects with $K$ for sufficiently
large $k$, say $\gamma_{k}(s_{k})\in K$ for some
$s_{k}\in[\alpha_{k},\beta_{k}]$. Therefore, $s_{k}\to\infty$, but

$s_{k}=k_{\Omega}(x_{0}=\gamma_{k}(0),\gamma_{k}(s_{k}))\leq\max_{x\in
K}k_{\Omega}(x_{0},x),$

which gives that $s_{k}$ does not tend to $\infty$, a contradiction. Hence,
$\xi=\xi_{\infty}$. This completes the proof. ∎

We further observe that in a visibility domain the uniform convergence of
geodesics on compacta basically turns into the uniform convergence.

######  Definition 4.3.

Let $\gamma_{k}:[0,T_{k}]\to\Omega$ be a sequence of curves with
$T_{k}\in(0,\infty]$, we say that $\gamma_{k}$ converges uniformly to a curve
$\gamma:[0,T]\to\Omega$ provided $T=\lim\limits_{k\to\infty}T_{k}$ and for
every $\epsilon>0$ there exists $k_{0}\in\mathbb{N}$ such that for every
$k\geq k_{0}$ and for every $t\in[0,T_{k}]$ we have

$|\gamma_{k}(t)-\gamma(t)|\leq\epsilon$

######  Proposition 4.6.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Suppose $\Omega$ is a
visibility domain. If $\\{\gamma_{k}\\}$ be a sequence of qh-geodesics (it can
be ray also) which converges uniformly on compacta to a qh-geodesic ray
$\gamma$, then $\gamma_{k}$ converges uniformly to $\gamma$.

Proof. Since $\Omega$ is a visibility domain, by Lemma 4.2, we set
$\gamma(\infty)=p\in\partial_{Euc}\Omega$. Now, assume by contradiction that
there exists $\epsilon$ and a sequence $t_{k}\in[0,T_{k}]$ such that

$|\gamma_{k}(t_{k})-\gamma(t_{k})|>\epsilon.$

Since $\\{\gamma_{k}\\}$ converges uniformly on compacta to $\gamma$, it
follows that either $t_{k}=\infty$ or $t_{k}$ converges to $\infty$. Hence
upto subsequence we can assume $\\{\gamma_{k}(t_{k})\\}$ converges to a point
$q\in\partial_{Euc}\Omega$ such that

$|p-q|>\epsilon$

Fix an open neighborhood $U_{p}$ of $p$ and $V_{q}$ of $q$ such that
$\overline{U_{p}}\cap\overline{V_{q}}=\phi$. Now, since $\Omega$ is a
visibility domain, there exists a compact set $K\subset\Omega$ such that each
qh-geodesic joining points in $U_{p}\cap\Omega$ to a point in
$V_{p}\cap\Omega$ has to intersect $K$. Since
$\lim\limits_{t\to\infty}\gamma(t)=\gamma(\infty)=p\in\partial_{Euc}\Omega$,
there exists $R>0$ such that for all $r\geq R$, we have $\gamma(r)\in
U_{p}\cap\Omega$. Hence for every $r\geq R$, there exists $k_{r}\in\mathbb{N}$
such that $\gamma_{k}(r)\in U_{p}\cap\Omega$, for every $k\geq k_{r}$. By
taking $k_{r}$ large enough we can also assume $\gamma_{k}(t_{k})\in
V_{q}\cap\overline{\Omega}^{Euc}$ for every $k\geq k_{r}$. Hence
$\gamma_{k}|_{[r,t_{k})}$ is a sequence of qh-geodesic joining points in
$U_{p}\cap\Omega$ to a point in $V_{p}\cap\Omega$. Thus, by visibility
property of $\Omega$, there exists a compact set $K\subset\Omega$ and
$s_{k}\in(r,t_{k})$ such that $\gamma_{k}(s_{k})\in K$. On the other hand, we
have

$r<s_{k}=k_{\Omega}(\gamma_{k}(0),\gamma_{k}(s_{k}))\leq\max_{z\in
K}k_{\Omega}(\gamma_{k}(0),z)<\infty.$

Which gives the contradiction to the fact that $r$ is large enough. ∎

######  Definition 4.4.

Let $\Omega\in\mathbb{R}^{n}$ be a bounded domain and $X$ be a
compactification of $\Omega$ (for instance $\overline{\Omega}^{Euc}$ and
$\overline{\Omega}^{G}$ are compactification of $\Omega$). We say that a
geodesic line $\gamma:(-\infty,\infty)\to\Omega$ is a geodesic loop in $X$ if

$\Gamma^{\infty}_{\gamma}=\Gamma^{-\infty}_{\gamma}$

######  Definition 4.5.

If there is no geodesic line in $\Omega$ which is a geodesic loop in $X$, then
we say that $\Omega$ has no geodesic loop in $X$.

In view of Remark 2.6, $\Omega$ has no geodesic loop in
$\overline{\Omega}^{G}$. On the other hand, $\Omega$ can have geodesic loop in
$\overline{\Omega}^{Euc}$. See the following example,

######  Example 4.1.

Let $\Omega=\mathbb{D}\setminus\\{(-1,0]\\}$. Then in view of Riemann mapping
theorem, consider the conformal (analytic, one-to-one) map
$f:\mathbb{D}\to\Omega$. Let $p,q\in\partial_{Euc}\mathbb{D}$, $p\neq q$, and
a geodesic line $\gamma$ joining $p$ and $q$. Since every Riemann map has
continuous extension on the boundary, we can assume that $f(p)=f(q)=-1/2$.
Hence, $f(\gamma)$ is a geodesic line in $\Omega$ which is a geodesic loop in
$\overline{\Omega}^{Euc}$.

Proof of Theorem 1.4. Suppose the identity map extends as a continuous
surjective map $\widehat{id}:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc}$.
It is clear that $\widehat{id}(\partial_{G}\Omega)=\partial_{Euc}\Omega$. Now
we show that if identity map $id:\Omega\to\Omega$ extends to a continuous
surjective map $\widehat{id}:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc}$,
then $\Omega$ is a visibility domain. Let $p,q\in\partial_{Euc}\Omega$, $p\neq
q$. Consider the two subsets $L_{p}=\widehat{id}^{-1}(p)$,
$L_{q}=\widehat{id}^{-1}(q)$ of $\partial_{G}\Omega$. Clearly both are compact
and $L_{p}\cap L_{q}=\phi$. Since $(\Omega,k_{\Omega})$ is Gromov hyperbolic,
by Theorem 2.6, it is easy to prove that there exist
$\overline{\Omega}^{G}$-open subset $V_{p}$ and $V_{q}$ of $p$ and $q$
respectively and compact set $K_{p,q}\subset\Omega$ such that $L_{p}\subset
V_{p}$, $L_{q}\subset V_{q}$ and $\overline{V_{p}}\cap\overline{V_{q}}=\phi$
with the property that if $\gamma:[0,T]\to\Omega$ is any quasihyperbolic
geodesic with $\gamma(0)\in V_{p}\cap\Omega$ and $\gamma(T)\in
V_{q}\cap\Omega$, then $\gamma([0,T])\cap K_{p,q}\neq\phi$.

Now suppose $x_{k}$ and $y_{k}$ are sequences in $\Omega$ converging in the
Euclidean topology to $p$ and $q$ respectively. Then there exists
$k_{0}\in\mathbb{N}$ such that $x_{k}\in V_{p}\cap\Omega$ and $y_{k}\in
V_{q}\cap\Omega$, for all $k\geq k_{0}$. Thus, for $k\geq k_{0}$ any qh-
geodesic $\gamma_{k}$ joining $x_{k}$ and $y_{k}$ intersects with $K_{p,q}$,
and hence $\Omega$ is a visibility domain.

Conversly, assume that $\Omega$ is a visibility domain. Fix a point
$z_{0}\in\Omega$. In view of Lemma 4.2, define a map
$\Phi:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc}$ as

$\Phi(z)=\begin{cases}z&\mbox{ if }z\in\Omega\\\\[8.53581pt] p&\mbox{ if
}\xi\in\partial_{G}\Omega\end{cases}$

In view of Lemma 4.3, the map $\Phi$ is well defined. Then Lemma 4.4 gives
that $\Phi$ is surjective. To show that $\Phi$ is continuous we need to
consider the following two cases

  * Case I.

If $\\{z_{k}\\}\subset\Omega$ is a sequence such that
$z_{k}\stackrel{{\scriptstyle Gromov}}{{\longrightarrow}}\xi$, where
$\xi\in\partial_{G}\Omega$. Since $z_{k}\stackrel{{\scriptstyle
Gromov}}{{\longrightarrow}}\xi$, which means that for every qh-geodesic
$\gamma_{k}:[0,R_{k}]\to\Omega$ with $\gamma_{k}(0)=z_{0}$ and
$\gamma_{k}(R_{k})=z_{k}$, every subsequence has a subsequence which converges
uniformly on compact subset to a geodesic ray $\gamma\in\xi$, which in view of
Lemma 4.2 lands at $p$. Thus, by Lemma 4.5, we have
$\Phi(z_{k})=z_{k}\longrightarrow p=\Phi(\xi)$.

  * Case II.

If $\\{\xi_{k}\\}\subset\partial_{G}\Omega$ is a sequence such that
$\xi_{k}\stackrel{{\scriptstyle Gromov}}{{\longrightarrow}}\xi$, where
$\xi\in\partial_{G}\Omega$
Let $\gamma_{k}$ be the representative from $\xi_{k}$. Then
$\xi_{k}\stackrel{{\scriptstyle Gromov}}{{\longrightarrow}}\xi$ gives that
every subsequence of $\gamma_{k}$ has a subsequence which converges uniformly
on compact subsets to a geodesic ray $\gamma\in\xi$. Thus, by Lemma 4.5, we
have that $\Phi(\xi_{k})\stackrel{{\scriptstyle
Euc}}{{\longrightarrow}}\Phi(\xi)$.

Hence in view of Case I and Case II, $\Phi$ is continuous. This completes the
proof. ∎

Proof of Theorem 1.5. Since $\Omega$ has no geodesic loop in
$\overline{\Omega}^{G}$ and $\Phi$ is a homeomorphism, it follows that
$\Omega$ has no geodesic loop in $\overline{\Omega}^{Euc}$.

Conversly, suppose $\Omega$ has no geodesic loop in $\overline{\Omega}^{Euc}$.
Since $\overline{\Omega}^{G}$ and $\overline{\Omega}^{Euc}$ are Hausdorff and
compact, therefore if we show that $\Phi$ is injective it will imply that
$\Phi$ is a homeomorphism. Therefore it is sufficient to prove that if
$\Omega$ has no geodesic loop in $\overline{\Omega}^{Euc}$, then $\Phi$ is
injective. This is equivalent to proving that if $\Phi$ is not injective, then
$\Omega$ has a geodesic loop in $\overline{\Omega}^{Euc}$. Therefore, assume
that $\Phi$ is not injective i.e., $\Phi(\xi)=\Phi(\theta)$, for some
$\xi,\theta\in\partial_{G}\Omega$, $\xi\neq\theta$. Let $\gamma\in\xi$ and
$\sigma\in\theta$ be two representatives; $\gamma$ and $\sigma$ are qh-
geodesic rays in $\Omega$ such that both $\gamma$ and $\sigma$

  * •

lands at $p\in\partial_{Euc}\Omega$

  * •

are not equivalent. In other words,

$\sup_{t\in[0,\infty)}k_{\Omega}(\gamma(t),\sigma(t))=\infty.$

Therefore, there exists a sequence $t_{k}\to\infty$ such that

$\lim_{k\to\infty}k_{\Omega}(\gamma(t_{k}),\sigma(t_{k}))=\infty$

Now let $\eta_{k}:[0,R_{k}]\to\Omega$ be a sequence of qh-geodesics joining
$\gamma(t_{k})$ and $\sigma(t_{k})$. Consider the geodesic triangle with sides
$\gamma|_{[0,t_{k}]}$, $\eta_{k}$ and $\sigma|_{[0,t_{k}]}$. Since
$(\Omega,k_{\Omega})$ is Gromov hyperbolic, this geodesic triangle is
$\delta$-thin, for some $\delta<\infty$, i.e.,

(4.7) $\forall k\in\mathbb{N},\,\,\forall m\leq
k,\,\min\\{k_{\Omega}(\gamma(t_{m}),\sigma([0,t_{k}])),k_{\Omega}(\gamma(t_{m}),\eta_{k}([0,R_{k}]))\\}\leq\delta$

It is easy to prove that

(4.8)
$\lim\limits_{k\to\infty}k_{\Omega}(\gamma(t_{k}),\mbox{range}(\sigma))=\infty.$

By (4.8), choose $k_{0}$ such that

(4.9) $\forall k\geq
k_{0},\,\,k_{\Omega}(\gamma(t_{k}),\mbox{range}(\sigma))>\delta.$

Using (4.7) and (4.9), we obtain

$\forall k\geq
k_{0},\,\,k_{\Omega}(\gamma(t_{k_{0}}),\eta_{k}([0,R_{k}]))\leq\delta$

Consider the following compact set $K$ of $\Omega$

$K:=\\{z\in\Omega:k_{\Omega}(z,\gamma(t_{k_{0}}))\leq\delta\\}.$

Then for every $k$, $\eta_{k}([0,R_{k}])\cap K\neq\phi$. We can parametrize
$\eta_{k}$ in such a way that $\eta_{k}:[-s_{k},t_{k}]\to\Omega$ such that
$\eta_{k}(0)=z_{0}$. Then, by visibility of $\Omega$ and in view of Lemma 4.4,
we get that upto subsequence $\eta_{k}$ converges to a geodesic line
$\eta:(-\infty,\infty)\to\Omega$. Since $\Phi(\xi)=\Phi(\theta)$, therefore
$\gamma$ and $\sigma$ lands at a same point point $p\in\partial_{Euc}\Omega$
i.e., for any sequences $t_{k}\to\infty$ and $s_{k}\to\infty$, we have

$\lim_{k\to\infty}\gamma(t_{k})=p\mbox{ and
}\lim_{k\to\infty}\sigma(s_{k})=p.$

Therefore, we have

$\lim_{t\to\infty}\eta(t)=p\mbox{ and }\lim_{t\to-\infty}\eta(t)=p.$

Therefore, $\eta$ is a geodesic line in $\Omega$ which is a geodesic loop in
$\overline{\Omega}^{Euc}$. This completes the proof. ∎

##  5\. Visibility domains via conformal and quasiconformal mappings

In this section, we characterize visibility domains using the boundary
extension of conformal and quasiconformal mappings, and as an application of
Theorem 1.4 and Theorem 1.5, we prove Theorem 1.6 and Theorem 1.7. To proceed
further we recall some results which requires the following definitions.

######  Definition 5.1.

A bounded domain $\Omega\in\mathbb{R}^{n}$, $n\geq 2$, is said to be

  * (i)

finitely connected along the boundary if every point
$p\in\partial_{Euc}\Omega$ has sufficiently small neighborhoods $U$ such that
$\Omega\cap U$ has finite number of components,

  * (ii)

locally connected along the boundary if every point $p\in\partial_{Euc}\Omega$
has sufficiently small neighborhoods $U$ such that $\Omega\cap U$ is
connected,

  * (iii)

Jordan domain if $\partial_{Euc}\Omega$ is homeomorphic to $S^{n-1}$.

######  Remark 5.1.

Every Jordan domain in $\mathbb{R}^{2}$ is simply connected domain. Moreover,
a proper simply connected domain in $\mathbb{R}^{2}$ is Jordan domain if, and
only if, $\Omega$ is locally conncted along its boundary.

We recall some results related to the extension of conformal mappings of the
unit disk. We will state the results in the context of bounded domains only.
For details we refer to [44, Chapter IX.4]. The following theorem is due to
Väisälä and Näkki (see [44, Theorem 4.7, Chapter IX.4]).

Theorem B. Let $\Omega$ be a bounded domain in $\mathbb{C}$, and let $f$ be a
conformal mapping from $\mathbb{D}\to\Omega$. Then $f$ extends to a continuous
mapping $\overline{f}$ of $\overline{\mathbb{D}}^{Euc}$ onto
$\overline{\Omega}^{Euc}$ if, and only if, $\Omega$ is finitely connected
along the boundary.

Carathéodory has proved the following variant of Theorem B (see [44, Chapter
IX.4]).

Theorem C. Let $\Omega$ be a bounded domain in $\mathbb{C}$, and let $f$ be a
conformal mapping from $\mathbb{D}\to\Omega$. Then $f$ extends to a continuous
mapping $\overline{f}$ of $\overline{\mathbb{D}}^{Euc}$ onto
$\overline{\Omega}^{Euc}$ if, and only if, $\partial_{Euc}\Omega$ is locally
connected.

Thus, in view of Theorem B and Theorem C, we have that in the case of simply
connected domains, locally connected along the boundary is equivalent to
boundary being locally connected. Further, if we merely replace the
requirement that the extension is homeomorphism, then we have the following.

Theorem D. Let $\Omega$ be a bounded domain in $\mathbb{C}$, and let $f$ be a
conformal mapping from $\mathbb{D}\to\Omega$. Then $f$ extends to a
homeomorphism $\overline{f}$ from $\overline{\mathbb{D}}^{Euc}$ to
$\overline{\Omega}^{Euc}$ if, and only if, $\Omega$ is locally connected along
the boundary.  As we know that every Jordan domain is simply connected and a
proper simply connected domain is Jordan domain if, and only if, it is locally
connected along the boundary. Thus, we have the following result due to
Carathéodory and Osgood.

Theorem E (Carathéodory-Osgood theorem). Let $\Omega$ be a bounded domain in
$\mathbb{C}$, and let $f$ be a conformal mapping from $\mathbb{D}$ to
$\Omega$. Then $f$ extends to a homeomorphism $\overline{f}$ from
$\overline{\mathbb{D}}^{Euc}$ to $\overline{\Omega}^{Euc}$ if, and only if,
$\Omega$ is a Jordan domain.

In the case of quasiconformal mappings, we have the following result (see
[49]).

Theorem F. Let $\Omega$ be a bounded domain. Let $\Omega$ is locally connected
along the boundary and quasiconformally equivalent to ball. Then every
quasiconformal mapping $f:\mathbb{B}^{n}\to\Omega$ extends to a homeomorphism
from $\overline{\mathbb{B}^{n}}^{Euc}$ to $\overline{\Omega}^{Euc}$.

Now we are ready to prove Theorem 1.6 and Theorem 1.7.

Proof of Theorem 1.6. Let $f:\mathbb{B}^{n}\to\Omega$ be quasiconformal map.
Since quasiconformal images between Euclidean domains are rough quasi-
isometries in the quasihyperbolic metric, each quasiconformal image of Gromov
hyperbolic domain is Gromov hyperbolic. Therefore, $(\Omega,k_{\Omega})$ is
Gromov hyperbolic. Suppose $\Omega$ is locally connected along the boundary,
in view of Theorem F, $f$ extends to a homeomorphism

$\widetilde{f}:\overline{\mathbb{B}^{n}}^{Euc}\to\overline{\Omega}^{Euc}.$

Moreover, from the theory of Gromov hyperbolic spaces $f^{-1}$ extends to a
homeomorphism

$\widehat{f^{-1}}:\overline{\Omega}^{G}\to\overline{\mathbb{B}^{n}}^{G}.$

The identity map

$id_{\mathbb{B}^{n}}:(\mathbb{B}^{n},k_{\mathbb{B}^{n}})\to(\mathbb{B}^{n},d_{Euc})$

extends to a homeomorphism

$\widehat{id_{\mathbb{B}^{n}}}:\overline{\mathbb{B}^{n}}^{G}\to\overline{\mathbb{B}^{n}}^{Euc}.$

Therefore, the identity map

$id_{\Omega}:(\Omega,k_{\Omega})\to(\Omega,d_{Euc})$

extends to a homeomorphism

$\widehat{id_{\Omega}}=\widetilde{f}\circ\widehat{id_{\mathbb{B}^{n}}}\circ\widehat{f^{-1}}:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc}.$

In view of Theorem 1.4 and Theorem 1.5, $\Omega$ is a visibility domain which
has no geodesic loops in $\overline{\Omega}^{Euc}$. This completes the proof.
∎

Proof of Theorem 1.7. Let $\Omega$ be a bounded simply connected domain. Since
every simply connected planar domain is Gromov hyperbolic with respect to
quasihyperbolic metric and since $\Omega$ is a visibility domain, in view of
Theorem 1.4, we have that the identity map

$id_{\Omega}:(\Omega,k_{\Omega}\to(\Omega,d_{Euc})$

extends to a continuous surjective map

$\widehat{id_{\Omega}}:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc}.$

Let $f:\mathbb{D}\to\Omega$ be the Riemann map. Then $f$ is $(c,0)$ rough
quasi-isometry and hence $f$ extends to a homeomorphism

$\widetilde{f}:\overline{\mathbb{D}}^{G}\to\overline{\Omega}^{G}.$

Since $\mathbb{D}$ is a visibility domain, the identity map

$id_{\mathbb{D}}:(\mathbb{D},k_{\mathbb{D}})\to(\mathbb{D},d_{Euc})$

extends to a homeomorphism

$\widehat{id_{\mathbb{D}}}:\overline{\mathbb{D}}^{G}\to\overline{\mathbb{D}}^{Euc}.$

Therefore, $f:\mathbb{D}\to\Omega$ extends to a continuous surjective map

$\widehat{f}=\widehat{id_{\Omega}}\circ\widetilde{f}\circ\widehat{id_{\mathbb{D}}}^{-1}:\overline{\mathbb{D}}^{Euc}\to\overline{\Omega}^{Euc}.$

Thus, in view of Theorem C, $\partial_{Euc}\Omega$ is locally connected.
Conversly, suppose $\partial_{Euc}\Omega$ is locally connected, following the
lines of the proof of Theorem 1.6, we have that $\Omega$ is a visibility
domain.

Moreover, suppose $\Omega$ has no geodesic loop in $\overline{\Omega}^{Euc}$,
then in view of Theorem 1.5, the identity map

$id_{\Omega}:(\Omega,k_{\Omega}\to(\Omega,d_{Euc})$

extends to a homeomorphism

$\widehat{id_{\Omega}}:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc},$

and hence $f:\mathbb{D}\to\Omega$ extends to a homeomorphism

$\widetilde{f}=\widehat{id_{\Omega}}\circ\widetilde{f}\circ\widehat{id_{\mathbb{D}}}^{-1}:\overline{\mathbb{D}}^{Euc}\to\overline{\Omega}^{Euc}.$

In view of Theorem D or Theorem E, $\Omega$ is locally connected along the
boundary, or simply, $\Omega$ is a Jordan domain. Conversly, suppose $\Omega$
is locally connected along the boundary, then in view of Theorem 1.6 and the
fact that every conformal map is $1$-quasiconformal map, we have that $\Omega$
is a visibility domain which has no geodesic loop in
$\overline{\Omega}^{Euc}$. This completes the proof. ∎

##  6\. Continuous extension of quasihyperbolic isometries and quasi-
isometries

This section is devoted to the proof of Theorem 1.8 and Theorem 1.9. To
proceed further we need some preparation. Let us first define the term
quasihyperbolically well behaved domain mentioned in the theorems.

######  Definition 6.1.

A bounded domain $\Omega\subset\mathbb{R}^{n}$ is said to be
quasihyperbolically well behaved if, $\\{x_{k}\\}$, $\\{y_{k}\\}$ are
sequences in $\Omega$ with

$\lim_{k\to\infty}x_{k}=\lim_{k\to\infty}y_{k}=p\in\partial_{Euc}\Omega,$

and $\gamma_{k}:[a_{k},b_{k}]\to\Omega$ be a sequence of qh-geodesic joining
$x_{k}$ and $y_{k}$, then we have

$\lim_{k\to\infty}k_{\Omega}(x_{0},\gamma_{k})=\infty.$

for some (hence all) $x_{0}\in\Omega$.

Next we study how quasihyperbolic well behaved domain is related to visibility
domain.

######  Proposition 6.1.

If $\Omega\subset\mathbb{R}^{n}$ is quasihyperbolically well behaved then
$\Omega$ has no geodesic loop in $\overline{\Omega}^{Euc}$.

Proof. By contradiction, suppose $\Omega$ has geodesic loop in
$\overline{\Omega}^{Euc}$ i.e, there exists a geodesic line
$\gamma:(-\infty,\infty)\to\Omega$ such that

$\lim_{t\to\infty}\gamma(t)=\lim_{t\to-\infty}\gamma(t)=p\in\partial_{Euc}\Omega.$

Let $x_{k}=\gamma(t_{k})$ and $y_{k}=\gamma(s_{k})$, where
$s_{k}\to-\infty,t_{k}\to\infty$, then

$\lim_{k\to\infty}x_{k}=\lim_{k\to\infty}y_{k}=p\in\partial_{Euc}\Omega,$

and we can parametrize $\gamma$ in such a way that $\gamma(0)=x_{0}$. Now the
sequence $\gamma_{n}$ converges uniformly on every compact subset to a
geodesic line $\sigma$. In this case

$\lim_{k\to\infty}k_{\Omega}(x_{0},\gamma_{k})\leq
k_{\Omega}(x_{0},\sigma)\leq M,$

which is a contradiction to the fact that $\Omega$ is quasihyperbolically well
behaved. ∎

Next we show that the converse of Proposition 6.1 holds under the visibility
assumption.

######  Proposition 6.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a visibility domain. Then
$\Omega\subset\mathbb{R}^{n}$ is quasihyperbolically well behaved if, and only
if, $\Omega$ has no geodesic loop in $\overline{\Omega}^{Euc}$.

Proof. Let $\Omega$ be a visibility domain. Suppose $\Omega$ has no geodesic
loop in $\overline{\Omega}^{Euc}$. Let $\\{x_{k}\\}$, $\\{y_{k}\\}$ be two
sequences in $\Omega$ with

$\lim_{k\to\infty}x_{k}=\lim_{k\to\infty}y_{k}=p\in\partial_{Euc}\Omega,$

and $\gamma_{k}:[a_{n},b_{n}]\to\Omega$ be a sequence of qh-geodesic joining
$x_{k}$ and $y_{k}$. We can parametrize $\gamma_{k}:[-s_{k},t_{k}]\to\Omega$
such that $\gamma_{k}(-s_{k})=x_{k}$ and $\gamma_{k}(t_{k})=y_{k}$. Then
$\gamma_{k}$ converges to a geodesic line $\gamma:(-\infty,\infty)\to\Omega$
such that

$\lim_{t\to\infty}\gamma(t)=\lim_{t\to-\infty}\gamma(t)=p\in\partial_{Euc}\Omega$

which is a contradiction to the fact that $\Omega$ has no geodesic loop in
$\overline{\Omega}^{Euc}$. The converse follows from Proposition 6.1. This
completes the proof. ∎

We are now ready to prove Theorem 1.8. For this we need the following lemma
which gives an equivalent visibility criteria.

######  Lemma 6.3.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Then $\Omega$ is a
visibility domain if, and only if, the following holds:
If $p,q\in\partial_{Euc}\Omega$, $p\neq q$, and $\\{x_{k}\\},\\{y_{k}\\}$ be
sequence in $\Omega$ with $x_{k}\to p$, $y_{k}\to q$, then for every qh-
geodesic $\gamma_{k}:[a,b]\to\Omega$ with $\gamma_{k}(a)=x_{k}$ and
$\gamma_{k}(b)=y_{k}$, there exists a positive constant $M>0$ such that

$\sup_{k\geq 1}{k_{\Omega}}(x_{0},\gamma_{k})\leq M$

for every $x_{0}\in\Omega$.

Proof. Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Fix
$x_{0}\in\Omega$. Suppose $\Omega$ is a visibility domain, then there exists a
compact set $k\subset\Omega$ such that $\gamma_{k}$ intersects with $K$ for
sufficiently large $k$. Let $z_{k}$ be the intersection points. Then for
sufficiently large $k$, we have the following

$k_{\Omega}(x_{0},z_{k})\leq k_{\Omega}(x_{0},z),\,\,\,\,\,z\in\gamma_{k}$

That is $k_{\Omega}(x_{0},\gamma_{k})=k_{\Omega}(x_{0},z_{k})$ for
suffieiently large $k$, and since $z_{k}$ lies in a compact set, we have

$\sup_{k\geq 1}{k_{\Omega}}(x_{0},\gamma_{k})\leq M.$

Conversly, Let $p,q\in\partial_{Euc}\Omega$, $p\neq q$, and
$\\{x_{k}\\},\\{y_{k}\\}$ be sequences in $\Omega$ with $x_{k}\to p$,
$y_{k}\to q$, then for every qh-geodesic $\gamma_{k}:[a,b]\to\Omega$ with
$\gamma_{k}(a)=x_{k}$ and $\gamma_{k}(b)=y_{k}$, we have

$\sup_{k\geq 1}{k_{\Omega}}(x_{0},\gamma_{k})\leq M$

for some $M>0$. Since $\gamma_{k}$ is compact in $(\Omega,k_{\Omega})$,
therefore there exits $z_{k}\in\gamma_{k}$ such that

$k_{\Omega}(x_{0},\gamma_{k})=k_{\Omega}(x_{0},z_{k}).$

By the assumption $\sup_{k\geq 1}k_{\Omega}(x_{0},z_{k})<M$, Lemma 2.8 gives
that the set $\\{z_{k}\\}$ is compactly contained in $\Omega$. Therefore,
there exists a compact set $K$ with $\\{z_{k}\\}\subset K$, and hence
$\gamma_{k}$ intersects with $K$, which gives $\Omega$ is a visibility domain.
This completes the proof. ∎

To prove Theorem 1.9, we first see the behavior of Gromov product on
visibility domains. The the proof of the following theorem is similar to that
of [15, Proposition 2.4 and Proposition 2.5].

######  Theorem 6.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain. Then the following are
equivalent:

  * (1)

$\Omega$ is a visibility domain.

  * (2)

For every pair of points $p,q\in\partial_{Euc}\Omega$, $p\neq q$, and
sequences $z_{k}\to p$, $w_{k}\to q$, we have

$\limsup_{k\to\infty}(z_{k}|w_{k})_{o}<\infty,$

for a fixed point (hence any) $o\in\Omega$

Proof. $(1)\implies(2)$: Let $\Omega$ be a visibility domain and $z_{k}$,
$w_{k}$ be any sequence in $\Omega$ with $z_{k}\to p$ and $w_{k}\to q$, where
$p,q\in\partial_{Euc}\Omega$, $p\neq q$. Let $\gamma_{k}$ be qh-geodesics
joining $z_{k}$ and $w_{k}$. Then by the visibility property there exists a
compact set $K$ and $k_{0}\in\mathbb{N}$ such that $\gamma_{k}$ intersects
with $K$ for all $k\geq k_{0}$. Let $\theta_{k}$ be the inetrsection points of
$\gamma_{k}$ and $K$. Fix any point $o\in\Omega$. then for all $k\geq k_{0}$,
we have

$\displaystyle k_{\Omega}(z_{k},w_{k})$ $\displaystyle=$ $\displaystyle
k_{\Omega}(z_{k},\theta_{k})+k_{\Omega}(\theta_{k},w_{k})$ $\displaystyle\geq$
$\displaystyle
k_{\Omega}(z_{k},o)+k_{\Omega}(o,w_{k})-2k_{\Omega}(o,\theta_{k})$
$\displaystyle\geq$ $\displaystyle
k_{\Omega}(z_{k},o)+k_{\Omega}(o,w_{k})-2\max_{\theta\in
K}k_{\Omega}(o,\theta)$

which gives that

$\limsup_{k\to\infty}(z_{k}|w_{k})_{o}=\lim_{k\to\infty}\frac{1}{2}\left(k_{\Omega}(z_{k},o)+k_{\Omega}(o,w_{k})-k_{\Omega}(z_{k},w_{k})\right)\leq\max_{\theta\in
K}k_{\Omega}(o,\theta)<\infty.$

$(2)\implies(1)$: Assume by contradiction that $\Omega$ is not a visibility
domain. Then by Lemma 6.3 there is a pair
$\\{p,q\\}\subset\partial_{Euc}\Omega$, a sequences $z_{k}\to p$ and $w_{k}\to
q$ and a sequence of qh-geodesics $\gamma_{k}$ joining $z_{k}$ and $w_{k}$
such that

$\sup_{k\geq 1}k_{\Omega}(o,\gamma_{k})=\infty$

Hence

(6.4)
$\limsup_{k\to\infty}k_{\Omega}(o,\gamma_{k})=\limsup_{k\to\infty}k_{\Omega}(o,\theta_{k})=\infty$

where $\theta_{k}\in\gamma_{k}$. Now by the completeness property of
quasihyperbolic metric it follows that $\theta_{k}\to
p^{*}\in\partial_{Euc}\Omega$. We can always choose some $z_{k}$ at the place
of $\theta_{k}$ so that $p^{*}$ will be different from $p$ and $q$. It is easy
to see that

$\displaystyle k_{\Omega}(z_{k},o)+k_{\Omega}(o,w_{k})$ $\displaystyle\geq$
$\displaystyle
k_{\Omega}(z_{k},w_{k})=k_{\Omega}(z_{k},\theta_{k})+k_{\Omega}(\theta_{k},w_{k})$
$\displaystyle=$
$\displaystyle-2(z_{k}|\theta_{k})_{o}-2(\theta_{k}|w_{k})_{o}+k_{\Omega}(z_{k},o)+k_{\Omega}(o,w_{k})+2k_{\Omega}(o,\theta_{k})$
$\displaystyle\geq$ $\displaystyle
k_{\Omega}(z_{k},o)+k_{\Omega}(o,w_{k})+2k_{\Omega}(o,\theta_{k})-C$

This gives that $k_{\Omega}(o,\theta_{k})<C$, which contradicts (6.4). This
completes the proof. ∎

Proof of Theorem 1.8. Fix some point $p\in\partial_{Euc}\Omega$. We claim that

$\lim_{x\to p}f(x)\mbox{ exists in }\partial_{Euc}\Omega^{\prime}.$

If $\lim_{x\to p}f(x)$ does not exist, then there exist sequences
$x_{k},y_{k}\in\Omega$ with

$\lim_{k\to\infty}x_{k}=p=\lim_{k\to\infty}y_{k}\mbox{ but
}\lim_{k\to\infty}f(x_{k})=\xi\neq\eta=\lim_{k\to\infty}f(y_{k}).$

Since $f$ is quasihyperbolic isometry,
$\eta,\xi\in\partial_{Euc}\Omega^{\prime}$. For each $k$, let $\gamma_{k}$ be
a sequence of geodesic in $\Omega$ joining $x_{k}$ and $y_{k}$. Then
$f\circ\gamma_{k}$ is also a quasihyperbolic geodesic in $\Omega^{\prime}$.
Fix $x_{0}\in\Omega$. Since $\Omega^{\prime}$ is a visibility domain, In view
of Lemma 6.3, we have

$\sup_{k\geq 1}k_{\Omega^{\prime}}(f(x_{0}),f\circ\gamma_{k})<\infty.$

Hence

$\sup_{k\geq 1}k_{\Omega}(x_{0},\gamma_{k})<\infty,$

which is a contradiction to the fact that $\Omega$ is quasihyperbolically well
behaved. Thus, $\lim_{x\to p}f(x)\mbox{ exists in
}\partial_{Euc}\Omega^{\prime}.$

Next we define the map
$F:\overline{\Omega}^{Euc}\to\overline{\Omega^{\prime}}^{Euc}$ by

$F(z)=\begin{cases}f(z)&\mbox{ if }z\in\Omega\\\\[14.22636pt]
\widetilde{f}(z)=\lim_{x\to z}f(z)&\mbox{ if
}z\in\partial_{Euc}\Omega\end{cases}$

In view of the definition of $F$, to show $F$ is continuous, it is sufficient
to show that
$\widetilde{f}:\partial_{Euc}\Omega\to\partial_{Euc}\Omega^{\prime}$ is
continuous. Let $p_{k}$ be a sequence in $\partial_{Euc}\Omega$ which
converges to $p\in\partial_{Euc}\Omega$, since $p\in\partial_{Euc}\Omega$, we
can find a sequence $z_{k}\in\Omega$ which is sufficiently close to $p_{k}$
such that $z_{k}\to p$. Then write
$\lim_{k\to\infty}f(z_{k})=\widetilde{f}(p)=q\in\partial_{Euc}\Omega^{\prime}$.
is a sequence in $\partial_{Euc}\Omega^{\prime}$. Since
$\partial_{Euc}\Omega^{\prime}$ is compact and $z_{k}$ is sufficiently close
to $p_{k}$, we have

$\lim_{k\to\infty}\widetilde{f}(p_{k})=\lim_{k\to\infty}f(z_{k})=q=\widetilde{f}(p).$

This completes the proof. ∎

Proof of Theorem 1.9. Fix some point $p\in\partial_{Euc}\Omega$. We claim that

$\lim_{x\to p}f(x)\mbox{ exists in }\partial_{Euc}\Omega^{\prime}.$

If $\lim_{x\to p}f(x)$ does not exist, then there exists sequence
$x_{k},y_{k}\in\Omega$ with

$\lim_{k\to\infty}x_{k}=p=\lim_{k\to\infty}y_{k}\mbox{ but
}\lim_{k\to\infty}f(x_{k})=\xi\neq\eta=\lim_{k\to\infty}f(y_{k})$

Since $f$ is isometry, $\eta,\xi\in\partial_{Euc}\Omega^{\prime}$. For each
$k$, let $\gamma_{k}$ be a sequence of geodesic in $\Omega$ joining $x_{k}$
and $y_{k}$. Then
$f\circ\gamma_{k}=\sigma_{k}:[a_{k},b_{k}]\to\Omega^{\prime}$ is a sequence of
$(1,\mu)$-quasi-isometric path in $\Omega^{\prime}$ joining $f(x_{k})=u_{k}$
and $f(y_{k})=v_{k}$.

### Claim 1:

There exists a compact set $K\subset\Omega$ such that for sufficiently large
$k$, $\sigma_{k}$ intersects with $K$.

### Proof of the Claim 1:

If $\sigma_{k}$ there does not exist such compact set, then

$\max_{t\in[a_{k},b_{k}]}\delta_{\Omega^{\prime}}(\sigma_{k}(t))\to
0\,\,\mbox{ as }\,\,n\to\infty$

Let $w_{k}=\sigma_{k}(t_{k})$ such that

$\max_{t\in[a_{k},b_{k}]}\delta_{\Omega^{\prime}}(\sigma_{k}(t))=\delta_{\Omega^{\prime}}(\sigma_{k}(t_{k})).$

Let $f(x_{0})=w_{0}$. Clearly, $z_{k}\to\xi\in\partial_{Euc}\Omega^{\prime}$,
therefore, $k_{\Omega^{\prime}}(w_{0},w_{k})\to\infty$ as $k\to\infty$. Now

$\displaystyle
k_{\Omega^{\prime}}(u_{k},w_{0})+k_{\Omega^{\prime}}(v_{k},w_{0})$
$\displaystyle\geq$ $\displaystyle k_{\Omega^{\prime}}(f(x_{k}),f(y_{k}))$
$\displaystyle=$ $\displaystyle
k_{\Omega^{\prime}}(\sigma_{k}(a_{k}),\sigma_{k}(b_{k}))$ $\displaystyle\geq$
$\displaystyle(b_{k}-a_{k})-\mu\,(\mbox{since }\sigma_{k}\mbox{ is a
}(1,\mu)-\mbox{quasi-isometric path}$ $\displaystyle=$
$\displaystyle(b_{k}-t_{k}+\mu)+(t_{k}-a_{k}+\mu)-3\mu$ $\displaystyle\geq$
$\displaystyle
k_{\Omega^{\prime}}(v_{k},w_{k})+k_{\Omega^{\prime}}(w_{k},u_{k})-3\mu$

Therefore, we have

$\displaystyle
k_{\Omega^{\prime}}(u_{k},w_{0})+k_{\Omega^{\prime}}(v_{k},w_{0})$
$\displaystyle\geq$
$\displaystyle-2(v_{k}|w_{k})_{w_{0}}-2(w_{k}|u_{k})_{w_{0}}+k_{\Omega^{\prime}}(u_{k},w_{0})+k_{\Omega^{\prime}}(w_{0},v_{k})$
$\displaystyle+2k_{\Omega^{\prime}}(w_{0},w_{k})-3\mu$

Since $\Omega^{\prime}$ is a visibility domain, we have

$k_{\Omega^{\prime}}(u_{k},w_{0})+k_{\Omega^{\prime}}(v_{k},w_{0})\geq-2C+k_{\Omega^{\prime}}(u_{k},w_{0})+k_{\Omega^{\prime}}(w_{0},v_{k})+2k_{\Omega}(w_{0},w_{k})-3\mu,$

which gives that

$2k_{\Omega^{\prime}}(w_{0},w_{k})\leq 2C+3\mu.$

This is a contradiction to the fact that
$k_{\Omega^{\prime}}(w_{0},w_{k})\to\infty$.

### Claim 2:

$\sup_{k\geq 1}k_{\Omega^{\prime}}(f(x_{0}),\sigma_{k})<\infty.$

### Proof of Claim 2:

Since there exists a compact set $K\subset\Omega$ such that $\sigma_{k}$
intersects with $K$ for sufficiently large $k$, let $z_{k}$ be one of the
intersection points. Then for sufficiently large $k$, we have the following

$k_{\Omega^{\prime}}(w_{0},z_{k})\leq
k_{\Omega^{\prime}}(w_{0},z),\,\,\,\,\,\mbox{ for all }z\in\sigma_{k}.$

That is,
$k_{\Omega^{\prime}}(w_{0},\sigma_{k})=k_{\Omega^{\prime}}(w_{0},z_{k})$ for
suffieiently large $k$. Since $z_{k}$ lies in a compact set, we have

$\sup_{k\geq 1}{k_{\Omega^{\prime}}}(w_{0},\sigma_{k})\leq M,$

and hence

$\sup_{k\geq 1}k_{\Omega}(x_{0},\gamma_{k})<\infty.$

This completes the proof of Claim 2. Now we note that Claim 2 gives a
contradiction to the fact that $\Omega$ is quasihyperbolically well behaved.
Thus, $\lim_{x\to p}f(x)$ exists in $\partial_{Euc}\Omega^{\prime}.$

Next define the map
$F:\overline{\Omega}^{Euc}\to\overline{\Omega^{\prime}}^{Euc}$ by

$F(z)=\begin{cases}f(z)&\mbox{ if }z\in\Omega\\\\[14.22636pt]
\widetilde{f}(z)=\lim_{x\to z}f(z)&\mbox{ if
}z\in\partial_{Euc}\Omega.\end{cases}$

In view of the definition of $F$, to show $F$ is continuous, it is sufficient
to show that
$\widetilde{f}:\partial_{Euc}\Omega\to\partial_{Euc}\Omega^{\prime}$ is
continuous. Let $p_{k}$ be a sequence in $\partial_{Euc}\Omega$ which
converges to $p\in\partial_{Euc}\Omega$. Since $p\in\partial_{Euc}\Omega$, we
can find a sequence $z_{k}\in\Omega$ which is sufficiently close to $p_{k}$
such that $z_{k}\to p$. Then write
$\lim_{k\to\infty}f(z_{k})=\widetilde{f}(p)=q\in\partial_{Euc}\Omega^{\prime}$.
is a sequence in $\partial_{Euc}\Omega^{\prime}$. Since
$\partial_{Euc}\Omega^{\prime}$ is compact and $z_{k}$ is sufficiently close
to $p_{k}$, we have

$\lim_{k\to\infty}\widetilde{f}(p_{k})=\lim_{k\to\infty}f(z_{k})=q=\widetilde{f}(p).$

This completes the proof of the theorem. ∎

Proof of Theorem 1.10. Since $\Omega$ and $D$ are Gromov hyperbolic, therefore
in view of Theorem 2.3, we have that $f$ extends to a homeomorphism
$\widetilde{f}:\overline{\Omega}^{G}\to\overline{D}^{G}$. Since $\Omega$ and
$D$ are Gromov hyperbolic and visibility domains with no geodesic loops,
therefore we have that the identity map

$id:(\Omega,k_{D})\to(\Omega,d_{Euc})$

extends as a homeomorphism

$\Phi_{\Omega}:\overline{\Omega}^{G}\to\overline{\Omega}^{Euc}$

and the identity map

$id_{D}:(D,k_{D})\to(D,d_{Euc})$

extends as a homeomorphism

$\Phi_{D}:\overline{D}^{G}\to\overline{D}^{Euc}.$

The map

$F=\Phi_{D}\circ\widetilde{f}\circ\Phi_{\Omega}^{-1}:\overline{\Omega}^{Euc}\to\overline{D}^{Euc}$

gives the desired homeomorphic extension. This completes the proof. ∎

##  7\. The end compactification, visibility and Gromov hyperbolicity

In this section, we prove Theorem 1.11, but before that we recall the
definition of end compactification. H. Freudenthal [23] introduced the notion
of end compactification for topological spaces, which also known as
Freudenthal compactification.

Given $\Omega\subset\mathbb{R}^{n}$, $\overline{\Omega}^{Euc}$ admits an
increasing sequence $K_{1}\subset K_{2}\subset K_{3}\subset\cdots$ of compact
subsets with $\overline{\Omega}^{Euc}=\cup_{n\geq 1}K_{n}$. By compactness
each $\overline{\Omega}^{Euc}\setminus K_{n}$ has finitely many components. An
end, say $e$, of $\overline{\Omega}^{Euc}$ is a decresing sequence
$U^{e}_{1}\supset U_{2}^{e}\supset U_{3}^{e}\supset\cdots$ of open connected
set (i.e., $e=\\{U_{n}^{e}\\}_{n\geq 1}$) such that

$U^{e}_{n}:=\mbox{ some connected component of
}\overline{\Omega}^{Euc}\setminus K_{n},\,\,\,n=1,2,3,\ldots$

Let $E\left(\overline{\Omega}^{Euc}\right)$ denote the set of ends of
$\overline{\Omega}^{Euc}$. The end compactification of
$\overline{\Omega}^{Euc}$ as a set is

$\overline{\Omega}^{End}:=\overline{\Omega}^{Euc}\cup
E\left(\overline{\Omega}^{Euc}\right),$

has a natural topology making it a compactification of
$\overline{\Omega}^{Euc}$.

######  Remark 7.1.

  * (1)

Note that the above construction does not depend upon the choice of compact
subsets $\\{K_{n}\\}$. In fact, if $\\{K_{n}^{\prime}\\}$ is another sequence
of compact sets with the desired property then there is a bijection between
the set of ends given by $\\{K_{n}\\}$ and the set of ends given by
$\\{K_{n}^{\prime}\\}$

  * (2)

If $\Omega$ is bounded, then it is easy to see that
$\overline{\Omega}^{End}=\overline{\Omega}^{Euc}$.

  * (3)

The space $\overline{\Omega}^{End}:=\overline{\Omega}^{Euc}\cup
E\left(\overline{\Omega}^{Euc}\right)$ is compact and Hausdorff. Further, the
subset $E\left(\overline{\Omega}^{Euc}\right)$ is closed and totally
disconnected.

Motivated by the work of Bracci et. al. [13, Section 6], we introduce the
following definition.

######  Definition 7.1.

Let $\Omega$ be an unbounded domain. Suppose $(\Omega,k_{\Omega})$ is Gromov
hyperbolic. $\Omega$ is said to be Gromov compactification and end
compactification (GCEC)-compatible domain if the following property holds:
Suppose $x$ is an end of $\overline{\Omega}$. Then there exists
$\xi_{x}\in\partial_{G}\Omega$ such that: if $z_{k}\in\Omega$ converges to $x$
in $\overline{\Omega}^{End}$, then

$z_{k}\stackrel{{\scriptstyle Gromov}}{{\longrightarrow}}\xi_{x}.$

Moreover, if $\overline{\Omega}$ has two ends $x$, $y$, then
$\xi_{x}\neq\xi_{y}$.

The Proof of Theorem 1.11 will be done in several lemmas. We first made the
following observation.

######  Observation 3.

Let $\Omega$ be an unbounded domain. If there are two sequence $\\{z_{k}\\}$,
$\\{w_{k}\\}\subset\Omega$ such that $\lim\limits_{k\to\infty}|z_{k}|=\infty$
and $\lim\limits_{k\to\infty}w_{k}\in\partial_{Euc}\Omega$, then

$k_{\Omega}(z_{k},w_{k})=\infty.$

Proof of Observation 3. This immediately follws from Equation 2.5. ∎

######  Lemma 7.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a domain (not necessarily bounded). If
there are two sequence $\\{z_{k}\\}$, $\\{w_{k}\\}\subset\Omega$ such that
$\lim\limits_{k\to\infty}z_{k}=p\in\partial_{Euc}\Omega$ and

$k_{\Omega}(z_{k},w_{k})<\infty,$

then $\lim\limits_{k\to\infty}w_{k}=p$

Proof. By Observation 3, it is clear that
$\lim\limits_{k\to\infty}w_{k}\in\partial_{Euc}\Omega$. By contradiction
assume that $\lim\limits_{k\to\infty}w_{k}=q$, $q\neq p$, and hence by
Observation 1 we get that $k_{\Omega}(z_{k},w_{k})=\infty,$ which is a
contradiction. ∎

######  Lemma 7.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a domain (not necessarily bounded).
Suppose $\Omega$ is a visibility domain. If $\gamma:[0,\infty)\to\Omega$ be a
qh-geodesic ray, then $\lim\limits_{t\to\infty}\gamma(t)$ exists in
$\partial\overline{\Omega}^{End}$

Proof. Consider the limit $\lim_{k\to\infty}\gamma(t_{k})$, where
$\\{t_{k}\\}$ is a sequence such that $t_{k}\to\infty$ Since $\gamma$ is a qh-
geodesic ray i.e., $\gamma:[0,\infty)\to(\Omega,k_{D})$ is an isometry,
therefore we have $|t_{k}|=k_{\Omega}(\gamma(0),\gamma(t_{k}))\to\infty$, and
hence $\lim_{k\to\infty}\gamma(t_{k})\in\partial\overline{\Omega}^{End}$. Now,
let us consider the following set

$L=\\{x\in\partial\overline{\Omega}^{End}:\mbox{ there exists
}t_{k}\to\infty\mbox{ such that }\gamma(t_{k})\to x\\}.$

Suppose for contradiction that $L$ is not singleton. Notice that $L$ is
connected and so $L$ contains at least one point in $\partial_{Euc}\Omega$
(since the space $\overline{\Omega}^{End}$ is Hausdorff). Then, again by
connectedness, $L$ must contain at least two points in $\partial_{Euc}\Omega$,
say, $p,q$. So we can find sequence $a_{k}\to\infty$ and $b_{k}\to\infty$ such
that $\gamma(a_{k})\to p$ and $\gamma(b_{k})\to q$. Consider the qh-geodesics
$\gamma_{k}=\gamma|_{[a_{k},b_{k}]}$, then by the visibility property, there
exists a compact set $K\subset\Omega$ such that for sufficiently large $k$,
there exists $s_{k}\in(a_{k},b_{k})$ such that $\gamma_{k}(s_{k})\in K$. Upto
subsequence we can always assume $a_{k}<s_{k}<b_{k}$ for all $k$. Therefore,
$s_{k}\to\infty$, but then $L\cap K\neq\phi$, which is a contradiction. This
completes the proof. ∎

######  Lemma 7.3.

Let $\Omega$ be Grmov hyperbolic and GCEC-compatible domain. Suppose $\Omega$
is a visibility domain. Fix $x_{0}\in\Omega$, and let $\gamma$ and $\sigma$ be
any two qh-geodesic rays with $\gamma(0)=x_{0}=\sigma(0)$ such that
$\gamma\sim\sigma$. Then

$\lim\limits_{t\to\infty}\gamma(t)=\lim\limits_{t\to\infty}\sigma(t).$

Proof. In view of Lemma 7.2, we have that
$\lim\limits_{t\to\infty}\gamma(t),\,\lim\limits_{t\to\infty}\sigma(t)\in\partial\overline{\Omega}^{End}$.
Suppose there are two distinct points $p,q\in\partial\overline{\Omega}^{End}$
such that

$\lim\limits_{t\to\infty}\gamma(t)=p\neq q=\lim\limits_{t\to\infty}\sigma(t).$

So there exists a sequence $\\{s_{k}\\}$ with $s_{k}\to\infty$ such that

$\lim_{k\to\infty}\gamma(s_{k})=p\mbox{ and
}\lim_{k\to\infty}\sigma(s_{k})=q.$

Write $z_{k}=\gamma(s_{k})$ and $w_{k}=\sigma(s_{k})$, if at least one of $p$
and $q$ belongs to $\partial_{Euc}\Omega$, then by Observation 3, both $p$ and
$q$ will belong to $\partial_{Euc}\Omega$. Hence $z_{k}\to p$ and $w_{k}\to q$
and since $p\neq q$ by Observation 1, we have that
$\lim_{k\to\infty}k_{\Omega}(z_{k},w_{k})=\infty$, which is a contradiction to
the fact that $\gamma\sim\sigma$. Now, if both $p,q$ are ends of
$\overline{\Omega}$, $\Omega$ being GCEC domain gives a contradiction to the
fact that $\gamma\sim\sigma$. This completes the proof. ∎

######  Lemma 7.4.

Let $\Omega$ be a domain (not necessairly bounded). Suppose $\Omega$ is a
visibility domain. Let $T_{k}\in(0,\infty]$, if
$\gamma_{k}:[0,T_{k})\to\Omega$ be a sequence of qh-geodesics converges
locally uniformly on compacta to a geodesic ray $\gamma:[0,\infty)\to\Omega$,
then

$\lim\limits_{t\to\infty}\gamma(t)=\lim\limits_{k\to\infty}\lim\limits_{t\to
T_{k}}\gamma_{k}(t)$

Proof. The proof is identical to the proof of the Lemma 4.5. So we skip it
here. It is clear that if $T_{k}=\infty$, then $\lim\limits_{t\to
T_{k}}\gamma_{k}(t)\in\partial\overline{\Omega}^{End}$, and hence
$\lim\limits_{k\to\infty}\lim\limits_{t\to
T_{k}}\gamma_{k}(t)\in\partial\overline{\Omega}^{End}$. Moreover, by Lemma
7.2, we have
$\lim\limits_{t\to\infty}\gamma(t)=\xi\in\partial\overline{\Omega}^{End}$. In
general, let $\lim\limits_{t\to T_{k}}\gamma_{k}(t)=\xi_{k}$, and

$\lim\limits_{k\to\infty}\lim\limits_{t\to
T_{k}}\gamma_{k}(t)=\xi_{\infty}\in\partial\overline{\Omega}^{End}.$

Assume by contradiction $\xi\neq\xi_{\infty}$. Since