†† 2020 Mathematics Subject Classification. Primary 53C23; Secondary 20F67,
20F65, 20F69 †† Key words and phrases. non-positively curved space, ideal
boundary, dimension raising map, coarse geometry.

# VISUAL MAPS BETWEEN COARSELY CONVEX SPACES

Yuuhei Ezawa, Tomohiro Fukaya

###### Abstract.

The class of coarsely convex spaces is a coarse geometric analogue of the
class of nonpositively curved Riemannian manifolds. It includes Gromov
hyperbolic spaces, CAT(0) spaces, proper injective metric spaces and systolic
complexes. It is well known that quasi-isometric embeddings of Gromov
hyperbolic spaces induce topological embeddings of their boundaries. Dydak and
Virk studied maps between Gromov hyperbolic spaces which induce continuous
maps between their boundaries. In this paper, we generalize their work to maps
between coarsely convex spaces.

T.Fukaya was supported by JSPS KAKENHI Grant Number JP19K03471

##  1\. Introduction

The class of coarsely convex spaces is introduced by the second author and
Shin-ichi Oguni [7]. This class includes Gromov hyperbolic spaces, and
Busemann nonpositively curved spaces, especially, CAT(0) spaces. Moreover, it
is shown that systolic complexes and proper injective metric spaces are
coarsely convex spaces, by Osajda and Przytycki [12], respectively, by
Descombes and Lang [3]. Recently, Chalopin, Chepoi, Genevois, Hirai and Osajda
[2] showed that Helly graphs and coarsely Helly spaces are coarsely dense in
their injective hulls, especially, they are coarsely convex. This result may
become a rich source of coarsely convex spaces. In fact, Osajda and Huang [11]
showed that Artin groups of type FC and Garside groups act geometrically on
Helly graphs, hence they do on coarsely convex spaces. Haettel, Hoda and Petyt
[8] showed that any hierarchically hyperbolic space admits a new metric that
is coarsely Helly, so it is coarsely convex.

Let $X$ be a proper coarsely convex space. In [7], the authors constructed the
ideal boundary $\partial X$ of $X$, and they showed that the open cone
$\mathcal{O}\partial X$ is coarsely homotopy equivalent to $X$ via the
“exponential map” and the “logarithmic map”. As a corollary, it follows that
the coarse Baum-Connes conjecture holds for coarsely convex spaces. In this
paper, we consider maps between coarsely convex spaces which induce continuous
maps between their boundaries.

Dydak and Virk [4] introduced the class of visual maps between Gromov
hyperbolic spaces that induce continuous maps between their boundaries. They
also introduced its subclass, called radial maps, and showed that these maps
induce Hölder maps between boundaries.

Then they showed that radial coarsely $n$-to-one maps between proper geodesic
Gromov hyperbolic spaces induces continuous $n$-to-one maps between
boundaries. As a consequence, they obtained dimension rising theorem for
radial maps, with respect to the topological dimensions of the Gromov
boundaries, and to the asymptotic dimensions of hyperbolic groups.

A metric space $(X,d)$ is coarsely convex if $X$ has a family of quasi-
geodesics ${{}_{X}\mathcal{L}}$, called the system of good quasi-geodesics,
satisfying some conditions. Especially, along quasi-geodesics in
${{}_{X}\mathcal{L}}$, the metric $d$ satisfies coarsely convex inequality
(Definition 2.1(ii)q).

In this paper, we introduce the class of visual maps between coarsely convex
spaces, that do induce continuous maps between their ideal boundaries
(Proposition 3.5 and Corollary 3.7).

We present a fixed point property of the maps induced by the visual
isometries. Let $M$ be a metric space and $f\colon M\to M$ be an isometry. We
say that $f$ is elliptic if the set $\\{f^{n}(x)\colon n\in\mathbb{N}\\}$ is
bounded for some $x\in M$.

######  Theorem 1.1.

Let $X$ be a proper coarsely convex space and let $f\colon X\to X$ be an
isometry. We suppose that $f$ is visual. Then, either $f$ is elliptic, or, the
induced map $\partial f\colon\partial X\to\partial X$ has a fixed point.

Let $X$ and $Y$ be coarsely convex spaces with systems of good quasi-geodesics
${{}_{X}\mathcal{L}}$ and ${{}_{Y}\mathcal{L}}$, respectively. We also
introduce the class of ${{}_{X}\mathcal{L}}$-radial and
${{}_{X}\mathcal{L}}$-${{}_{Y}\mathcal{L}}$-equivariant maps, which is in fact
a subclass of visual maps (Theorem 3.11).

The following is a generalization of the large scale dimension raising theorem
for hyperbolic spaces by Dydak and Virk [4, Theorem 1.1].

######  Theorem 1.2.

Let $X$ and $Y$ be proper coarsely convex spaces, and let
${{}_{X}\mathcal{L}}$ and ${{}_{Y}\mathcal{L}}$ be systems of good quasi-
geodesics of $X$ and $Y$, respectively. Let $f\colon X\to Y$ be a large scale
Lipschitz map which is ${{}_{X}\mathcal{L}}$-radial and
${{}_{X}\mathcal{L}}$-${{}_{Y}\mathcal{L}}$-equivariant. If $f$ is coarsely
$(n+1)$-to-one and coarsely surjective then

$\displaystyle\mathop{\mathrm{dim}}\nolimits(\partial
Y)\leq\mathop{\mathrm{dim}}\nolimits(\partial X)+n.$

We also study extension problems. Let $f\colon\partial X\to\partial Y$ be a
continuous map between ideal boundaries of coarsely convex spaces. Then we
construct a visual map $\mathop{\mathrm{rad}}\nolimits f\colon X\to Y$, called
radial extension of $f$, which induces the given map $f$ of the boundaries. We
also show that for visual
${{}_{X}\mathcal{L}}^{\infty}$-${{}_{Y}\mathcal{L}}^{\infty}$-equivariant map
$F\colon X\to Y$, the radial extension of its induced map $\partial
F\colon\partial X\to\partial Y$ is coarsely homotopic to the given map $F$.
These results are summarized in Theorem 7.7.

The organization of the paper is as follows. In Section 2, we review the
definition and basic properties of coarsely convex spaces. In Section 3, we
introduce the class of visual maps and its subclasses. At the end of the
section, we give a proof of Theorem 1.1.

In Section 4, we study on coarsely surjective maps. In Section 5, we show that
coarsely $n$-to-one maps induce $n$-to-one maps between boundaries (Theorem
5.4), and give a proof of Theorem 1.2. In Section 6, we discuss maps between
direct products of coarsely convex spaces and give an explicit example of a
map satisfying the assumptions in Theorem 1.2 whose domain and range are not
Gromov hyperbolic spaces.

In Section 7, we give the construction of the radial extension, and give a
proof of Theorem 7.7.

##  2\. Coarsely convex space

In this section, we review the theory of coarsely convex spaces and their
boundaries.

Throughout this paper, we use the following notations. For points $x,y$ in a
metric space $X$, we denote by $\,\overline{x,y}\,$ the distance between $x$
and $y$. For a subset $A\subset X$ and a point $x\in X$, we denote by
$\,\overline{x,A}\,$ the distance between $x$ and $A$. We denote by
$\mathop{\mathrm{diam}}\nolimits A$ the diameter of $A$. For a map $F\colon
P\to Q$, we denote by $\operatorname{Dom}F$ the domain of $F$, that is,
$\operatorname{Dom}F=P$. For $a,b\in\mathbb{R}$, we denote by $a\vee b$ the
maximum of $\\{a,b\\}$.

###  2.1. Coarse map

Let $X,Y$ be metric spaces. Let $f\colon X\to Y$ be a map.

  1. (1)

The map $f$ is bornologous if there exists a non-decreasing function
$\psi\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ such that for all
$x,x^{\prime}\in X$, we have

$\,\overline{f(x),f(x^{\prime})}\,\leq\psi(\,\overline{x,x^{\prime}}\,).$

Especially, for constants $A\geq 1,\,B\geq 0$, $f$ is a $(A,B)$-large scale
Lipschitz map if for all $x,x^{\prime}\in X$, we have
$\,\overline{f(x),f(x^{\prime})}\,\leq A\,\overline{x,x^{\prime}}\,+B$.

  2. (2)

The map $f$ is metrically proper if for each bounded subset $B\subset Y$, the
inverse image $f^{-1}(B)$ is bounded.

  3. (3)

The map $f$ is coarse if it is bornologous and metrically proper.

  4. (4)

The map $f$ is a coarse embedding if it is bornologous and there exists a non-
decreasing function $\psi^{\prime}\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}$ such that $\psi^{\prime}(t)\to\infty\,(t\to\infty)$ and for all
$x,x^{\prime}\in X$, we have

$\psi^{\prime}(\,\overline{x,x^{\prime}}\,)\leq\,\overline{f(x),f(x^{\prime})}\,.$

  5. (5)

The map $f$ is coarsely surjective if there exists $C\geq 0$ such that
$C$-neighbourhood of $f(X)$ is $Y$.

  6. (6)

The map $f$ is a coarse equivalence map if it is a coarse embedding and
coarsely surjective.

For more details on the above notions, we refer [13].

###  2.2. Coarse homotopy

Let $f,g\colon X\to Y$ be coarse maps between metric spaces. The maps $f$ and
$g$ are coarsely homotopic if there exists a metric subspace $Z=\\{(x,t):0\leq
t\leq T_{x}\\}$ of $X\times\mathbb{R}_{\geq 0}$ and a coarse map $h\colon Z\to
Y$, such that

  1. (1)

the map $X\ni x\mapsto T_{x}\in\mathbb{R}_{\geq 0}$ is bornologous,

  2. (2)

$h(x,0)=f(x)$, and

  3. (3)

$h(x,T_{x})=g(x)$.

Here we equip $X\times\mathbb{R}_{\geq 0}$ with the $l_{1}$-metric, that is,
$d_{X\times\mathbb{R}_{\geq 0}}((x,t),(y,s)):=\,\overline{x,y}\,+\left\lvert
t-s\right\rvert$ for $(x,t),(y,s)\in X\times\mathbb{R}_{\geq 0}$.

###  2.3. Coarsely convex space

######  Definition 2.1.

Let $X$ be a metric space. Let $\lambda\geq 1$, $k\geq 0$, $E\geq 1$, and
$C\geq 0$ be constants. Let $\theta\colon\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ be a non-decreasing function. Let $\mathcal{L}$ be a
family of $(\lambda,k)$-quasi-geodesic segments. The metric space $X$ is
$(\lambda,k,E,C,\theta,\mathcal{L})$-coarsely convex, if $\mathcal{L}$
satisfies the following.

  1. (i)q

For $v,w\in X$, there exists a quasi-geodesic segment $\gamma\in\mathcal{L}$
with $\operatorname{Dom}\gamma=[0,a]$, $\gamma(0)=v$ and $\gamma(a)=w$.

  2. (ii)q

Let $\gamma,\eta\in\mathcal{L}$ be quasi-geodesic segments with
$\operatorname{Dom}\gamma=[0,a]$ and $\operatorname{Dom}\eta=[0,b]$. Then for
$t\in[0,a]$, $s\in[0,b]$, and $0\leq c\leq 1$, we have that

$\displaystyle\,\overline{\gamma(ct),\eta(cs)}\,\leq
cE\,\overline{\gamma(t),\eta(s)}\,+(1-c)E\,\overline{\gamma(0),\eta(0)}\,+C.$

  3. (iii)q

Let $\gamma,\eta\in\mathcal{L}$ be quasi-geodesic segments with
$\operatorname{Dom}\gamma=[0,a]$ and $\operatorname{Dom}\eta=[0,b]$. Then for
$t\in[0,a]$ and $s\in[0,b]$, we have

$\displaystyle\left\lvert
t-s\right\rvert\leq\theta(\,\overline{\gamma(0),\eta(0)}\,+\,\overline{\gamma(t),\eta(s)}\,).$

The family $\mathcal{L}$ satisfying (i)q, (ii)q, and (iii)q is called a system
of good quasi-geodesic segments, and elements $\gamma\in\mathcal{L}$ are
called good quasi-geodesic segments. We call the inequality in (ii)q the
coarsely convex inequality.

We say that a metric space $X$ is a coarsely convex space if there exist
constants $\lambda,k,E,C$, a non-decreasing function
$\theta\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$, and a family of
$(\lambda,k)$-quasi-geodesic segments $\mathcal{L}$ such that $X$ is
$(\lambda,k,E,C,\theta,\mathcal{L})$-coarsely convex.

We remark that if $\mathcal{L}$ consists of only geodesic segments, then
$\mathcal{L}$ satisfies (iii)q by the triangle inequality. We say that a
metric space is geodesic $(C,\mathcal{L})$-coarsely convex if it is
$(1,0,1,C,\mathrm{id}_{\mathbb{R}_{\geq 0}},\mathcal{L})$-coarsely convex.

The class of coarsely convex spaces is closed under direct products. Let $X$
and $Y$ be metric spaces and $\gamma\colon[0,a]\to X$ and $\eta\colon[0,b]\to
Y$ be maps. Then we define the map $\gamma\oplus\eta\colon[0,a+b]\to X\times
Y$ by setting

$\displaystyle\gamma\oplus\eta(t):=\left(\gamma\left(\frac{a}{a+b}t\right),\eta\left(\frac{b}{a+b}t\right)\right),\quad
t\in[0,a+b].$

If $\gamma$ and $\eta$ are quasi-geodesic segments, so is $\gamma\oplus\eta$.

######  Proposition 2.2 ([7]).

Let $(X,d_{X})$ and $(Y,d_{Y})$ be coarsely convex metric spaces with systems
of good quasi-geodesic segments ${{}_{X}\mathcal{L}}$ and
${{}_{Y}\mathcal{L}}$, respectively. Then the product with the
$\ell_{1}$-metric $(X\times Y,d_{X\times Y})$ is coarsely convex, whose system
of good quasi-geodesic segments ${{}_{X\times Y}\mathcal{L}}$ consists of all
$\gamma\oplus\eta$ for $\gamma\in{{}_{X}\mathcal{L}}$ and
$\eta\in{{}_{Y}\mathcal{L}}$.

In the rest of this section, let $X$ be a
$(\lambda,k,E,C,\theta,\mathcal{L})$-coarsely convex space.

###  2.4. Ideal boundary

Let $\gamma\colon\mathbb{R}_{\geq 0}\to X$ be a map. Let
$\gamma_{n}\colon[0,a_{n}]\to X$ be quasi-geodesic segments in $X$. A sequence
$\\{(\gamma_{n},a_{n})\\}_{n}$ with $a_{n}\to\infty$ is an
$\mathcal{L}$-approximate sequence for $\gamma$ if for all $n$, we have
$\gamma_{n}\in\mathcal{L}$, $\gamma_{n}(0)=\gamma(0)$ and
$\\{\gamma_{n}\\}_{n}$ converges to $\gamma$ pointwise on $\mathbb{N}$. A map
$\gamma\colon\mathbb{R}_{\geq 0}\to X$ is $\mathcal{L}$-approximatable if
there exists an $\mathcal{L}$-approximate sequence for $\gamma$.

We define a family of quasi-geodesic rays, denoted by $\mathcal{L}^{\infty}$,
as a family consisting of all $\mathcal{L}$-approximatable maps
$\gamma\colon\mathbb{R}_{\geq 0}\to X$ such that $\gamma(t)=\gamma(\lfloor
t\rfloor)$ for all $t\in\mathbb{R}_{\geq 0}$. We set
$\bar{\mathcal{L}}:=\mathcal{L}\cup\mathcal{L}^{\infty}$. We remark that
$\gamma\in\mathcal{L}^{\infty}$ is a $(\lambda,k_{1})$-quasi-geodesic ray,
where $k_{1}:=\lambda+k$. See [7, Lemma 4.1].

######  Proposition 2.3 ([7, Proposition 4.2]).

The family $\bar{\mathcal{L}}$ satisfies the following.

  1. $(1)$

Let $\gamma,\eta\in\bar{\mathcal{L}}$ be quasi-geodesics. Then for
$t\in\operatorname{Dom}\gamma$, $s\in\operatorname{Dom}\eta$ and $0\leq c\leq
1$, we have

$\displaystyle\,\overline{\gamma(ct),\eta(cs)}\,\leq
cE\,\overline{\gamma(t),\eta(s)}\,+(1-c)E\,\overline{\gamma(0),\eta(0)}\,+D,$

where $D:=2(1+E)k_{1}+C$.

  2. $(2)$

We define a non-decreasing function $\tilde{\theta}\colon\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ by $\tilde{\theta}(t):=\theta(t+1)+1$. For
$\gamma,\eta\in\bar{\mathcal{L}}$ and for $t\in\operatorname{Dom}\gamma$,
$s\in\operatorname{Dom}\eta$, we have

$\displaystyle\left\lvert
t-s\right\rvert\leq\tilde{\theta}(\,\overline{\gamma(0),\eta(0)}\,+\,\overline{\gamma(t),\eta(s)}\,).$

######  Lemma 2.4 ([7, Lemma 4.3]).

Let $\gamma,\eta\in\bar{\mathcal{L}}$ be quasi-geodesics such that
$\gamma(0)=\eta(0)$. For all $a\in\operatorname{Dom}\gamma$,
$b\in\operatorname{Dom}\eta$ and $0\leq t\leq\min\\{a,b\\}$, we have

$\displaystyle\,\overline{\gamma(t),\eta(t)}\,$ $\displaystyle\leq
E(\,\overline{\gamma(a),\eta(b)}\,+\lambda\tilde{\theta}(\,\overline{\gamma(a),\eta(b)}\,)+k_{1})+D.$

For quasi-geodesic rays $\gamma$ and $\eta$ in $\mathcal{L}^{\infty}$, we say
that $\gamma$ and $\eta$ are equivalent if

$\displaystyle\sup\\{\,\overline{\gamma(t),\eta(t)}\,:t\in\mathbb{R}_{\geq
0}\\}<\infty,$

and we denote by $\gamma\sim\eta$. For $\gamma\in\mathcal{L}^{\infty}$, we
denote by $[\gamma]$ its equivalence class. The ideal boundary of $X$ is the
set $\partial X:=\mathcal{L}^{\infty}/\sim$ of equivalence classes of quasi-
geodesic rays in $\mathcal{L}^{\infty}$.

Let $O\in X$ be a base point. We define $\mathcal{L}_{O}^{\infty}$ as the
subset of $\mathcal{L}^{\infty}$ consisting of all quasi-geodesic rays in
$\mathcal{L}^{\infty}$ stating at $O$. The ideal boundary of $X$ with respect
to $O$ is the set $\partial_{O}X:=\mathcal{L}_{O}^{\infty}/\sim$ of
equivalence classes of quasi-geodesic rays in $\mathcal{L}_{O}^{\infty}$.

######  Lemma 2.5 ([7, Lemma 4.5]).

For $\gamma,\eta\in\mathcal{L}_{O}^{\infty}$, $\gamma$ and $\eta$ are
equivalent if and only if $\,\overline{\gamma(t),\eta(t)}\,\leq D$ for all
$t\in\mathbb{R}_{\geq 0}$.

We define a subset $\mathcal{L}_{O}\subset\mathcal{L}$ as the set of all
$\gamma\in\mathcal{L}$ with $\operatorname{Dom}\gamma=[0,a_{\gamma}]$,
$a_{\gamma}\geq 2\theta(0)$, and $\gamma(0)=O$. Set
$\bar{\mathcal{L}}_{O}:=\mathcal{L}_{O}\cup\mathcal{L}_{O}^{\infty}$.

For $x\in X\setminus\bar{B}(O,2\lambda\theta(0)+k)$, set

$\displaystyle\bar{\mathcal{L}}_{O}(x):=\\{\gamma\in\mathcal{L}_{O}:\operatorname{Dom}\gamma=[0,a_{\gamma}],\,\gamma(a_{\gamma})=x\\},$

and for $x\in\partial_{O}X$, set

$\displaystyle\bar{\mathcal{L}}_{O}(x):=\\{\gamma\in\mathcal{L}_{O}^{\infty}:\gamma\in
x\\}.$

######  Definition 2.6.

We define a product
$(\cdot\mid\cdot)_{O}\colon\bar{\mathcal{L}}_{O}\times\bar{\mathcal{L}}_{O}\to\mathbb{R}_{\geq
0}\cup\\{\infty\\}$ as follows. For $\gamma,\eta\in\bar{\mathcal{L}}_{O}$,

$\displaystyle(\gamma\mid\eta)_{O}:=\sup\\{t\in\operatorname{Dom}\gamma\cap\operatorname{Dom}\eta:\,\overline{\gamma(t),\eta(t)}\,\leq
2D+2\\}.$

We also define a product
$(\cdot\mid\cdot)_{O}\colon(X\cup\partial_{O}X)\times(X\cup\partial_{O}X)\to\mathbb{R}_{\geq
0}\cup\\{\infty\\}$ as follows.

  1. (1)

If $x\in\bar{B}(O,2\lambda\theta(0)+k))$ or
$y\in\bar{B}(O,2\lambda\theta(0)+k))$, set

$\displaystyle(x\mid y)_{O}:=0.$

  2. (2)

If $x,y\in(X\setminus\bar{B}(O,2\lambda\theta(0)+k))\cup\partial_{O}X$, set

$\displaystyle(x\mid
y)_{O}:=\sup\\{(\gamma\mid\eta)_{O}:\gamma\in\bar{\mathcal{L}}_{O}(x),\,\eta\in\bar{\mathcal{L}}_{O}(y)\\}.$

When the choice of the base point is clear, we write $(x\mid y)$ instead of
$(x\mid y)_{O}$.

We quote some lemmas from [7] which we need later.

######  Lemma 2.7 ([7, Lemma 4.14]).

Let $\gamma\in\mathcal{L}_{O}^{\infty}$ be a quasi-geodesic ray and let
$\\{(\gamma_{n},a_{n})\\}_{n}$ be an $\mathcal{L}$-approximate sequence for
$\gamma$. Then we have $\liminf_{n\to\infty}(\gamma\mid\gamma_{n})=\infty$.

######  Lemma 2.8.

There exists a constant $\Omega\geq 1$ depending on $\lambda,k,E,C,\theta(0)$
such that the following holds:

  1. $($i$)$

For $x,y\in X\cup\partial_{O}X$ and for $\gamma\in\bar{\mathcal{L}}_{O}(x)$
and $\eta\in\bar{\mathcal{L}}_{O}(y)$, we have

$\displaystyle(\gamma\mid\eta)\leq(x\mid y)\leq\Omega(\gamma\mid\eta).$

  2. $($ii$)$

For triplet $\gamma,\eta,\xi\in\bar{\mathcal{L}}_{O}$, we have

$\displaystyle(\gamma\mid\xi)\geq\Omega^{-1}\min\\{(\gamma\mid\eta),(\eta\mid\xi)\\}.$

  3. $($iii$)$

For triplet $x,y,z\in X\cup\partial_{O}X$, we have

$\displaystyle(x\mid z)\geq\Omega^{-1}\min\\{(x\mid y),(y\mid z)\\}.$

  4. $($iv$)$

Let $\gamma,\eta\in\bar{\mathcal{L}}_{O}$. For all $t\in\mathbb{R}_{\geq 0}$
with $t\leq(\gamma\mid\eta)$, we have

$\displaystyle\,\overline{\gamma(t),\eta(t)}\,\leq\Omega.$

  5. $($v$)$

Let $\gamma,\eta\in\bar{\mathcal{L}}_{O}$. If $\gamma(a)=\eta(b)$ for some
$a\in\operatorname{Dom}\gamma$ and $b\in\operatorname{Dom}\eta$, then for all
$t\in\operatorname{Dom}\gamma\cap\operatorname{Dom}\eta$ with $0\leq
t\leq\max\\{a,b\\}$, we have

$\displaystyle\,\overline{\gamma(t),\eta(t)}\,\leq\Omega.$

###### Proof..

(i),(ii) and (iii) follow from [7, Lemma 4.12], [7, Lemma 4.8] and [7,
Corollary 4.13], respectively.

First we give a proof of (iv). Set $a:=(\gamma\mid\eta)$. If $a=\infty$, by
Lemma 2.5, we have $\,\overline{\gamma(t),\eta(t)}\,\leq D$ for all
$t\in\mathbb{R}_{\geq 0}$. We suppose $a<\infty$. By [7, Lemma 4.7], we have
$\,\overline{\gamma(a),\eta(a)}\,\leq D_{1}+2k_{1}$. Then by Proposition 2.3,

$\displaystyle\,\overline{\gamma(t),\eta(t)}\,\leq
E(D_{1}+2k_{1})+D\quad(\forall t\in[0,a]).$

Next we give a proof of (v). Let
$t\in\operatorname{Dom}\gamma\cap\operatorname{Dom}\eta$ with $0\leq
t\leq\max\\{a,b\\}$. First we suppose $t\leq\min\\{a,b\\}$. Then by Lemma 2.4,
$\,\overline{\gamma(t),\eta(t)}\,\leq E(\lambda\tilde{\theta}(0)+k_{1})+D$.

Now we suppose $\min\\{a,b\\}<t\leq\max\\{a,b\\}$. Since $\gamma(a)=\eta(b)$,
we have $\left\lvert a-b\right\rvert\leq\tilde{\theta}(0)$. Then

$\displaystyle\,\overline{\gamma(t),\eta(t)}\,\leq\,\overline{\gamma(t),\gamma(a)}\,+\,\overline{\gamma(a),\eta(b)}\,+\,\overline{\eta(b),\eta(t)}\,\leq
2(\lambda\tilde{\theta}(0)+k_{1}).$

∎

######  Lemma 2.9.

For $\gamma,\gamma^{\prime}\in\bar{\mathcal{L}}_{O}$,
$t\in\operatorname{Dom}\gamma$ and
$t^{\prime}\in\operatorname{Dom}\gamma^{\prime}$, set
$\alpha:=\,\overline{\gamma(t),\gamma^{\prime}(t^{\prime})}\,$. Then we have

$\displaystyle(\gamma\mid\gamma^{\prime})\geq\frac{\max\\{t,t^{\prime}\\}-\tilde{\theta}(\alpha)}{E(\alpha+\lambda\tilde{\theta}(\alpha)+k_{1})}.$

###### Proof..

By Proposition 2.3 we have

$\displaystyle\left\lvert
t-t^{\prime}\right\rvert\leq\tilde{\theta}(\,\overline{\gamma(0),\gamma^{\prime}(0)}\,+\,\overline{\gamma(t),\gamma^{\prime}(t^{\prime})}\,)=\tilde{\theta}(\alpha).$

Set $u:=\min\\{t,t^{\prime}\\}$. We have

$\displaystyle\,\overline{\gamma(u),\gamma^{\prime}(u)}\,\leq\,\overline{\gamma(t),\gamma^{\prime}(t^{\prime})}\,+\lambda\left\lvert
t-t^{\prime}\right\rvert+k_{1}\leq\alpha+\lambda\tilde{\theta}(\alpha)+k_{1}.$

Set $c:=(E(\alpha+\lambda\tilde{\theta}(\alpha)+k_{1}))^{-1}$. Then we have

$\displaystyle\,\overline{\gamma(cu),\gamma^{\prime}(cu)}\,\leq
cE\,\overline{\gamma(u),\gamma^{\prime}(u)}\,+D\leq D+1.$

Therefore

$\displaystyle(\gamma\mid\gamma^{\prime})\geq
cu\geq\frac{\max\\{t,t^{\prime}\\}-\tilde{\theta}(\alpha)}{E(\alpha+\lambda\tilde{\theta}(\alpha)+k_{1})}.$

∎

In the rest of this section, we suppose that $X$ is proper, that is, all
closed bounded subset is compact. By a diagonal argument, we have the
following.

######  Lemma 2.10 ([7, Proposition 4.17]).

Let $\\{v_{n}\\}_{n}$ be a sequence in $X$ such that

$\displaystyle\lim_{n\to\infty}\,\overline{O,v_{n}}\,=\infty.$

Then there exists a $(\lambda,k_{1})$-quasi-geodesic ray
$\gamma\in\mathcal{L}_{O}^{\infty}$ starting at $O$, and a sequence $(N_{n})$
in $\mathbb{N}$ such that
$\liminf_{n\to\infty}(v_{N_{n}}\mid[\gamma])=\infty$.

######  Corollary 2.11.

Let $(x_{n})_{n}$ be a sequence in $X$ such that
$\lim_{n}\,\overline{O,x_{n}}\,=\infty$. Then there exists a subsequence
$(x_{N_{n}})_{n}$ such that $\lim_{k,l}(x_{N_{k}}\mid x_{N_{l}})=\infty$.

###### Proof..

By Lemma 2.10, there exists $\gamma\in\mathcal{L}_{O}^{\infty}$ and sequence
$(N_{n})$ in $\mathbb{N}$ such that
$\liminf_{n\to\infty}(x_{N_{n}}\mid[\gamma])=\infty$. So by Lemma 2.8 (iii),
we have.

$\displaystyle(x_{N_{k}}\mid
x_{N_{l}})\geq\Omega^{-1}\min\\{(x_{N_{k}}\mid[\gamma]),([\gamma]\mid
x_{N_{l}})\\}\to\infty.$

∎

By [7, Corollary 4.21], the inclusion
$\mathcal{L}_{O}^{\infty}\hookrightarrow\mathcal{L}^{\infty}$ induces the
bijection $\partial_{O}X\rightarrow\partial X$. Thus we identify
$\partial_{O}X$ with $\partial X$.

###  2.5. Topology on the boundary

We define a topology on $X\cup\partial_{O}X$ as follows. For $n\in\mathbb{N}$,
set

$\displaystyle V_{n}:=\\{(x,y)\in(X\cup\partial_{O}X)^{2}:(x\mid
y)>n\\}\cup\\{(x,y)\in X^{2}:\,\overline{x,y}\,<n^{-1}\\}.$

The family $\\{V_{n}\\}_{n\in\mathbb{N}}$ forms an uniform structure on
$X\cup\partial_{O}X$, which is metrizable. For $x\in X\cup\partial_{O}X$, set
$V_{n}(x):=\\{y\in X\cup\partial_{O}X:(x,y)\in V_{n}\\}$. Then
$\\{V_{n}(x)\\}_{n}$ is a fundamental neighbourhood of $x$ and the inclusion
$X\hookrightarrow X\cup\partial_{O}X$ is a topological embedding. Lemma 2.10
implies the following.

######  Proposition 2.12 ([7, Proposition 4.18]).

Let $X$ be a coarsely convex space. If $X$ is proper, then
$X\cup\partial_{O}X$ is compact.

###  2.6. Metric on the boundary

######  Proposition 2.13.

For sufficiently small $\epsilon>0$, there exists a metric
$d_{\epsilon,\partial X}$ on $\partial X$ and a constant $0<K<1$ such that

$\displaystyle\frac{1}{K}(x\mid y)^{-\epsilon}\leq d_{\epsilon,\partial
X}(x,y)\leq(x\mid y)^{-\epsilon}$

for all $x,y\in\partial_{O}X$.

We refer [7, Section 4.5] for details.

###  2.7. Sequential boundary

Let $(x_{n})_{n}$ be a sequence in $X$. We say that $(x_{n})_{n}$ tends to
infinity if

$\displaystyle\lim_{n,m\to\infty}(x_{n}\mid x_{m})=\infty.$

######  Lemma 2.14.

Suppose that $X$ is proper. Then for a sequence $(x_{n})_{n}$ in $X$ tending
to infinity, there exists $x\in\partial X$ such that $x_{n}\to x$.

###### Proof..

By Lemma 2.10, there exists $\gamma\in\mathcal{L}_{O}$ and
$(N_{n})_{n}\subset\mathbb{N}$ such that
$\lim_{n\to\infty}(x_{N_{n}}\mid[\gamma])=\infty$. So set $x:=[\gamma]$ and we
have $x_{N_{n}}\to x$. Since for all $m,n\in\mathbb{N}$

$\displaystyle(x_{m}\mid x)\geq\Omega^{-1}\min\\{(x_{m}\mid
x_{N_{n}}),\,(x_{N_{n}}\mid x)\\},$

we have $(x_{m}\mid x)\to\infty$. ∎

######  Lemma 2.15.

For $\gamma\in\mathcal{L}_{O}^{\infty}$ and a sequence $(x_{n})_{n}\subset X$
satisfying

$\displaystyle\limsup_{n\rightarrow\infty}\,\overline{x_{n},\mathop{\mathrm{Im}}\nolimits\gamma}\,<\infty,$

we have $\lim(x_{n}\mid[\gamma])=\infty$.

###### Proof..

Set
$S:=\limsup_{n\rightarrow\infty}\,\overline{x_{n},\mathop{\mathrm{Im}}\nolimits\gamma}\,<\infty$.
There exists $N$ such that for all $n>N$,
$\,\overline{x_{n},\mathop{\mathrm{Im}}\nolimits\gamma}\,\leq S$. We choose
$\gamma_{n}\in\mathcal{L}_{O}$ joining $O$ and $x_{n}$. Set
$[0,a_{n}]=\operatorname{Dom}\gamma_{n}$. For each $n>N$, there exists
$t_{n}\in\mathbb{R}_{\geq 0}$ such that
$\,\overline{\gamma_{n}(a_{n}),\gamma(t_{n})}\,<S+1$. Thus By Lemma 2.9, we
have

$\displaystyle(x_{n}\mid[\gamma])\geq(\gamma_{n}\mid\gamma)\geq\frac{a_{n}-\tilde{\theta}(S+1)}{E(S+1+\lambda\tilde{\theta}(S+1)+k_{1})}.$

Since $a_{n}\to\infty$, we have $(x_{n}\mid[\gamma])\to\infty.$ ∎

##  3\. Visual map and radial map

Throughout this section, let $X$ be a proper
$(\lambda,k,E,C,\theta,{{}_{X}\mathcal{L}})$-coarsely convex space, and let
$Y$ be a proper
$(\lambda^{\prime},k^{\prime},E^{\prime},C^{\prime},\theta^{\prime},{{}_{Y}\mathcal{L}})$-coarsely
convex space. Let $a\in X$ and $b\in Y$ be base points of $X$ and $Y$,
respectively. Moreover, let $\Omega$ be a constant satisfying the statements
of Lemma 2.8 for both $X$ and $Y$.

###  3.1. Visual map

######  Definition 3.1.

We say that a large scale Lipschitz map $f\colon X\to Y$ is visual if for
every pair of sequences $(x_{n})_{n},(y_{n})_{n}$ in $X$ with $(x_{n}\mid
y_{n})_{a}\to\infty$, we have $(f(x_{n})\mid f(y_{n}))_{b}\to\infty$.

######  Lemma 3.2.

Let $f\colon X\to Y$ be a large scale Lipschitz map. $f$ is visual if and only
if for all $r>0$ there exists $s>0$ such that for $x,y\in X$ with $(x\mid
y)_{a}>s$, we have $(f(x)\mid f(y))_{b}>r$.

###### Proof..

Suppose that there exists $r>0$ such that for all $s>0$, there exists $x,y\in
X$ satisfying $(x\mid y)_{a}>s$ and $(f(x)\mid f(y))_{b}\leq r$. Then we can
find sequences $(x_{n})_{n\in\mathbb{N}}$ and $(y_{n})_{n\in\mathbb{N}}$ such
that

$\displaystyle(x_{n}\mid y_{n})_{a}>n,\quad(f(x_{n})\mid f(y_{n}))_{b}\leq
r\quad(\forall n\in\mathbb{N}).$

Then $(x_{n}\mid y_{n})_{a}\to\infty$ and $\limsup_{n}(f(x_{n})\mid
f(y_{n}))_{b}\leq r$. So $f$ is not visual. The converse is clear. ∎

######  Proposition 3.3.

Let $f\colon X\to Y$ be a visual map. Then $f$ is metrically proper.

###### Proof..

We prove the contraposition. Suppose $f$ is not metrically proper. Then there
exists a sequence $(x_{n})_{n}$ in $X$ such that
$\,\overline{x_{n},a}\,\to\infty$ and
$\sup_{n}\,\overline{f(x_{n}),b}\,<\infty$. By Corollary 2.11 there exists
$(N_{n})_{n}\subset\mathbb{N}$ such that $(x_{N_{n}}\mid x_{N_{n}})\to\infty$.

For each $n\in\mathbb{N}$, we choose $\gamma_{n}\in{{}_{Y}\mathcal{L}}_{b}$
with $\operatorname{Dom}\gamma_{n}=[0,L_{n}]$ such that
$\gamma_{n}(L_{n})=f(x_{N_{n}})$. By Lemma 2.8 (i) we have

$\displaystyle(f(x_{N_{n}})\mid f(x_{N_{n}}))_{b}$
$\displaystyle\leq\Omega(\gamma_{n}\mid\gamma_{n})_{b}\leq\Omega L_{n}$
$\displaystyle\leq\Omega(\lambda^{\prime}(\,\overline{f(x_{N_{n}}),b}\,+k^{\prime}))\leq\lambda^{\prime}\Omega(\sup_{n}\,\overline{f(x_{n}),b}\,+k^{\prime})<\infty.$

Therefore $f$ is not visual. ∎

######  Definition 3.4.

Let $f\colon X\to Y$ be a visual map. We define a map $\partial
f\colon\partial X\to\partial Y$ as follows. For $x\in\partial X$, we choose a
representative $\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ of $x$, that is,
$\gamma\in x$. Then we define $\partial f(x):=\lim f(\gamma(n))$. The
following proposition says that the limit does exists. We say that $\partial
f$ is induced by $f$.

######  Proposition 3.5.

Let $f\colon X\to Y$ be a visual map. The map $\partial f$ is well-defined,
and for $x\in\partial X$, $\partial f(x)$ does not depend on the choice of the
representative $\gamma\in x$.

###### Proof..

For $x\in\partial X$, we choose a representative
$\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ of $x$, that is, $\gamma\in x$.
For $n\in\mathbb{N}$, set $x_{n}:=\gamma(n)$, $y_{n}:=f(x_{n})$. By Lemma
2.15, we have $(x_{n}\mid x_{m})_{a}\to\infty$. Since $f$ is visual , we have
$(y_{n}\mid y_{m})_{b}\to\infty$. Then by Lemma 2.14, there exists
$y\in\partial Y$ such that $y_{n}\to y$. Therefore $\partial f(x)=y$ is well-
defined.

We will show that $y$ does not depend on the choice of $\gamma$. Let $\eta\in
x$ and set $x^{\prime}_{n}:=\eta(n)$, $y^{\prime}_{n}:=f(x^{\prime}_{n})$,
$y:=\lim_{n\to\infty}y^{\prime}_{n}$. Since

$\displaystyle(x_{n}\mid x^{\prime}_{n})_{a}\geq\Omega^{-1}\min\\{(x_{n}\mid
x)_{a},\,(x\mid x^{\prime}_{n})_{a}\\}\to\infty,$

we have $(y_{n}\mid y^{\prime}_{n})_{b}\to\infty$. So

$\displaystyle(y\mid y^{\prime})_{b}$
$\displaystyle\geq\Omega^{-1}\min\\{(y\mid y_{n})_{b},\,(y_{n}\mid
y^{\prime})_{b}\\}$ $\displaystyle\geq\Omega^{-2}\min\\{(y\mid
y_{n})_{b},\,(y_{n}\mid y^{\prime}_{n})_{b},\,(y^{\prime}_{n}\mid
y^{\prime})_{b}\\}\to\infty.$

Therefore $y=y^{\prime}$. ∎

Lemma 3.2 and following Lemma 3.6 imply that $\partial f$ is continuous on
$\partial X$.

######  Lemma 3.6.

Let $f\colon X\to Y$ be a visual map and $\partial f\colon\partial
X\to\partial Y$ be a map induced by $f$. For $r>0$ there exists $s>0$ such
that for $x,y\in\partial X$ with $(x\mid y)>s$, we have $(\partial
f(x)\mid\partial f(y))>r$.

###### Proof..

We fix $r>0$. By Lemma 3.2, there exists $s^{\prime}>0$ such that for all
$p,q\in X$

$\displaystyle(p\mid q)_{a}>s^{\prime}\Rightarrow(f(p)\mid
f(q))_{b}>\Omega^{2}r.$

Set $s:=\Omega^{2}s^{\prime}$. Let $x,y\in\partial X$ with $(x\mid y)_{a}>s$.
We choose representative $\gamma\in x$ and $\eta\in y$. For $n\in\mathbb{N}$,
set $x_{n}:=\gamma(n)$ and $y_{n}:=\eta(n)$. Since

$\displaystyle(x_{n}\mid y_{n})_{a}\geq\Omega^{-2}\min\\{(x_{n}\mid
x)_{a},(x\mid y)_{a},(y\mid y_{n})_{a}\\}$

and $(x_{n}\mid x)_{a}\to\infty$ , $(y\mid y_{n})_{a}\to\infty$, for
sufficiently large $n$, we have $(x_{n}\mid y_{n})_{a}>s^{\prime}$. Then
$(f(x_{n})\mid f(y_{n}))_{b}>\Omega^{2}r$. Now we have

$\displaystyle(\partial f(x)\mid\partial
f(y))_{b}\geq\Omega^{-2}\min\\{(\partial f(x)\mid
f(x_{n}))_{b},\,(f(x_{n})\mid f(y_{n}))_{b},\,(f(y_{n})\mid\partial
f(y))_{b}\\}.$

Since $(\partial f(x)\mid f(x_{n}))_{b}\to\infty$ and $(f(y_{n})\mid\partial
f(y))_{b}\to\infty$, we have $(\partial f(x)\mid\partial f(y))_{b}>r$. ∎

######  Corollary 3.7.

Let $f\colon X\to Y$ be a visual map. Then the map

$\displaystyle f\cup\partial f\colon X\cup\partial X\to Y\cup\partial Y$

is continuous at every point in $\partial X$.

###### Proof..

We fix $x\in\partial X$. We will show that for $r>0$, there exists $s>0$ such
that, for $y\in X\cup\partial X$,

$\displaystyle(x\mid y)_{a}>\Omega s\Rightarrow(\partial
f(x)\mid(f\cup\partial f)(y))_{b}>\Omega^{-1}r.$

For $r>0$, we choose $s>0$ satisfying the statements of both Lemma 3.2 and
Lemma 3.6 for $f$ and $r$.

First let $y\in\partial X$ with $(x\mid y)_{a}>s$. Then by Lemma 3.6, we have
$(\partial f(x)\mid\partial f(y))_{b}>r$.

Now let $y\in X$ with $(x\mid y)_{a}>\Omega s$. Let
$\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ with $\gamma\in x$. By the proof
of Proposition 3.5, there exists $n\in\mathbb{N}$ such that
$(x\mid\gamma(n))_{a}>\Omega s$ and $(\partial f(x)\mid f(\gamma(n)))_{b}>r$.
Then

$\displaystyle(y\mid\gamma(n))_{a}\geq\Omega^{-1}\min\\{(y\mid
x)_{a},(x\mid\gamma(n))_{a}\\}>s.$

Thus by Lemma 3.2, $(f(\gamma(n))\mid f(y))_{b}>r$. Therefore

$\displaystyle(\partial f(x)\mid f(y))_{b}\geq\Omega^{-1}\min\\{(\partial
f(x)\mid f(\gamma(n))_{b},(f(\gamma(n))\mid f(y))_{b}\\}>\Omega^{-1}r.$

∎

###  3.2. Maps compatible with systems of good quasi-geodesics

Let $\tau\geq 0$ be a constant. We say that a map
$\sigma\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ is a $\tau$-rough
contraction if $\sigma(0)=0$, $\lim_{t\to\infty}\sigma(t)=\infty$, and

$\displaystyle\rho(t)\leq t\quad(\forall t\in\mathbb{R}_{\geq 0}),$
$\displaystyle\left\lvert\rho(t)-\rho(s)\right\rvert\leq\left\lvert
t-s\right\rvert+\tau\quad(\forall t,s\in\mathbb{R}_{\geq 0}),$ $\displaystyle
t\leq s\Rightarrow\rho(t)\leq\rho(s)\quad(\forall t,s\in\mathbb{R}_{\geq
0},).$

We say that $\sigma$ is a rough contraction if it is $\tau$-rough contraction
for some $\tau\geq 0$.

Dydak and Virk [4, Definition 3.2] introduced a class of radial function, in
the setting of Gromov hyperbolic spaces. The following is its analogue in the
setting of coarsely convex spaces. We remark that the following is slightly
weaker than the one defined by Dydak and Virk.

######  Definition 3.8.

Let $\sigma\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a rough
contraction. We say that a map $f\colon X\to Y$ is
$\sigma$-${{}_{X}\mathcal{L}}$-radial if for
$\gamma\in{{}_{X}\mathcal{L}}_{a}$, we have

$\displaystyle\sigma(t)\leq\,\overline{f(\gamma(t)),f(a)}\,\quad(\forall
t\in\operatorname{Dom}\gamma).$

We say that $f$ is ${{}_{X}\mathcal{L}}$-radial if it is
$\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial for some $\sigma$ and $a\in X$.

The above condition is in fact equivalent to the following. Let
$\sigma\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a rough
contraction. We say that a map $f\colon X\to Y$ is weakly
$\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial if for all $x\in X$, there exists
$\gamma_{x}\in{{}_{X}\mathcal{L}}_{a}$ and $T_{x}\in\mathbb{R}_{\geq 0}$ such
that $x=\gamma_{x}(T_{x})$ and

$\displaystyle\sigma(T_{x})\leq\,\overline{f(x),f(a)}\,.$

######  Lemma 3.9.

Let $\sigma\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a rough
contraction. If a large scale Lipschitz map $f\colon X\to Y$ is weakly
$\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial, then it is
$\hat{\sigma}$-${{}_{X}\mathcal{L}}_{a}$-radial for some rough contraction
$\hat{\sigma}$.

###### Proof..

Let $\gamma\in{{}_{X}\mathcal{L}}_{a}$. We fix $t\in\operatorname{Dom}\gamma$
and set $x=\gamma(t)$. Since $f$ is weakly
$\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial, there exists
$\gamma_{x}\in{{}_{X}\mathcal{L}}_{a}$ and $T_{x}\in\mathbb{R}_{\geq 0}$ such
that $x=\gamma_{x}(T_{x})$ and

$\displaystyle\sigma(T_{x})\leq\,\overline{f(\gamma_{x}(t)),f(a)}\,.$

Since $\gamma(t)=x=\gamma_{x}(T_{x})$, we have $\left\lvert
t-T_{x}\right\rvert\leq\theta(0)$. We suppose that $\sigma$ is $\tau$-rough
contraction. Then we have

$\displaystyle\left\lvert\sigma(T_{x})-\sigma(t)\right\rvert\leq\left\lvert
T_{x}-t\right\rvert+\tau\leq\theta(0)+\tau.$

Thus we have

$\displaystyle\,\overline{f(\gamma(t)),f(a)}\,$
$\displaystyle=\,\overline{f(\gamma_{x}(T_{x})),f(a)}\,$
$\displaystyle\geq\sigma(T_{x})$ $\displaystyle\geq\sigma(t)-\theta(0)-\tau.$

Set $\hat{\sigma}(t):=\left(\sigma(t)-\theta(0)-\tau\right)\vee 0$. Then $f$
is $\hat{\sigma}$-${{}_{X}\mathcal{L}}_{a}$-radial. ∎

In the study of maps between Gromov hyperbolic spaces, the Morse lemma plays
essential roles. In the case of coarsely convex spaces, in general, the Morse
lemma does not holds. Instead, we introduce the following.

######  Definition 3.10.

Let $\rho\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a rough
contraction. Let $H>0$. We say that a map $f\colon X\to Y$ is
$\rho$-$H$-${{}_{X}\mathcal{L}}_{a}$-${{}_{Y}\mathcal{L}}_{b}$-equivariant if
$f(a)=b$, and, for $\gamma\in{{}_{X}\mathcal{L}}_{a}$ with
$\operatorname{Dom}\gamma=[0,L_{\gamma}]$ and for
$\eta\in{{}_{Y}\mathcal{L}}_{b}$ with $\operatorname{Dom}\eta=[0,L_{\eta}]$
satisfying $\eta(L_{\eta})=f(\gamma(L_{\gamma}))$, we have

$\displaystyle\,\overline{f\circ\gamma(t),\eta(\rho(t))}\,<H\quad(0\leq\forall
t\leq\min\\{L_{\gamma},\rho^{-1}(L_{\eta})\\}).$

We say that $f$ is ${{}_{X}\mathcal{L}}$-${{}_{Y}\mathcal{L}}$-equivariant if
it is
$\rho$-$H$-${{}_{X}\mathcal{L}}_{a}$-${{}_{Y}\mathcal{L}}_{b}$-equivariant for
some $\rho$, $H$, $a\in X$ and $b\in Y$.

######  Theorem 3.11.

For $\lambda_{1}\geq 1$, $\nu_{1},H,\tau>0$, and $\tau$-rough contractions
$\sigma,\rho\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$, there exists a
metrically proper map $\hat{\rho}\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}$ such that the following holds; let $f\colon X\to Y$ be a
$(\lambda_{1},\nu_{1})$-large scale Lipschitz,
$\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial, and
$\rho$-$H$-${{}_{X}\mathcal{L}}_{a}$-${{}_{Y}\mathcal{L}}_{b}$-equivariant map
satisfying $f(a)=b$. Then for $x,y\in X$, we have

$\displaystyle(f(x)\mid f(y))_{b}\geq\hat{\rho}((x\mid y)_{a}).$

Especially, $f$ is visual.

###### Proof..

Let $\gamma,\eta\in{{}_{X}\mathcal{L}}_{a}$, be a good quasi-geodesics joining
the base point $a$ and $x,y$, respectively. Set
$[0,L_{\gamma}]=\operatorname{Dom}\gamma$,
$[0,L_{\eta}]=\operatorname{Dom}\eta$. We remark that
$(\gamma\mid\eta)_{a}\geq\Omega^{-1}(x\mid y)_{a}$.

Let $\gamma^{\prime},\eta^{\prime}\in{{}_{Y}\mathcal{L}}_{b}$ be a good quasi-
geodesics joining the base point $b$ and $f(x),f(y)$, respectively. Set
$[0,L_{\gamma^{\prime}}]=\operatorname{Dom}\gamma^{\prime}$,
$[0,L_{\eta^{\prime}}]=\operatorname{Dom}\eta^{\prime}$. We have

$\displaystyle\min\\{L_{\gamma},L_{\eta}\\}\geq(\gamma\mid\eta)_{a}\geq\Omega^{-1}(x\mid
y)_{a}.$

Since $\gamma^{\prime}$ is $(\lambda^{\prime},k^{\prime})$-quasi geodesic and
$f$ is $\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial, it follows that

$\displaystyle\lambda^{\prime}L_{\gamma^{\prime}}+k^{\prime}$
$\displaystyle\geq\,\overline{b,f(x)}\,\geq\sigma(L_{\gamma})\geq\sigma(\Omega^{-1}(x\mid
y)).$

By switching $x$ and $y$, and applying the same argument, we obtain

$\displaystyle\lambda^{\prime}L_{\eta^{\prime}}+k^{\prime}\geq\sigma(\Omega^{-1}(x\mid
y)_{a}).$

So we have

(1)
$\displaystyle\min\\{L_{\gamma^{\prime}},L_{\eta^{\prime}}\\}\geq\frac{1}{\lambda^{\prime}}\sigma(\Omega^{-1}(x\mid
y)_{a})-\frac{k^{\prime}}{\lambda^{\prime}}.$

Here we introduce the following constants,

$\displaystyle T:=\lambda_{1}\Omega+\nu_{1}+2H,\quad c:=(E^{\prime}T)^{-1}$

and we define a function $\delta\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}$ by

$\displaystyle\delta(t):=\sup\left\\{u\in\mathbb{R}_{\geq
0}:\rho(u)\leq\left(\frac{1}{\lambda^{\prime}}\min\\{\sigma(\Omega^{-1}t),\,\rho(\Omega^{-1}t)\\}-\frac{k^{\prime}}{\lambda^{\prime}}-\tau\right)\vee
0\right\\}.$

We remark that, since $\rho$ is a $\tau$-rough contraction, for any
$\alpha\in\mathbb{R}_{\geq 0}$, we have

$\displaystyle\rho(\sup\\{u\in\mathbb{R}_{\geq
0}:\rho(u)\leq\alpha\\})\leq\alpha+\tau.$

We will show that

(2) $\displaystyle(f(x)\mid f(y))\geq c\left\\{\rho\circ\delta((x\mid
y)_{a})-\tau\right\\}.$

We set

$\displaystyle S$
$\displaystyle:=\frac{1}{\lambda^{\prime}}\min\\{\sigma(\Omega^{-1}(x\mid
y)_{a}),\,\rho(\Omega^{-1}(x\mid
y)_{a})\\}-\frac{k^{\prime}}{\lambda^{\prime}},$ $\displaystyle s$
$\displaystyle:=\delta((x\mid y)_{a})=\sup\\{u\in\mathbb{R}_{\geq
0}:\rho(u)\leq(S-\tau)\vee 0\\}.$

Then we have $\rho(s)\leq S\vee\tau$ and $\rho\circ\delta((x\mid
y)_{a})=\rho(s)$.

First we assume that $S<\tau$. Then $\rho\circ\delta((x\mid y)_{a})\leq\tau$,
so clearly (2) holds, since $(f(x)\mid f(y))\geq 0$.

Now we assume that $S\geq\tau$. Since $\rho(s)\leq S$ and
$\lambda^{\prime}\geq 1$, we have

$\displaystyle\rho(s)\leq\rho(\Omega^{-1}(x\mid
y)_{a})\leq\rho((\gamma\mid\eta)_{a}).$

Thus

$\displaystyle s\leq(\gamma\mid\eta)_{a}\leq\min\\{L_{\gamma},L_{\eta}\\}.$

By (1) and $\rho(s)\leq S$, we have

$\displaystyle
0\leq\rho(s)\leq\min\\{L_{\gamma^{\prime}},L_{\eta^{\prime}}\\}.$

Since $f$ is
$\rho$-$H$-${{}_{X}\mathcal{L}}_{a}$-${{}_{Y}\mathcal{L}}_{b}$-equivariant, we
have

$\displaystyle\max\left\\{\,\overline{\gamma^{\prime}(\rho(s)),f\circ\gamma(s)}\,,\,\overline{f\circ\eta(s),\eta^{\prime}(\rho(s))}\,\right\\}<H.$

By Lemma 2.8 (iv) we have

$\displaystyle\,\overline{\gamma(s),\eta(s)}\,\leq\Omega.$

Thus,

$\displaystyle\,\overline{\gamma^{\prime}(\rho(s)),\eta^{\prime}(\rho(s))}\,$
$\displaystyle\leq\,\overline{\gamma^{\prime}(\rho(s)),f\circ\gamma(s)}\,+\,\overline{f\circ\gamma(s),f\circ\eta(s)}\,+\,\overline{f\circ\eta(s),\eta^{\prime}(\rho(s))}\,$
$\displaystyle\leq H+\lambda_{1}\,\overline{\gamma(s),\eta(s)}\,+\nu_{1}+H$
$\displaystyle\leq\lambda_{1}\Omega+\nu_{1}+2H=T.$

Since $c=1/(E^{\prime}T)$, we have

$\displaystyle\,\overline{\gamma^{\prime}(c\rho(s)),\eta^{\prime}(c\rho(s))}\,\leq
1+C^{\prime}\leq 2D^{\prime}+2.$

Thus we have

$\displaystyle(\gamma^{\prime}\mid\eta^{\prime})_{b}\geq
c\rho(s)=c\rho\circ\delta((x\mid y)_{a}).$

Therefore (2) holds. ∎

######  Lemma 3.12.

Let $f\colon X\to Y$ be a map. Suppose that there exists a non-decreasing map
$\hat{\rho}\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ such that for all
$x,y\in X$, $(f(x)\mid f(y))_{b}\geq\hat{\rho}((x\mid y)_{a})$. Then for all
$u,v\in\partial X$, we have

$\displaystyle(\partial f(u)\mid\partial
f(v))_{b}\geq\Omega^{-2}\hat{\rho}(\Omega^{-2}(u\mid v)_{a}).$

###### Proof..

Let $u,v\in\partial_{a}X$. We choose representatives $\gamma\in u$ and
$\eta\in v$. For $n\in\mathbb{N}$, set $x_{n}:=\gamma(n)$ and
$y_{n}:=\eta(n)$. We have

$\displaystyle(x_{n}\mid y_{n})_{a}\geq\Omega^{-2}\min\\{(x_{n}\mid
u)_{a},(u\mid v)_{a},(v\mid y_{n})_{a}\\}$

Since $(x_{n}\mid u)_{a}\to\infty$ and $(v\mid y_{n})_{a}\to\infty$, we have
$(x_{n}\mid y_{n})_{a}\geq\Omega^{-2}(u\mid v)_{a}$ for sufficiently large
$n$. Then $(f(x_{n})\mid f(y_{n}))_{b}\geq\hat{\rho}(\Omega^{-2}(u\mid
v)_{a})$. Now we have

$\displaystyle(\partial f(u)\mid\partial
f(v))_{b}\geq\Omega^{-2}\min\\{(\partial f(u)\mid
f(x_{n}))_{b},\,(f(x_{n})\mid f(y_{n}))_{b},\,(f(y_{n})\mid\partial
f(v))_{b}\\}.$

Since $(\partial f(u)\mid f(x_{n}))_{b}\to\infty$ and $(f(y_{n})\mid\partial
f(v))_{b}\to\infty$, we have

$\displaystyle(\partial f(u)\mid\partial
f(v))_{b}>\Omega^{-2}\hat{\rho}(\Omega^{-2}(u\mid v)_{a}).$

∎

We say that a map $\sigma\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ is a
rough-isometric contraction if it is a rough contraction and there exist
constants $A,B\geq 0$ such that
$\left\lvert\sigma(t)-\sigma(s)\right\rvert\geq A\left\lvert
t-s\right\rvert-B$ for all $s,t\in\mathbb{R}_{\geq 0}$.

######  Corollary 3.13.

Let $\sigma,\rho\colon\mathbb{R}\to\mathbb{R}$ be rough-isometric
contractions. Let $f\colon X\to Y$ be a large scale Lipschitz,
$\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial, and
$\rho$-$H$-${{}_{X}\mathcal{L}}_{a}$-${{}_{Y}\mathcal{L}}_{b}$ map satisfying
$f(a)=b$. Then the induced map $\partial f\colon\partial X\to\partial Y$ is a
Lipschitz map.

###### Proof..

Let $\epsilon>0$ be a sufficiently small constant and, $d_{\epsilon,\partial
X}$ and $d_{\epsilon,\partial Y}$ be a metric on $\partial X$ and on $\partial
Y$, respectively, constructed in Proposition 2.13. Let $K$ be a constant
satisfying the statement of Proposition 2.13 for both $\partial X$ and
$\partial Y$.

Since $\sigma$ and $\rho$ are rough-isometric contractions, by the proof of
Theorem 3.11 and Lemma 3.12, there exists constants $\kappa,\nu$ such that for
all $x,y\in\partial X$, we have

$\displaystyle(\partial f(x)\mid\partial f(y))_{b}\geq\kappa(x\mid
y)_{a}-\nu.$

We use the following constant

$\displaystyle\displaystyle
V:=\frac{1}{K}\left(\frac{\kappa}{2\nu}\right)^{\epsilon}.$

For $x,y\in\partial X$ with $d_{\epsilon,\partial X}(x,y)\geq V$, we have

$\displaystyle d_{\epsilon,\partial Y}(\partial f(x),\partial
f(y))\leq\mathop{\mathrm{diam}}\nolimits\partial
Y\leq\frac{\mathop{\mathrm{diam}}\nolimits\partial Y}{V}d_{\epsilon,\partial
X}(x,y).$

For $x,y\in\partial X$ with $d_{\epsilon,\partial X}(x,y)\leq V$, we have

$\displaystyle(x\mid y)_{a}\geq\left(\frac{1}{Kd_{\epsilon,\partial
X}(x,y)}\right)^{1/\epsilon}\geq\left(\frac{1}{KV}\right)^{1/\epsilon}\geq\frac{2\nu}{\kappa}.$

Thus $\nu\leq(1/2)\kappa(x\mid y)_{b}$. By Theorem 3.11,

$\displaystyle d_{\epsilon,\partial Y}(\partial f(x),\partial f(x^{\prime}))$
$\displaystyle\leq\left(\frac{1}{(f(x)\mid
f(y))_{b}}\right)^{\epsilon}\leq\left(\frac{1}{\kappa(x\mid
y)_{a}-\nu}\right)^{\epsilon}$ $\displaystyle\leq\left(\frac{2}{\kappa(x\mid
y)_{a}}\right)^{\epsilon}\leq
K\left(\frac{2}{\kappa}\right)^{\epsilon}d_{\epsilon,\partial
X}(x,x^{\prime}).$

∎

######  Definition 3.14.

Let $\rho\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a rough
contraction. Let $f\colon X\to Y$ be a map. Let $T>0$. We say that $f$ is
$\rho$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant
if $f(a)=b$, and for $\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ there exists
$\eta\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that

$\displaystyle\,\overline{f\circ\gamma(t),\eta(\rho(t))}\,<T\quad(\forall
t\in\mathbb{R}_{\geq 0}).$

######  Lemma 3.15.

Let $\rho\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a rough
contraction. Let $f\colon X\to Y$ be a visual map. Let
$\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ and
$\eta\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ be quasi-geodesic rays satisfying
$\,\overline{f\circ\gamma(t),\eta(\rho(t))}\,<T$ for all $t\in\mathbb{R}_{\geq
0}$. Then we have $\partial f([\gamma])=[\eta]$.

###### Proof..

For $n\in\mathbb{N}$, set $x_{n}:=\gamma(n)$. By Lemma 2.15,
$x_{n}\to[\gamma]$. By Corollary 3.7, $f(x_{n})\to\partial f([\gamma])$, that
is, $(f(x_{n})\mid\partial f([\gamma]))_{b}\to\infty$. By the assumption of
$\gamma$ and $\eta$, for all $n\in\mathbb{N}$,
$\,\overline{f(x_{n}),\eta(\rho(n))}\,<T$. Thus by Lemma 2.15,
$(f(x_{n})\mid[\eta])_{b}\to\infty$. Then

$\displaystyle(\partial
f([\gamma])\mid[\eta])_{b}\geq\Omega^{-1}\min\\{(\partial f([\gamma])\mid
f(x_{n}))_{b},(f(x_{n})\mid[\eta])_{b}\\}\to\infty.$

Therefore $\partial f([\gamma])=[\eta]$. ∎

######  Lemma 3.16.

For $H>0$ there exists $T>0$ such that if a large scale Lipschitz map $f\colon
X\to Y$ is ${{}_{X}\mathcal{L}}$-radial and
$\rho$-$H$-${{}_{X}\mathcal{L}}_{a}$-${{}_{Y}\mathcal{L}}_{b}$-equivariant,
then it is
$\rho$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant.

###### Proof..

Let $\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$. There exists
${{}_{X}\mathcal{L}}$-approximate sequence
$\\{(\gamma_{n},\alpha^{\prime}_{n})\\}$ of $\gamma$. Set
$\alpha_{n}:=(\gamma\mid\gamma_{n})_{a}$. By Lemma 2.8 (iv) for all
$t\in[0,\alpha_{n}]$, we have

$\displaystyle\,\overline{\gamma(t),\gamma_{n}(t)}\,\leq\Omega.$

We suppose that $f$ is a $(\lambda_{1},\nu_{1})$-large scale Lipschitz map.
Then

(3)
$\displaystyle\,\overline{f(\gamma(t)),f(\gamma_{n}(t))}\,\leq\lambda_{1}\Omega+\nu_{1}.$

Set $x_{n}:=\gamma_{n}(\alpha_{n}^{\prime})$ and $y_{n}:=f(x_{n})$. Let
$\eta_{n}\in{{}_{Y}\mathcal{L}}_{b}$ be a good quasi-geodesic from $b$ to
$y_{n}$. Since $f$ is ${{}_{X}\mathcal{L}}$-radial,
$\,\overline{b,y_{n}}\,\to\infty$. Then there exists
$\eta\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ and $(N_{n})_{N}\subset\mathbb{N}$
such that ${\eta_{N_{n}}}$ is a ${{}_{Y}\mathcal{L}}$-approximate sequence of
$\eta$. Set $\beta_{n}:=(\eta_{N_{n}}\mid\eta)_{b}$.

Now we fix $t\in\mathbb{R}_{\geq 0}$. By Lemma 2.7, there exists $n$ such that
$\rho(t)\leq t\leq\min\\{\alpha_{N_{n}},\beta_{n}\\}$. By Lemma 2.8 (iv),

(4)
$\displaystyle\,\overline{\eta_{N_{n}}(\rho(t)),\eta(\rho(t))}\,\leq\Omega.$

By (3), (4) and the assumption that $f$ is a
$\rho$-$H$-${{}_{X}\mathcal{L}}_{a}$-${{}_{Y}\mathcal{L}}_{b}$-equivariant
map,

$\displaystyle\,\overline{f(\gamma(t)),\eta(\rho(t))}\,$
$\displaystyle\leq\,\overline{f(\gamma(t)),f(\gamma_{N_{n}}(t))}\,+\,\overline{f(\gamma_{N_{n}}(t)),\eta_{N_{n}}(\rho(t))}\,+\,\overline{\eta_{N_{n}}(\rho(t)),\eta(\rho(t))}\,$
$\displaystyle\leq\lambda_{1}\Omega+\nu_{1}+H+\Omega.$

We denote by $T$, the right most side of the above inequality. Then we have
that $f$ is
$\rho$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant.
∎

###  3.3. Proof of Theorem 1.1

Let $X$ be a proper coarsely convex space. Set $\bar{X}:=X\cup\partial X$. By
Theorem 3.7, $f$ extends to the continuous map $f\cup\partial
f\colon\bar{X}\to\bar{X}$. By [7, Proposition 6.6], $\bar{X}$ is a coarse
compactification. So Theorem 1.1 follows from [6, Corollary 1.7].

##  4\. Coarsely surjective

By Proposition 3.3, visual maps are coarse maps, so they induce continuous
maps on Higson coronae. Austin and Virk [1] showed that coarsely surjective
maps induce surjections on Higson coronae. In [7, Section 6.2], it is shown
that for a proper coarsely convex space $X$, $X\cup\partial X$ is a coarse
compactification. Then by the universality of the Higson compactification [13,
Proposition 2.39], we have the following.

######  Proposition 4.1.

Let $X$ and $Y$ be proper coarsely convex spaces. Let $f\colon X\to Y$ be a
visual map. If $f$ is coarsely surjective, then the induced map $\partial
f\colon\partial X\to\partial Y$ is surjective.

For reader’s convenience, we give a proof of Proposition 4.1 without using
Higson coronae.

###### Proof..

We use same setting for $X$ and $Y$ as those stated in the beginning of
Section 3. Since $f$ is coarsely surjective, there exists $S>0$ such that
$S$-neighbourhood of $f(X)$ is equal to $Y$. Then for
$\eta\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ and $n\in\mathbb{N}$, there exists
$x_{n}\in X$ such that $\,\overline{\eta(n),f(x_{n})}\,\leq S$. We suppose
that $f$ is a $(\lambda_{1},\nu_{1})$-large scale Lipschitz map. We have

$\displaystyle\,\overline{x_{n},a}\,$
$\displaystyle\geq\frac{1}{\lambda_{1}}(\,\overline{f(x_{n}),b}\,-\nu_{1})\geq\frac{1}{\lambda_{1}}(\,\overline{\eta(n),b}\,-\,\overline{\eta(n),f(x_{n})}\,-\nu_{1})$
$\displaystyle\geq\frac{1}{\lambda_{1}}(\,\overline{\eta(n),b}\,-S-\nu_{1})\to\infty,$

Thus there exists $\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ and
$(N_{n})_{n}\subset\mathbb{N}$ such that

$\displaystyle\lim_{n\to\infty}(x_{N_{n}}\mid[\gamma])_{a}=\infty.$

Then we have

(5) $\displaystyle(\gamma(n)\mid
x_{N_{n}})_{a}\geq\Omega^{-1}\min\\{(\gamma(n)\mid[\gamma])_{a},([\gamma],x_{N_{n}})_{a}\\}\to\infty.$

We will show that $\partial f([\gamma])=[\eta]$. Set $y:=\partial f([\gamma])$
and $y_{n}=f(\gamma(n))$. Since $f\cup\partial f$ is continuous at
$[\gamma]\in\partial_{a}X$, we have $(y\mid y_{n})_{b}\to\infty$. Now we have

$\displaystyle(y\mid[\eta])_{b}\geq\Omega^{-2}\min\\{(y\mid
y_{n})_{b},(y_{n}\mid f(x_{N_{n}}))_{b},(f(x_{N_{n}})\mid[\eta])_{b}\\}$

By Lemma 2.15, we have $(f(x_{N_{n}})\mid[\eta])_{b}\to\infty$. Since $f$ is
visual, by (5), we have

$\displaystyle(y_{n}\mid f(x_{N_{n}}))_{b}=(f(\gamma(n))\mid
f(x_{N_{n}}))_{b}\to\infty.$

Thus $(y\mid[\eta])_{b}=\infty$, therefore $\partial f([\gamma])=y=[\eta]$. ∎

##  5\. Coarsely n-to-one map

We say that a map is $n$-to-1 if its point-inverses contain at most $n$
points.

The following proposition is a generalization of [4, Theorem 7.1] in the case
of Gromov hyperbolic spaces, to that of coarsely convex spaces.

######  Proposition 5.1.

Let $X$ and $Y$ be proper coarsely convex spaces, with base points $a\in X$
and $b\in Y$, respectively. Let $f\colon X\to Y$ be a visual map, and
$\partial f\colon\partial X\to\partial Y$ be the map induced by $f$. The
following is equivalent.

  1. $($A$)$

$\partial f$ is $n$-to-one.

  2. $($B$)$

For all $R>0$, there exists $S>0$ such that, for all $x_{1},\dots,x_{n+1}\in
X$ with $(x_{i}\mid x_{j})<R\,(\forall i,j,\,i\neq j)$, there exists $1\leq
k<m\leq n+1$ satisfying $(f(x_{k})\mid f(x_{m}))_{b}<S$.

We say that the map $f$ is visually $n$-to-one if $f$ satisfies the statement
(B) of Proposition 5.1.

###### Proof..

First, we suppose that (B) is negative, and we will shows that (A) is
negative.

Since (B) is negative, there exists $R>0$ such that for all $l\in\mathbb{N}$,
there exists $x_{1}^{l},\dots,x_{n+1}^{l}\in X$ with $(x_{i}^{l}\mid
x_{j}^{l})_{a}<R$ and $(f(x_{i}^{l})\mid f(x_{j}^{l}))_{b}>l$ for all
$i,j\in\\{1,\dots,n+1\\}$ with $i\neq j$. Since $X$ is proper and by
Proposition 2.11, by replacing subsequence, we can assume that for each $i$,
the sequence $(x_{i}^{l})_{l}$ converges to some $x_{i}\in\partial X$ when
$l\to\infty$. Since $(x_{i}^{l}\mid x_{j}^{l})_{a}<R$, we have $x_{i}\neq
x_{j}$. Since $(f(x_{i}^{l})\mid f(x_{j}^{l}))_{b}\geq l\to\infty$, we have
$\partial f(x_{i})=\partial f(x_{j})$. Therefore $\partial f$ is not $n$-to-
one.

Next, we suppose that (A) is negative, and we will show that (B) is negative.
There exist $n+1$ points $a_{1},\dots,a_{n+1}\in\partial X$, which are
mutually different, such that $\partial f(a_{i})=\partial f(a_{j})$ for all
$i,j$. Set $z:=\partial f(a_{1})$. For $t>0$, we define

$\displaystyle U(a_{i},t):=\\{x\in X\cup\partial X:(x\mid a_{i})_{a}\geq
t\\}.$

Let $\Omega$ be a constant satisfying the statement of Lemma 2.8 for both $X$
and $Y$. We take $R>0$ satisfying $U(a_{i},\Omega^{-1}R)\cap
U(a_{j},\Omega^{-1}R)=\emptyset$ for all $i\neq j$. We will see that for all
$S>0$, we have $U(a_{i},R)\cap f^{-1}(U(z,\Omega S))\neq\emptyset$.

Let $\gamma_{i}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ with $\gamma_{i}\in
a_{i}$. By Corollary 3.7, we have $z=\partial
f(a_{i})=\lim_{n\to\infty}f(\gamma_{i}(n))$. Since
$(a_{i}\mid\gamma_{i}(n))_{a}\to\infty$, for sufficiently large $n$, we can
assume that $\gamma_{i}(n)\in U(a_{i},R)$. Similarly, by $(z\mid
f(\gamma_{i}(n)))_{b}\to\infty$, for sufficiently large $n$, we can assume
that $f(\gamma_{i}(n))\in U(z,\Omega S)$. Then $\gamma_{i}(n)\in
U(a_{i},R)\cap f^{-1}(U(z,\Omega S))\neq\emptyset$.

For each $i$, we choose $x_{i}\in U(a_{i},R)\cap f^{-1}(U(z,\Omega S))$. Then
we have $(x_{i}\mid x_{j})<R$ for all $i\neq j$. Indeed, if we suppose
$(x_{i}\mid x_{j})\geq R$, then

$\displaystyle(a_{i}\mid x_{j})_{a}\geq\Omega^{-1}\min\\{(a_{i}\mid
x_{i})_{a},(x_{i}\mid x_{j})_{a}\\}\geq\Omega^{-1}R,$

so $x_{j}\in U(a_{i},\Omega^{-1}R)$, and this contradicts the assumption of
$R$.

On the other hand, since $f(x_{i})\in U(z,\Omega S)$, we have

$\displaystyle(f(x_{i})\mid f(x_{j}))_{b}\geq\Omega^{-1}\min\\{(f(x_{i})\mid
z)_{b},(z\mid f(x_{j}))_{b}\\}\geq S.$

Therefore (B) is negative. ∎

Miyata and Virk [10] introduced the following notion, which they called
$(\mathrm{B})_{n}$. Later, Dydak and Virk [4] reintroduced and they named it
coarsely $n$-to-one.

######  Definition 5.2.

Let $X$ and $Y$ be metric spaces. We say that a coarse map $f\colon X\to Y$ is
coarsely $n$-to-one if for all $R>0$, there exists $S>0$ such that for all
subset $A\subset Y$ with $\mathop{\mathrm{diam}}\nolimits(A)<R$, there exists
$B_{1},\dots,B_{n}\subset X$ satisfying
$\mathop{\mathrm{diam}}\nolimits(B_{i})<S$ for all $i$ and
$f^{-1}(A)=B_{1}\cup\dots\cup B_{n}$.

We use the following characterisation of coarsely $n$-to-one maps due to Dydak
and Virk.

######  Proposition 5.3 ([4, Proposition 7.3.]).

Let $X$ and $Y$ be metric spaces and let $f\colon X\to Y$ be a bornologous
map. Then the following is equivalent.

  1. $($A$)$

$f$ is coarsely $n$-to-one.

  2. $($B$)$

For all $R>0$, there exists $S>0$ such that, for all $n+1$ points
$x_{1},\dots,x_{n+1}\in X$ with $\,\overline{x_{i},x_{j}}\,>S$ for all $i\neq
j$, there exists $1\leq k<m\leq n+1$ such that
$\,\overline{f(x_{k}),f(x_{m})}\,>R$.

######  Theorem 5.4.

Let $X$ and $Y$ be proper coarsely convex spaces and let $f\colon X\to Y$ be a
visual,
$\rho$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant
map. If $f$ is coarsely $n$-to-one, then the induced map $\partial
f\colon\partial X\to\partial Y$ is $n$-to-one.

###### Proof..

We use same setting for $X$ and $Y$ as those stated in the beginning of
Section 3.

We suppose that $\partial f$ is not $n$-to-one. Thus there exist mutually
different $n+1$ elements $a_{1},\dots,a_{n+1}\in\partial X$ such that
$\partial f(a_{1})=\partial f(a_{2})=\dots=\partial f(a_{n+1})$.

We choose $\gamma_{i}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ satisfying
$\gamma_{i}\in a_{i}$ for each $i=1,\dots,n+1$. Since $f$ is
$\rho$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant,
for each $\gamma_{i}$, there exists
$\eta_{i}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that for all
$t\in\mathbb{R}_{\geq 0}$,

$\displaystyle\,\overline{f\circ\gamma_{i}(t),\eta_{i}(\rho(t))}\,<T.$

Now set $R:=D^{\prime}+2T$ and let $S>0$ be a constant satisfying (B) in
Proposition 5.3 for the coarsely $n$-to-one map $f$. Since
$a_{1},\dots,a_{n+1}$ are mutually different, we can take
$s\in\mathbb{R}_{\geq 0}$ such that
$\,\overline{\gamma_{i}(s),\gamma_{j}(s)}\,>S$ for all $i\neq j$. Then by
Proposition 5.3, there exists a pair of indices $k,m$ such that

$\displaystyle\,\overline{f\circ\gamma_{k}(s),f\circ\gamma_{m}(s)}\,>R=D^{\prime}+2T.$

Therefore

$\displaystyle\,\overline{\eta_{k}(\rho(s)),\eta_{m}(\rho(s))}\,$
$\displaystyle\geq\,\overline{f\circ\gamma_{k}(s),f\circ\gamma_{m}(s)}\,-\,\overline{f\circ\gamma_{k}(s),\eta_{k}(\rho(s))}\,-\,\overline{\eta_{m}(\rho(s)),f\circ\gamma_{m}(s)}\,$
$\displaystyle>D^{\prime}+2T-2T=D^{\prime}.$

So by Lemma 2.5, we have $[\eta_{k}]\neq[\eta_{m}]$. However, by Lemma 3.15,
we have $[\eta_{k}]=\partial f(a_{k})=\partial f(a_{m})=[\eta_{m}]$. This is a
contradiction. ∎

######  Corollary 5.5.

Let $X$ and $Y$ be proper coarsely convex spaces. Suppose that there exists a
visual
$\sigma$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant
map $f\colon X\to Y$ which is coarsely surjective and coarsely $(n+1)$-to-one.
Then

$\displaystyle\mathop{\mathrm{dim}}\nolimits\partial
Y\leq\mathop{\mathrm{dim}}\nolimits\partial X+n.$

###### Proof..

By Proposition 4.1 and Theorem 5.4, the induced map $\partial f\colon\partial
X\to\partial Y$ is surjective and $(n+1)$-to-one. Since both $\partial X$ and
$\partial Y$ are compact Hausdorff spaces, $\partial f$ is a closed map. Then
we obtain the desired inequality by dimension raising theorem [5, Theorem
4.3.3]. ∎

###### Proof of Theorem 1.2.

By Theorem 3.11, $f$ is visual. By Lemma 3.16, $f$ is
$\sigma$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant.
Then Theorem 1.2 follows from Corollary 5.5. ∎

######  Corollary 5.6.

Let $X$ and $Y$ be proper coarsely convex spaces, and let $f\colon X\to Y$ be
a visual
$\sigma$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant
coarse embedding map. Then $\partial f$ is a topological embedding. Moreover,
if $f$ is coarsely surjective, then $\partial f$ is a homeomorphism.

###### Proof..

Since $f$ is coarsely one-to-one, by Theorem 5.4, $\partial f$ is one-to-one.
Since $\partial X$ and $\partial Y$ are compact Hausdorff spaces, $\partial f$
is a topological embedding. If $f$ is coarsely surjective, by Proposition 4.1,
$\partial f$ is surjective, thus $\partial f$ is a homeomorphism. ∎

##  6\. Example

Dydak and Virk constructed a example of coarsely 2-to-1 map from the Cayley
graph of the free group of rank 2 with respect to standard generating set to
the hyperbolic plane. In this section, by taking direct products of their
examples and some coarsely convex spaces, we construct examples of coarsely
2-to-1 map between coarsely convex spaces satisfying the assumptions in
Theorem 5.4. First we prepare some lemmas.

###  6.1. Lemmas for direct products of coarsely convex spaces

######  Lemma 6.1.

For $i=1,2$, let $X_{i}$ and $Y_{i}$ be metric spaces and $f_{i}\colon
X_{i}\to Y_{i}$ be maps. The following holds.

  1. $($i$)$

If $f_{1}$ and $f_{2}$ are coarsely surjective, so is $f_{1}\times f_{2}$.

  2. $($ii$)$

If $f_{1}$ and $f_{2}$ are large scale Lipschitz maps, so is $f_{1}\times
f_{2}$.

  3. $($iii$)$

If $f_{1}$ is coarsely $n$-to-one and $f_{2}$ is coarsely $m$-to-one, then
$f_{1}\times f_{2}$ is coarsely $nm$-to-one.

Moreover, if $X_{i}$ and $Y_{i}$ are coarsely convex and
${{}_{X_{i}}\mathcal{L}}$ are systems of good quasi-geodesics of $X_{i}$, the
following holds.

  1. $($iv$)$

If $f_{1}$ is $\sigma_{1}$-${{}_{X_{i}}\mathcal{L}}_{a_{1}}$-radial and
$f_{2}$ is $\sigma_{2}$-${{}_{X_{2}}\mathcal{L}}_{a_{2}}$-radial, where
$\sigma_{1},\sigma_{2}$ are affine maps, then $f_{1}\times f_{2}$ is
$\min\\{\sigma_{1},\sigma_{2}\\}$-${{}_{X_{1}\times
X_{2}}\mathcal{L}}_{(a_{1},a_{2})}$-radial.

###### Proof..

(i) and (ii) are clear and we leave the proof to the reader. (iii) is due to
Miyata and Virk [10, Proposition 3.4]. We suppose that $X_{i}$ and $Y_{i}$ are
coarsely convex. Let ${{}_{X_{i}}\mathcal{L}}$ be the systems of good quasi-
geodesics of $X_{i}$. Then by [7, Proposition 3.3], all element in the system
of good quasi-geodesics ${{}_{X_{1}\times X_{2}}\mathcal{L}}$ of $X_{1}\times
X_{2}$ is of the following form:

$\displaystyle\alpha_{1}\gamma_{1}\oplus\alpha_{2}\gamma_{2}:[0,a_{1}+a_{2}]\ni
t\mapsto\left(\gamma_{1}\left(\alpha_{1}t\right),\gamma_{2}\left(\alpha_{2}t\right)\right)\in
X_{1}\times X_{2}$

where $\gamma_{i}\in{{}_{X_{i}}\mathcal{L}}$,
$\operatorname{Dom}\gamma_{i}=[0,a_{i}]$ and $\alpha_{i}=a_{i}/(a_{1}+a_{2})$.
Now it is easy to see that (iv) holds. ∎

For the topological structure of the ideal boundaries of products of coarsely
convex spaces, the following is known.

######  Proposition 6.2 ([7, Proposition 4.26]).

Let $X$ and $Y$ be proper coarsely convex spaces. Then the ideal boundary
$\partial(X\times Y)$ of the product $X\times Y$ is homeomorphic to the join
$\partial X\star\partial Y$.

Since the ideal boundary of the hyperbolic plane $\mathbb{H}$ is the circle
$S^{1}$ and that of the real line $\mathbb{R}$ is the two point space $S^{0}$,
the ideal boundary of the product $\mathbb{H}\times\mathbb{R}$ is the
suspension $\Sigma S^{1}$ which is homeomorphic to the 2-dimensional sphere
$S^{2}$. Similarly, let $\mathrm{Cay}(F_{2})$ be the Cayley graph of the free
group of rank 2 with the standard generating set. Then the ideal boundary of
$\mathrm{Cay}(F_{2})\times\mathbb{R}$ is the suspension of the Cantor set.

###  6.2. Lemma for product of geodesic coarsely convex spaces

For technical reason, in the rest of this section we only consider geodesic
coarsely convex spaces. For $i=1,2$, Let $X_{i}$ be geodesic
$(C_{i},{{}_{X_{i}}\mathcal{L}})$-coarsely convex spaces and $Y_{i}$ be
geodesic $(C^{\prime}_{i},{{}_{Y_{i}}\mathcal{L}})$-coarsely convex spaces.
Let $a_{i}\in X_{i}$ and $b_{i}\in Y_{i}$ be base points.

######  Lemma 6.3.

Let $f_{i}\colon X_{i}\to Y_{i}$ be maps preserving base points and distances
from base points, that is, $f(a_{i})=b_{i}$ and
$\,\overline{b_{i},f(x)}\,=\,\overline{a_{i},x}\,$ for all $x\in X_{i}$. If
$f_{i}$ is $\mathop{\mathrm{id}}_{\mathbb{R}_{\geq
0}}$-$H$-${{}_{X_{i}}\mathcal{L}}_{a_{i}}$-${{}_{Y_{i}}\mathcal{L}}_{b_{i}}$-equivariant,
then $f_{1}\times f_{2}$ is $\mathop{\mathrm{id}}_{\mathbb{R}_{\geq
0}}$-$2H$-${{}_{X_{1}\times
X_{2}}\mathcal{L}}_{(a_{1},a_{2})}$-${{}_{Y_{1}\times
Y_{2}}\mathcal{L}}_{(b_{1},b_{2})}$-equivariant.

###### Proof..

Let $\gamma_{1}\oplus\gamma_{2}\in{{}_{X_{1}\times
X_{2}}\mathcal{L}}_{(a_{1},a_{2})}$. Set
$[0,c_{i}]=\operatorname{Dom}\gamma_{i}$. Then
$\operatorname{Dom}\gamma_{1}\oplus\gamma_{2}=[0,c_{1}+c_{2}]$. We choose
$\eta_{i}\in{{}_{Y_{i}}\mathcal{L}}_{b_{i}}$ such that
$\operatorname{Dom}\eta_{i}=[0,d_{i}]$ and
$f_{i}(\gamma_{i}(c_{i}))=\eta_{i}(d_{i})$. Then we have
$\eta_{1}\oplus\eta_{2}\in{{}_{Y_{1}\times Y_{2}}\mathcal{L}}_{(b_{1},b_{2})}$
and $f_{1}\times
f_{2}(\gamma_{1}\oplus\gamma_{2}(c_{1}+c_{2}))=\eta_{1}\oplus\eta_{2}(d_{1}+d_{2})$.
Since $\gamma_{i}$ and $\eta_{i}$ are geodesics, and $f_{i}$ preserve the
distance from base points,

$\displaystyle
c_{i}=\,\overline{\gamma_{i}(c_{i}),a_{i}}\,=\,\overline{f_{i}(\gamma_{i}(c_{i})),b_{i}}\,=\,\overline{\eta_{i}(d_{i}),b_{i}}\,=d_{i}.$

Moreover, since $f_{i}$ is $\mathop{\mathrm{id}}_{\mathbb{R}_{\geq
0}}$-$H$-${{}_{X_{i}}\mathcal{L}}$-${{}_{Y_{i}}\mathcal{L}}$-equivariant,

$\displaystyle\,\overline{f_{i}(\gamma_{i}(s)),\eta_{i}(s)}\,<H\quad(\forall
s\in[0,c_{i}]).$

Now set $e_{i}:=c_{i}/(c_{1}+c_{2})$. For all $t\in[0,c_{1}+c_{1}]$, we have

$\displaystyle\,\overline{f_{1}\times
f_{2}(\gamma_{1}\oplus\gamma_{2}(t)),\eta_{1}\oplus\eta_{2}(t)}\,$
$\displaystyle=\,\overline{(f_{1}\circ\gamma_{1}(e_{1}t),f_{2}\circ\gamma_{2}(e_{2}t)),(\eta_{1}(e_{1}t),\eta_{2}(e_{2}t))}\,$
$\displaystyle=\,\overline{(f_{1}\circ\gamma_{1}(e_{1}t),\eta_{1}(e_{1}t))}\,+\,\overline{(f_{2}\circ\gamma_{2}(e_{2}t),\eta_{2}(e_{2}t))}\,$
$\displaystyle<2H.$

∎

###  6.3. Example

Dydak and Virk constructed a map $f\colon\mathrm{Cay}(F_{2})\to\mathbb{H}^{2}$
from the Cayley graph of the free group of rank 2 with respect to the standers
generating set to the hyperbolic plane [4]. The map $f$ satisfies the
following:

  1. (i)

$f$ is a large scale Lipschitz map.

  2. (ii)

$f(e)=O$, here $e\in F_{2}$ is the unit and $O\in\mathbb{H}^{2}$ is the
origin.

  3. (iii)

For all $x\in\mathrm{Cay}(F_{2})$, we have
$\,\overline{x,e}\,=\,\overline{f(x),O}\,$.

  4. (iv)

$f$ is coarsely two-to-one.

By (iii), $f$ is $\mathop{\mathrm{id}}_{\mathbb{R}_{\geq 0}}$-radial. Since
both $\mathbb{H}^{2}$ and $\mathbb{R}$ is geodesic coarsely convex space, by
Proposition 2.2 so is the product $\mathbb{H}^{2}\times\mathbb{R}$. Then by
Lemma 6.1,
$f\times\mathop{\mathrm{id}}\colon\mathrm{Cay}(F_{2})\times\mathbb{R}\to\mathbb{H}^{2}\times\mathbb{R}$
is a large scale Lipschitz, coarsely two-to-one map. By Lemma 6.3 and Theorem
3.11, $f\times\mathop{\mathrm{id}}$ is visual, and by Theorem 5.4, the induced
map
$\partial(f\times\mathop{\mathrm{id}})\colon\partial(\mathrm{Cay}(F_{2})\times\mathbb{R})\to
S^{2}$ is two-to-one, where $\partial(\mathrm{Cay}(F_{2})\times\mathbb{R})$ is
homeomorphic to the suspension of the Cantor set. Similarly,

$\displaystyle f\times
f\colon\mathrm{Cay}(F_{2})\times\mathrm{Cay}(F_{2})\to\mathbb{H}^{2}\times\mathbb{H}^{2}$

is a large scale Lipschitz, coarsely four-to-one map, and induced map

$\displaystyle\partial(f\times
f)\colon\partial(\mathrm{Cay}(F_{2})\times\mathrm{Cay}(F_{2}))\to S^{3}$

is four-to-one map, where
$\partial(\mathrm{Cay}(F_{2})\times\mathrm{Cay}(F_{2}))$ is homeomorphic to
the topological join of two Cantor sets.

##  7\. Radial extension

In this section, we will show that continuous maps between the boundaries of
coarsely convex spaces extend to visual maps between those coarsely convex
spaces. The idea is the following. Let $X$ and $Y$ be coarsely convex spaces
and let $f\colon\partial X\to\partial Y$ be a continuous map. We first
construct a map $\mathcal{O}f\colon\mathcal{O}\partial X\to\mathcal{O}\partial
Y$, where $\mathcal{O}\partial X$ and $\mathcal{O}\partial Y$ are open cones
over $\partial X$, and $\partial Y$, respectively. Then the extension might be
obtained by a composition of the “logarithmic map” $\log^{\epsilon}\colon
X\to\mathcal{O}\partial X$, $\mathcal{O}f$, and the “exponential map”
$\exp_{\epsilon}\colon\mathcal{O}\partial Y\to Y$. However, in general, the
map $\log^{\epsilon}$ is defined only on the subset of $X$, and the composite
$\exp_{\epsilon}\circ\mathcal{O}f$ is not necessarily bornologous. So we need
some modifications.

###  7.1. Open cone

Let $Z$ be a compact metrizable space. The open cone over $Z$, denoted by
$\mathcal{O}Z$, is the quotient $\mathbb{R}_{\geq 0}\times Z/(\\{0\\}\times
Z)$. For $(t,z)\in\mathbb{R}_{\geq 0}\times Z$, we denote by $tz$ the point in
$\mathcal{O}Z$ represented by $(t,z)$.

Let $d_{Z}$ be a metric on $Z$. We assume that the diameter of $Z$ is at most
2. We define a metric $d_{\mathcal{O}Z}$ on $\mathcal{O}Z$ by

$\displaystyle d_{\mathcal{O}Z}(tx,sy):=\left\lvert
t-s\right\rvert+\min\\{t,s\\}d_{Z}(x,y).$

The function $d_{\mathcal{O}Z}$ does satisfy the triangle inequality. See [14]
for details. We call $d_{\mathcal{O}Z}$ the induced metric by $d_{Z}$.

######  Definition 7.1.

Let $Z$ and $W$ be compact metric spaces and let $f\colon Z\to W$ be a
continuous map. We define the map
$\mathcal{O}f\colon\mathcal{O}Z\to\mathcal{O}W$ by $\mathcal{O}f(tz):=tf(z)$.

We remark that $\mathcal{O}f$ is proper, but not necessarily bornologous.
However, by composing with the following radial contraction, we will obtain a
coarse map.

######  Definition 7.2.

Let $r\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a rough contraction.
The radial contraction associated to $r$ is the map
$\phi_{r}\colon\mathcal{O}Z\to\mathcal{O}Z$ defined by $\phi_{r}(tz):=r(t)z$
for $tz\in\mathcal{O}Z$.

The following proposition says that for any proper continuous map from the
open cone, by composing an appropriate radial contraction, we obtain a coarse
map. In fact, being metrically proper and pseudocontinuous is enough to this
result. We refer [9, Section 4] and [7, Definition 5.2] for the definition of
pseudocontinuous maps. We remark that continuous maps are pseudocontinuous.

######  Proposition 7.3 ([9, Lemma 4.12]).

Let $V$ be a metric space, and let $F\colon\mathcal{O}Z\to V$ be a metrically
proper pseudocontinuous map. Then there exists a 0-rough contraction
$r\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ such that, for the radial
contraction $\phi_{r}\colon\mathcal{O}Z\to\mathcal{O}Z$ associated to $r$, the
composite $F\circ\phi_{r}\colon\mathcal{O}Z\to V$ is a coarse map.

######  Remark 7.4.

Let $\epsilon>0$ be a constant which will be fixed in the beginning of Section
7.2. We can assume without loss of generality that the function $r$ in
Proposition 7.3 satisfies the following. For $t,t^{\prime}\in\mathbb{R}_{\geq
0}$ with $t^{\epsilon},t^{\prime\epsilon}\in r^{-1}([1,\infty))$, we have

$\displaystyle\left\lvert
r(t^{\epsilon})^{1/\epsilon}-r(t^{\prime\epsilon})^{1/\epsilon}\right\rvert\leq\left\lvert
t-t^{\prime}\right\rvert.$

Indeed, we define the map $r^{\prime}\colon\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ by $r^{\prime}(t):=r(t)$ for $t\in r^{-1}([0,1))$,
and $r^{\prime}(t):=r(t)^{\epsilon}$ for $t\in r^{-1}([1,\infty))$. Then
$r^{\prime}(t)\leq r(t)$ for all $t\in\mathbb{R}_{\geq 0}$ and for
$t,t^{\prime}\in\mathbb{R}_{\geq 0}$ with $t^{\epsilon},t^{\prime\epsilon}\in
r^{-1}([1,\infty))$,

$\displaystyle\left\lvert
r^{\prime}(t^{\epsilon})^{1/\epsilon}-r^{\prime}(t^{\prime\epsilon})^{1/\epsilon}\right\rvert=\left\lvert
r(t^{\epsilon})-r(t^{\prime\epsilon})\right\rvert\leq\left\lvert
t^{\epsilon}-t^{\prime\epsilon}\right\rvert\leq\left\lvert
t-t^{\prime}\right\rvert.$

If the radial contraction associated to $r$ satisfies the statement of
Proposition 7.3, so does the one associated to $r^{\prime}$. Therefore we can
replace $r$ by $r^{\prime}$.

######  Lemma 7.5.

Let $r\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be a 0-rough
contraction satisfying Remark 7.4. Set $\delta:=\sup\\{u\in\mathbb{R}_{\geq
0}:r(u^{\epsilon})\leq 1\\}$. For $u,u^{\prime}\in\mathbb{R}_{\geq 0}$, we
have

$\displaystyle\left\lvert
r(u^{\epsilon})^{1/\epsilon}-r(u^{\prime\epsilon})^{1/\epsilon}\right\rvert\leq\left\lvert
u-u^{\prime}\right\rvert+2\delta.$

Especially, the map $t\mapsto r(t^{\epsilon})^{1/\epsilon}$ is a
$2\delta$-rough contraction.

###### Proof..

If $\min\\{r(u^{\epsilon}),r(u^{\prime\epsilon})\\}\leq 1$, then

$\displaystyle\left\lvert
r(u^{\epsilon})^{1/\epsilon}-r(u^{\prime\epsilon})^{1/\epsilon}\right\rvert\leq
r(u^{\epsilon})^{1/\epsilon}+r(u^{\prime\epsilon})^{1/\epsilon}\leq
u+u^{\prime}\leq\left\lvert u-u^{\prime}\right\rvert+2\delta.$

If $\min\\{r(u^{\epsilon}),r(u^{\prime\epsilon})\\}>1$, then by Remark 7.4,

$\displaystyle\left\lvert
r(u^{\epsilon})^{1/\epsilon}-r(u^{\prime\epsilon})^{1/\epsilon}\right\rvert\leq\left\lvert
u-u^{\prime}\right\rvert.$

∎

###  7.2. Exponential and logarithmic map

In the rest of this section, let $X$ and $Y$ be proper
$(\lambda,k,E,C,\theta,\mathcal{L})$-coarsely convex spaces. We fix a positive
number $0<\epsilon<1$ satisfying the statement of Proposition 2.13. Let
$d_{\epsilon,\partial_{a}X}$ and $d_{\epsilon,\partial_{b}Y}$ be a metric
given by Proposition 2.13 for $\partial_{a}X$ and $\partial_{b}Y$,
respectively. Let $K$ be the constant in the statement of Proposition 2.13.

First, we review the construction of the exponential map and the logarithmic
map for coarsely convex spaces. We refer [7, Section 5] for details.

The exponential map $\exp_{\epsilon}\colon\mathcal{O}\partial_{b}Y\to Y$ is
defined as follows. For each $y\in\partial_{b}Y$, we choose a quasi-geodesic
ray $\eta_{y}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ with $\eta_{y}\in y$. Then
for $t\in\mathbb{R}_{\geq 0}$, we define
$\exp_{\epsilon}(ty):=\eta_{y}(t^{\frac{1}{\epsilon}})$.

We remark that $\exp_{\epsilon}$ is proper, however, not necessarily
bornologous. It is shown that $\exp_{\epsilon}$ is pseudocontinuous ([7, Lemma
5.4]).

Now we introduce the logarithmic map. We define the subspace
$X^{\text{Vis}}\subset X$ as follows.

$\displaystyle
X^{\text{Vis}}:=\bigcup_{\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}}\gamma(\mathbb{R}_{\geq
0}).$

The logarithmic map $\log^{\epsilon}\colon
X^{\text{Vis}}\to\mathcal{O}\partial_{a}X$ is defined as follows. For $v\in
X^{\text{Vis}}$, we choose a geodesic ray
$\gamma_{v}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ and a parameter
$t_{v}\in\mathbb{R}_{\geq 0}$ such that $\gamma_{v}(t_{v})=v$. Then we define
$\log^{\epsilon}(v):=t^{\epsilon}[\gamma_{v}]$. The map $\log^{\epsilon}$ is a
coarse map ([7, Proposition 5.6]).

Since the domain of $\log^{\epsilon}$ is a subspace of $X$, we need to deform
$X$ to the subspace $X^{\text{Vis}}$.

######  Proposition 7.6 ([7, Section 5.5]).

There exists a 1-rough contraction $\chi\colon\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ such that the map $\varphi\colon X\to X$ given by

$\displaystyle\varphi(v):=\gamma_{v}(\chi(T_{v}))\quad(v\in X).$

is a coarse map, and $\varphi(X)\subset X^{\text{Vis}}$. Here $v\in X$,
$\gamma_{v}\in{{}_{X}\mathcal{L}}_{a}$, $T_{v}\in\mathbb{R}_{\geq 0}$, and
$v=\gamma_{v}(T_{v})$. Moreover, $\varphi$ is coarsely homotopic to the
identity.

###  7.3. Radial extension

In the rest of this section, let $f\colon\partial_{a}X\to\partial_{b}Y$ be a
continuous map.

Let $\mathcal{O}f\colon\mathcal{O}\partial_{a}X\to\mathcal{O}\partial_{b}Y$ be
the map given by Definition 7.1. Since $\mathcal{O}f$ is continuous and
$\exp_{\epsilon}\colon\mathcal{O}\partial_{b}Y\to Y$ is pseudocontinuous, the
composite $\exp_{\epsilon}\circ\mathcal{O}f$ is pseudocontinuous. Thus by
Proposition 7.3, there exists a 0-rough contraction $r\colon\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ such that the composite
$\exp_{\epsilon}\circ\mathcal{O}f\circ\phi_{r}$ is a coarse map, where
$\phi_{r}$ is a radial contraction
$\phi_{r}\colon\mathcal{O}\partial_{a}X\to\mathcal{O}\partial_{a}X$ associated
to $r$.

Now we define the map $\mathop{\mathrm{rad}}\nolimits f\colon X\to Y$ by the
composite

(6) $\displaystyle\mathop{\mathrm{rad}}\nolimits
f:=\exp_{\epsilon}\circ\mathcal{O}f\circ\phi_{r}\circ\log^{\epsilon}\circ\varphi$

where $\varphi$ is the map given in Proposition 7.6. We call it the radial
extension of $f$ associated to the map $r$. Since
$\mathop{\mathrm{rad}}\nolimits f$ is the composite of coarse maps, it is a
coarse map. In Proposition 7.11, we will show that the coarse homotopy class
of $\mathop{\mathrm{rad}}\nolimits f$ does not depend on the choice of the map
$r$.

The main purpose of this section is to prove the following.

######  Theorem 7.7.

Let $X$ and $Y$ be coarsely convex space with system of good quasi-geodesics
${{}_{X}\mathcal{L}}$ and ${{}_{Y}\mathcal{L}}$, respectively. Let
$f\colon\partial X\rightarrow\partial Y$ be a continuous map. Then there
exists a 0-rough contraction $r\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}$ such that the radial extension $\mathop{\mathrm{rad}}\nolimits f$ of $f$
associated to $r$ is ${{}_{X}\mathcal{L}}$-radial and
${{}_{X}\mathcal{L}}$-${{}_{Y}\mathcal{L}}$-equivariant. Moreover,

  1. $($i$)$

For a continuous map $f\colon\partial X\rightarrow\partial Y$, the induced map
$\partial\mathop{\mathrm{rad}}\nolimits f\colon\partial X\to\partial Y$ is
equal to $f$.

  2. $($ii$)$

Let $F\colon X\to Y$ be a visual
$\rho$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant
map. Then $\mathop{\mathrm{rad}}\nolimits\partial F$ is coarsely homotopic to
$F$.

In the rest of this section, let $\Omega$ be a constant satisfying the
statement of Lemma 2.8 for both $X$ and $Y$.

######  Lemma 7.8.

There exists a non-decreasing map $\psi\colon\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}\cup\\{\infty\\}$ with
$\lim_{t\to\infty}\psi(t)=\infty$ such that for $p,q\in\partial_{a}X$ with
$p\neq q$, we have

$\displaystyle(f(p)\mid f(q))_{b}\geq\psi((p\mid q)_{a}).$

###### Proof..

We define a map $\psi\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}\cup\\{\infty\\}$ by

$\displaystyle\psi(t):=\inf\\{(f(p)\mid
f(q))_{b}:(p,q)\in(\partial_{a}X)^{2},\,(p\mid q)_{a}\geq t\\},\quad(t\geq
0).$

By the definition, $\psi$ is non-decreasing. Since $\partial_{a}X$ is compact,
$f$ is uniformly continuous. So by Proposition 2.13, we have
$\lim_{t\to\infty}\psi(t)=\infty$. ∎

We remark that if $\partial_{a}X$ has at least one accumulation point, then
the range of the map $\psi$ in Lemma 7.8 does not contain $\infty$.

Set $\mathop{\mathrm{rad^{\prime}}}\nolimits
f:=\exp_{\epsilon}\circ\mathcal{O}f\circ\phi_{r}\circ\log^{\epsilon}\colon
X^{\text{Vis}}\to Y$. First we consider this map.

######  Proposition 7.9.

Suppose that the map $r\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$
satisfies

$\displaystyle
r\left(t^{\epsilon}\right)^{1/\epsilon}\leq\Omega^{-1}\psi\left(\frac{\Omega^{-1}(t-\bar{\theta}(0))}{E(\lambda\bar{\theta}(0)+k_{1})}\right)\quad(\forall
t\in\mathbb{R}_{\geq 0}).$

Then the map $\mathop{\mathrm{rad^{\prime}}}\nolimits f$ is
${{}_{X}\mathcal{L}}$-radial and
${{}_{X}\mathcal{L}}$-${{}_{Y}\mathcal{L}}$-equivariant.

We remark that $X^{\text{Vis}}$ is a subspace of the coarsely convex space,
however, both being $\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial and being
$\rho$-$H$-${{}_{X}\mathcal{L}}_{a}$-${{}_{Y}\mathcal{L}}_{b}$-equivariant
make sense for the map $\mathop{\mathrm{rad^{\prime}}}\nolimits f$.

###### Proof..

Let $x\in X^{\text{Vis}}$. We choose
$\gamma_{x}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$, $T_{x}\in\mathbb{R}_{\geq 0}$
such that $x=\gamma_{x}(T_{x})$ and
$\log^{\epsilon}(x)=T_{x}^{\epsilon}[\gamma_{x}]$. For $[\gamma_{x}]$, we
choose $\eta_{x}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that
$f([\gamma_{x}])=[\eta_{x}]$ and
$\exp_{\epsilon}(s[\eta_{x}])=\eta_{x}(s^{1/\epsilon})$ for
$s\in\mathbb{R}_{\geq 0}$. Then we can compute
$\mathop{\mathrm{rad^{\prime}}}\nolimits f(x)$ as follows.

$\displaystyle
x\xmapsto{\log^{\epsilon}}T_{x}^{\epsilon}[\gamma_{x}]\xmapsto{\phi_{r}}r(T_{x}^{\epsilon})[\gamma_{x}]\xmapsto{\mathcal{O}f}r(T_{x}^{\epsilon})f([\gamma_{x}])\xmapsto{\exp_{\epsilon}}\eta_{x}(r(T_{x}^{\epsilon})^{1/\epsilon}).$

We have

(7)
$\displaystyle\,\overline{\eta_{x}(r(T_{x}^{\epsilon})^{1/\epsilon}),b}\,\geq\frac{1}{\lambda}r(T_{x}^{\epsilon})^{1/\epsilon}-k_{1}.$

Set

$\displaystyle\rho(t):=r(t^{\epsilon})^{1/\epsilon}\quad\text{and}\quad\sigma(t):=\left(\frac{1}{\lambda}\rho(t)-k_{1}\right)\vee
0.$

By (7), $\mathop{\mathrm{rad^{\prime}}}\nolimits f$ is weakly
$\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial, so it is
${{}_{X}\mathcal{L}}$-radial, by Lemma 3.9.

Now let $\xi\in{{}_{X}\mathcal{L}}_{a}$ with
$\operatorname{Dom}\xi=[0,L_{\xi}]$. We put $x:=\xi(L_{\xi})$. We use the
above computation for $x$, that is, $\mathop{\mathrm{rad^{\prime}}}\nolimits
f(x)=\eta_{x}(r(T_{x}^{\epsilon})^{1/\epsilon})$. Let
$\eta\in{{}_{Y}\mathcal{L}}_{b}$ with $\operatorname{Dom}\eta=[0,L_{\eta}]$
such that
$\eta(L_{\eta})=\eta_{x}(\rho(T_{x}))=\mathop{\mathrm{rad^{\prime}}}\nolimits
f(\xi(L_{\xi}))$. Since $\operatorname{Dom}\eta_{x}=\mathbb{R}_{\geq 0}$, by
Lemma 2.8 (v),

(8)
$\displaystyle\,\overline{\eta_{x}(s),\eta(s)}\,\leq\Omega,\quad(0\leq\forall
s\leq L_{\eta}).$

We fix $t\in\mathbb{R}_{\geq 0}$ with
$t\leq\min\\{L_{\xi},\rho^{-1}(L_{\eta})\\}$ and set $w:=\xi(t)$. We choose
$\gamma_{w}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ and $T_{w}\in\mathbb{R}_{\geq
0}$ such that $w=\gamma_{w}(T_{w})$ and
$\log^{\epsilon}(w)=T_{w}^{\epsilon}[\gamma_{w}]$. Since
$\gamma_{w}(T_{w})=\xi(t)$, by Lemma 2.9,

$\displaystyle(\gamma_{w}\mid\xi)_{a}\geq\frac{t-\tilde{\theta}(0)}{E(\lambda\tilde{\theta}(0)+k_{1})}.$

Applying the same argument for $x=\gamma_{x}(T_{x})=\xi(L_{\xi})$, we have

$\displaystyle(\gamma_{x}\mid\xi)_{a}\geq\frac{L_{\xi}-\tilde{\theta}(0)}{E(\lambda\tilde{\theta}(0)+k_{1})}.$

So we have

$\displaystyle(\gamma_{x}\mid\gamma_{w})_{a}\geq\Omega^{-1}\min\\{(\gamma_{x}\mid\xi)_{a},(\gamma_{w}\mid\xi)_{a}\\}\geq\frac{\Omega^{-1}(t-\tilde{\theta}(0))}{E(\lambda\tilde{\theta}(0)+k_{1})}.$

We choose $\eta_{w}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that
$f([\gamma_{w}])=[\eta_{w}]$ and
$\exp_{\epsilon}(s[\eta_{w}])=\eta_{w}(s^{1/\epsilon})$ for all
$s\in\mathbb{R}_{\geq 0}$. By Lemma 7.8,

$\displaystyle([\eta_{x}]\mid[\eta_{w}])_{b}$
$\displaystyle\geq\psi(([\gamma_{x}]\mid[\gamma_{w}])_{a})\geq\psi\left(\frac{\Omega^{-1}(t-\tilde{\theta}(0))}{E(\lambda\tilde{\theta}(0)+k_{1})}\right).$

Since $\gamma_{w}(T_{w})=\xi(t)$, we have $\left\lvert
T_{w}-t\right\rvert\leq\tilde{\theta}(0)$. By the assumption on $r$, we have

$\displaystyle\rho(t)\leq\Omega^{-1}\psi\left(\frac{\Omega^{-1}(t-\tilde{\theta}(0))}{E(\lambda\tilde{\theta}(0)+k_{1})}\right)\leq(\eta_{x}\mid\eta_{w})_{b}.$

Then by Lemma 2.8 (iv), we have

$\displaystyle\,\overline{\eta_{w}(\rho(t)),\eta_{x}(\rho(t))}\,\leq\Omega.$

Set $\delta:=\sup\\{u\in\mathbb{R}_{\geq 0}:r(u^{\epsilon})\leq 1\\}$. By
Lemma 7.5,

$\displaystyle\left\lvert\rho(T_{w})-\rho(t)\right\rvert\leq\left\lvert
T_{w}-t\right\rvert+2\delta\leq\tilde{\theta(0)}+2\delta.$

Therefore

(9) $\displaystyle\,\overline{\mathop{\mathrm{rad^{\prime}}}\nolimits
f(\xi(t)),\eta_{x}(\rho(t))}\,$
$\displaystyle=\,\overline{\eta_{w}(\rho(T_{w})),\eta_{x}(\rho(t))}\,\leq\,\overline{\eta_{w}(\rho(t)),\eta_{x}(\rho(t))}\,+\lambda(\tilde{\theta}(0)+2\delta)+k_{1}$
$\displaystyle\leq\Omega+\lambda(\tilde{\theta}(0)+2\delta)+k_{1}.$

Set $H:=\Omega+\lambda(\tilde{\theta}(0)+2\delta)+k_{1}+\Omega$. Since
$\rho(t)\leq L_{\eta}$, by (9) and (8), we have

$\displaystyle\,\overline{\mathop{\mathrm{rad^{\prime}}}\nolimits
f(\xi(t)),\eta(\rho(t))}\,\leq H.$

Hence the proof is done. ∎

######  Proposition 7.10.

The map $\varphi\colon X\to X$ is ${{}_{X}\mathcal{L}}$-radial and
${{}_{X}\mathcal{L}}$-${{}_{Y}\mathcal{L}}$-equivariant.

###### Proof..

Let $x\in X$. We choose $\gamma_{x}\in{{}_{X}\mathcal{L}}$ and
$T_{x}\in\mathbb{R}_{\geq 0}$ such that $x=\gamma_{x}(T_{x})$,
$\varphi(x)=\gamma_{x}(\chi(T_{x}))$. Then

$\displaystyle\,\overline{\varphi(x),a}\,=\,\overline{\gamma_{x}(\chi(T_{x})),\gamma_{x}(0)}\,\geq\frac{1}{\lambda}\chi(T_{x})-k.$

Set $\sigma(t):=(\lambda^{-1}\chi(t)-k)\vee 0$. Then $\varphi$ is weakly
$\sigma$-${{}_{X}\mathcal{L}}_{a}$-radial, so it is
${{}_{X}\mathcal{L}}$-radial, by Lemma 3.9.

Let $\gamma\in{{}_{X}\mathcal{L}}$. Set
$\operatorname{Dom}\gamma=[0,L_{\gamma}]$ and $x:=\gamma(L_{\gamma})$. Let
$\eta\in{{}_{X}\mathcal{L}}_{a}$ with $\operatorname{Dom}\eta=[0,L_{\eta}]$
such that $\eta(L_{\eta})=\varphi(x)$.

We fix $t\in\mathbb{R}_{\geq 0}$ with
$t\leq\min\\{L_{\gamma},\chi^{-1}(L_{\eta})\\}$, and set $w:=\gamma(t)$. We
choose $\gamma_{x},\gamma_{w}\in{{}_{X}\mathcal{L}}_{a}$ and
$T_{x},T_{w}\in\mathbb{R}_{\geq 0}$ such that $x=\gamma_{x}(T_{x})$,
$w=\gamma_{w}(T_{w})$, $\varphi(x)=\gamma_{x}(\chi(T_{x}))$, and
$\varphi(w)=\gamma_{w}(\chi(T_{w}))$.

Since $\gamma(t)=\gamma_{w}(T_{w})$, we have $\left\lvert
T_{w}-t\right\rvert\leq\theta(0)$. Set $u:=\min\\{T_{w},t\\}$. We remark

$\displaystyle\left\lvert\chi(T_{w})-\chi(u)\right\rvert$
$\displaystyle\leq\left\lvert T_{w}-u\right\rvert+1\leq\theta(0)+1,$
$\displaystyle\left\lvert\chi(t)-\chi(u)\right\rvert$
$\displaystyle\leq\left\lvert t-u\right\rvert+1\leq\theta(0)+1.$

By Lemma 2.8 (v),

$\displaystyle\,\overline{\gamma_{w}(\chi(u)),\gamma(\chi(u))}\,\leq\Omega.$

Therefore we have

(10) $\displaystyle\,\overline{\gamma_{w}(\chi(T_{w})),\gamma(\chi(t))}\,$
$\displaystyle\leq\,\overline{\gamma_{w}(\chi(u)),\gamma(\chi(u))}\,+\lambda(\theta(0)+1)+k$
$\displaystyle\leq\Omega+\lambda\theta(0)+\lambda+k.$

Now, first we suppose $\chi(t)\leq T_{x}$. Then, since
$\gamma(L_{\gamma})=x=\gamma_{x}(T_{x})$, by Lemma 2.8 (v),

$\displaystyle\,\overline{\gamma(\chi(t)),\gamma_{x}(\chi(t))}\,\leq\Omega.$

Since $\gamma_{x}(\chi(T_{x}))=\eta(L_{\eta})$ and $\chi(t)\leq L_{\eta}$, by
Lemma 2.8 (v) we have

$\displaystyle\,\overline{\gamma_{x}(\chi(t)),\eta(\chi(t))}\,\leq\Omega.$

Thus we have

(11) $\displaystyle\,\overline{\gamma_{w}(\chi(T_{w})),\eta(\chi(t))}\,\leq
3\Omega+\lambda\theta(0)+k+\lambda.$

Next we suppose $\chi(t)>T_{x}$. Then, we have

$\displaystyle T_{x}<\chi(t)\leq t\leq
L_{\gamma}\quad\text{and}\quad\chi(T_{x})\leq T_{x}<\chi(t)\leq L_{\eta}.$

Since $\gamma_{x}(T_{x})=\gamma(L_{\gamma})$ and
$\gamma_{x}(\chi(T_{x}))=\eta(L_{\eta})$,

$\displaystyle\left\lvert
T_{x}-L_{\gamma}\right\rvert\leq\theta(0)\quad\text{and}\quad\left\lvert\chi(T_{x})-L_{\eta}\right\rvert\leq\theta(0).$

Then we have

$\displaystyle\,\overline{\gamma(\chi(t)),\eta(\chi(t))}\,$
$\displaystyle\leq\,\overline{\gamma(\chi(t)),\gamma(L_{\gamma})}\,+\,\overline{\gamma_{x}(T_{x}),\gamma_{x}(\chi(T_{x}))}\,+\,\overline{\eta(L_{\eta}),\eta(\chi(t))}\,$
$\displaystyle\leq\lambda(\left\lvert\chi(t)-L_{\gamma}\right\rvert+\left\lvert
T_{x}-\chi(T_{x})\right\rvert+\left\lvert L_{\eta}-\chi(t)\right\rvert)+3k$
$\displaystyle\leq\lambda(\left\lvert
T_{x}-L_{\gamma}\right\rvert+2\left\lvert\chi(T_{x})-L_{\eta}\right\rvert)+3k$
$\displaystyle\leq 3\lambda\theta(0)+3k.$

Thus by (10) and the above inequality, we have

(12)
$\displaystyle\,\overline{\gamma_{w}(\chi(T_{w})),\eta(\chi(t))}\,\leq\Omega+4(\lambda\theta(0)+k)+\lambda.$

Finally by (11) and (12) we have

$\displaystyle\,\overline{\varphi(\gamma(t)),\eta(\chi(t))}\,$
$\displaystyle=\,\overline{\varphi(w),\eta(\chi(t))}\,=\,\overline{\gamma_{w}(\chi(T_{w})),\eta(\chi(t))}\,$
$\displaystyle\leq 3\Omega+4(\lambda\theta(0)+k)+\lambda.$

Hence the proof is done. ∎

The first half of Theorem 7.7 follows from Proposition 7.9 and Proposition
7.10.

######  Proposition 7.11.

For $i=1,2$, let $r_{i}\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ be
0-rough contractions satisfying Remark 7.4. Let
$\mathop{\mathrm{rad}}\nolimits_{i}f$ be the radial extension of $f$
associated to $r_{i}$. Then $\mathop{\mathrm{rad}}\nolimits_{1}f$ and
$\mathop{\mathrm{rad}}\nolimits_{2}f$ are coarsely homotopic.

###### Proof..

Let $\phi_{i}$ be the radial contractions
$\phi_{i}\colon\mathcal{O}\partial_{a}X\to\mathcal{O}\partial_{a}X$ associated
to $r_{i}$ such that
$\mathop{\mathrm{rad}}\nolimits_{i}=\exp_{\epsilon}\circ\mathcal{O}f\circ\phi_{i}\circ\log^{\epsilon}\circ\varphi$.

We can assume without loss of generality that

(13) $\displaystyle r_{1}(t)\leq r_{2}(t)\quad(\forall t\in\mathbb{R}_{\geq
0}).$

Indeed, set $r_{3}(t):=\min\\{r_{1}(t),r_{2}(t)\\}$ and let
$\mathop{\mathrm{rad}}\nolimits_{3}f$ be the radial extension of $f$
associated to $r_{3}$. If we showed that $\mathop{\mathrm{rad}}\nolimits_{i}f$
and $\mathop{\mathrm{rad}}\nolimits_{3}f$ are coarsely homotopic, then by the
transitivity of coarse homotopy, $\mathop{\mathrm{rad}}\nolimits_{1}f$ and
$\mathop{\mathrm{rad}}\nolimits_{2}f$ are coarsely homotopic.

Now set
$\mathop{\mathrm{rad^{\prime}}}\nolimits_{i}:=\exp_{\epsilon}\circ\mathcal{O}f\circ\phi_{i}\circ\log^{\epsilon}\colon
X^{\text{Vis}}\to Y$. It is enough to show that
$\mathop{\mathrm{rad^{\prime}}}\nolimits_{1}f$ and
$\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f$ are coarsely homotopic.

For each $x\in X^{\text{Vis}}$, we choose
$\gamma_{x}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ and $T_{x}\in\mathbb{R}_{\geq
0}$ such that $\log^{\epsilon}(x)=T_{x}^{\epsilon}[\gamma_{x}]$. We also
choose $\eta_{x}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that
$f([\gamma_{x}])=[\eta_{x}]$ and
$\exp_{\epsilon}(s[\eta_{x}])=\eta_{x}(s^{1/\epsilon})$ for all
$s\in\mathbb{R}_{\geq 0}$.

Set $Z:=\\{(x,t)\in X^{\text{Vis}}\times\mathbb{R}_{\geq 0}\colon t\leq
T_{x}\\}$. We define a coarse homotopy $H\colon Z\to Y$ by

$\displaystyle
H(x,t)=\eta_{x}(r_{1}((T_{x}-t)^{\epsilon})^{1/\epsilon}+r_{2}(t^{\epsilon})^{1/\epsilon}).$

Then for $x\in X^{\text{Vis}}$,
$H(x,0)=\mathop{\mathrm{rad^{\prime}}}\nolimits_{1}f(x)$ and
$H(x,T_{x})=\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f(x)$. So all we need
to show is that $H$ is a coarse map. It is easy to show that $H$ is metrically
proper. We will show that $H$ is bornologous.

Let $(x,t),(y,t^{\prime})\in Z$. For $y$, we choose
$\gamma_{y}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$, $T_{y}\in\mathbb{R}_{\geq 0}$
and $\eta_{y}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ as above. Set
$\delta:=\sup\\{u\in\mathbb{R}_{\geq 0}:r_{1}(u^{\epsilon})\leq 1\\}$. By
Lemma 7.5, for $i=1,2$, we have

(14) $\displaystyle\left\lvert
r_{i}(u^{\epsilon})^{1/\epsilon}-r_{i}(u^{\prime\epsilon})^{1/\epsilon}\right\rvert\leq\left\lvert
u-u^{\prime}\right\rvert+2\delta,\quad(\forall u,u^{\prime}\in\mathbb{R}_{\geq
0}).$

Now set
$s:=r_{1}((T_{x}-t)^{\epsilon})^{1/\epsilon}+r_{2}(t^{\epsilon})^{1/\epsilon}$
and
$s^{\prime}:=r_{1}((T_{y}-t^{\prime})^{\epsilon})^{1/\epsilon}+r_{2}(t^{\prime\epsilon})^{1/\epsilon}$.
By (14), we have

$\displaystyle\left\lvert s-s^{\prime}\right\rvert$
$\displaystyle\leq\left\lvert
r_{1}((T_{x}-t)^{\epsilon})^{1/\epsilon}-r_{1}((T_{y}-t^{\prime})^{\epsilon})^{1/\epsilon}\right\rvert+\left\lvert
r_{2}(t^{\epsilon})^{1/\epsilon}-r_{2}(t^{\prime\epsilon})^{1/\epsilon}\right\rvert$
$\displaystyle\leq\left\lvert T_{x}-T_{y}\right\rvert+2\left\lvert
t-t^{\prime}\right\rvert+4\delta$
$\displaystyle\leq\tilde{\theta}(\,\overline{x,y}\,)+2\left\lvert
t-t^{\prime}\right\rvert+4\delta.$

Set $q:=\min\\{s,s^{\prime}\\}$. Then by Lemma 2.4 and (13), we have

$\displaystyle\,\overline{H(x,t),H(y,t^{\prime})}\,=$
$\displaystyle\,\overline{\eta_{x}(s),\eta_{y}(s^{\prime})}\,$
$\displaystyle=$
$\displaystyle\,\overline{\eta_{x}(q),\eta_{y}(q)}\,+\lambda\left\lvert
s-s^{\prime}\right\rvert+k_{1}$ $\displaystyle\leq$ $\displaystyle
E^{\prime}\left(\,\overline{\eta_{x}(r_{2}(T_{x}^{\epsilon})^{1/\epsilon}),\eta_{y}(r_{2}(T_{y}^{\epsilon})^{1/\epsilon})}\,+\lambda^{\prime}\tilde{\theta}(\,\overline{\eta_{x}(r_{2}(T_{x}^{\epsilon})^{1/\epsilon}),\eta_{y}(r_{2}(T_{y}^{\epsilon})^{1/\epsilon}}\,)\right)$
$\displaystyle+D^{\prime}+\lambda(\tilde{\theta}(\,\overline{x,y}\,)+2\left\lvert
t-t^{\prime}\right\rvert+4\delta)+k^{\prime}_{1}.$ $\displaystyle=$
$\displaystyle
E^{\prime}\left(\,\overline{\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f(x),\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f(y)}\,+\lambda^{\prime}\tilde{\theta}(\,\overline{\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f(x),\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f(y)}\,)\right)$
$\displaystyle+D^{\prime}+\lambda(\tilde{\theta}(\,\overline{x,y}\,)+2\left\lvert
t-t^{\prime}\right\rvert+4\delta)+k^{\prime}_{1}.$

Since $\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f$ is bornologous,
$\,\overline{\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f(x),\mathop{\mathrm{rad^{\prime}}}\nolimits_{2}f(y)}\,$
is bounded from the above by a constant depending only on
$\,\overline{x,y}\,$. Therefore $H$ is bornologous. This complete the proof. ∎

###  7.4. Induced map of the radial extension

In this section, we give a proof of the statement (i) of Theorem 7.7. Let
$f\colon\partial_{a}X\to\partial_{b}Y$ be a continuous map and let
$\mathop{\mathrm{rad}}\nolimits f$ be its radial extension. We will show that
the induced map $\partial\mathop{\mathrm{rad}}\nolimits f$ is equal to $f$.

######  Lemma 7.12.

We have $\partial\mathop{\mathrm{rad^{\prime}}}\nolimits f=f$.

###### Proof..

Let $\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$. We choose
$\eta\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that $[\eta]=f([\gamma])$ and
$\eta(s)=\exp_{\epsilon}(s[\eta])$ for all $s\in\mathbb{R}_{\geq 0}$.

Set $x_{n}:=\gamma(n)$ for $n\in\mathbb{N}$. We choose
$\gamma_{n}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ and $T_{n}\in\mathbb{R}_{\geq
0}$ such that $\gamma_{n}(T_{n})=x_{n}=\gamma(n)$ and
$\log^{\epsilon}(x_{n})=T_{n}[\gamma_{n}]$. By Lemma 2.9,

$\displaystyle(\gamma_{n}\mid\gamma)_{a}\geq\frac{n-\tilde{\theta}(0)}{E(\tilde{\theta}(0)+k_{1})}.$

We choose $\eta_{n}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that
$[\eta_{n}]=f([\gamma_{n}])$ and $\exp_{\epsilon}(s[\eta_{n}])=\eta_{n}(s)$
for $s\in\mathbb{R}_{\geq 0}$. Then $\mathop{\mathrm{rad^{\prime}}}\nolimits
f(x_{n})=\eta_{n}(r(T_{n}^{\epsilon})^{1/\epsilon})$.

By Lemma 7.8,

$\displaystyle([\eta_{n}]\mid[\eta])_{b}$ $\displaystyle=(f([\gamma_{n}])\mid
f([\gamma]))_{b}$ $\displaystyle\geq\psi(([\gamma_{n}]\mid[\gamma])_{a})$
$\displaystyle\geq\psi\left(\frac{n-\tilde{\theta}(0)}{E(\tilde{\theta}(0)+k_{1})}\right)\rightarrow\infty.$

Let $\eta^{\prime}_{n}\in{{}_{Y}\mathcal{L}}_{b}$ with
$\operatorname{Dom}\eta^{\prime}_{n}=[0,L_{\eta^{\prime}_{n}}]$ such that
$\eta^{\prime}_{n}(L_{\eta^{\prime}_{n}})=\mathop{\mathrm{rad^{\prime}}}\nolimits
f(x_{n})$. By Lemma 2.9,

$\displaystyle(\mathop{\mathrm{rad^{\prime}}}\nolimits
f(x_{n})\mid[\eta_{n}])_{b}\geq(\eta^{\prime}_{n}\mid\eta_{n})_{b}\geq\frac{r(T_{n}^{\epsilon})^{1/\epsilon}-\tilde{\theta}(0)}{E(\lambda\tilde{\theta}(0)+k_{1})}\to\infty.$

So we have

$\displaystyle(\mathop{\mathrm{rad^{\prime}}}\nolimits
f(x_{n})\mid[\eta])_{b}\geq\Omega^{-1}\min\\{(\mathop{\mathrm{rad^{\prime}}}\nolimits
f(x_{n})\mid[\eta_{n}])_{b},([\eta_{n}]\mid[\eta])_{b}\\}\to\infty.$

Thus $\mathop{\mathrm{rad^{\prime}}}\nolimits f(x_{n})\to[\eta]$. Since
$\mathop{\mathrm{rad^{\prime}}}\nolimits
f\cup\partial\mathop{\mathrm{rad^{\prime}}}\nolimits f$ is continuous at
$\partial_{a}X$, we have $\partial\mathop{\mathrm{rad^{\prime}}}\nolimits
f([\gamma])=[\eta]=f([\gamma])$. ∎

######  Lemma 7.13.

We have $\partial\varphi=\mathop{\mathrm{id}}_{\partial_{a}X}$.

###### Proof..

Let $\gamma\in{{}_{X}\mathcal{L}}_{a}^{\infty}$. Set $x_{n}:=\gamma(n)$ for
$n\in\mathbb{N}$. We choose $\gamma_{n}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$
and $T_{n}\in\mathbb{R}_{\geq 0}$ such that
$\gamma_{n}(T_{n})=x_{n}=\gamma(n)$ and
$\varphi(x_{n})=\gamma_{n}(\chi(T_{n}))$. Since $\gamma_{n}(T_{n})=\gamma(n)$,
by Lemma 2.9,

$\displaystyle(\gamma_{n}\mid\gamma)_{a}\geq\frac{n-\tilde{\theta}(0)}{E(\tilde{\theta}(0)+k_{1})}\to\infty.$

Let $\eta_{n}\in{{}_{X}\mathcal{L}}_{a}$ with
$\operatorname{Dom}\eta_{n}=[0,S_{n}]$ such that
$\eta_{n}(S_{n})=\varphi(x_{n})=\gamma_{n}(\chi(T_{n}))$. Then, by Lemma 2.9,

$\displaystyle(\eta_{n}\mid\gamma_{n})_{a}\geq\frac{\chi(T_{n})-\tilde{\theta}(0)}{E(\tilde{\theta}(0)+k_{1})}\to\infty.$

So we have

$\displaystyle(\varphi(x_{n})\mid[\gamma])_{a}\geq(\eta_{n}\mid\gamma)\geq\Omega^{-1}\min\\{(\eta_{n}\mid\gamma_{n})_{a},(\gamma_{n}\mid\gamma)_{a}\\}\to\infty.$

Thus $\varphi(x_{n})\to[\gamma]$. Since $\varphi\cup\partial\varphi$ is
continuous at $\partial_{a}X$, we have $\partial\varphi([\gamma])=[\gamma]$. ∎

######  Corollary 7.14.

$\partial\mathop{\mathrm{rad}}\nolimits f=f$. Especially,
$\mathop{\mathrm{rad}}\nolimits f$ is visually $n$-to-one if and only if $f$
is $n$-to-one.

###### Proof..

The second half of the statement follows from Proposition 5.1. ∎

###  7.5. Radial extension of the induced map

In this section, we give a proof of the statement (ii) of Theorem 7.7. Let
$F\colon X\to Y$ be a visual
$\rho$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant
map. We will show the that the radial extension
$\mathop{\mathrm{rad}}\nolimits\partial F$ of the induced map $\partial
F\colon\partial X\to\partial Y$ is coarsely homotopic to $F$.

First, we will show that $\mathop{\mathrm{rad^{\prime}}}\nolimits\partial
F\colon X^{\text{Vis}}\to Y$ is coarsely homotopic to the restriction
$F|_{X^{\text{Vis}}}$. For each $x\in X^{\text{Vis}}$, we choose
$\gamma_{x}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$ and $T_{x}\in\mathbb{R}_{\geq
0}$ such that $\log^{\epsilon}(x)=T_{x}^{\epsilon}[\gamma_{x}]$. We also
choose $\eta_{x}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that $\partial
F([\gamma_{x}])=[\eta_{x}]$ and
$\exp_{\epsilon}(s[\eta_{x}])=\eta_{x}(s^{1/\epsilon})$.

Set $Z^{\text{Vis}}:=\\{(x,t)\in X^{\text{Vis}}\times\mathbb{R}_{\geq 0}\colon
t\leq T_{x}\\}$. We define a coarse homotopy $H\colon Z^{\text{Vis}}\to Y$ by

$\displaystyle H(x,t)=\eta_{x}(\rho(T_{x}-t)+r(t^{\epsilon})^{1/\epsilon}).$

By Proposition 7.11, we can assume without loss of generality that
$r(t^{\epsilon})^{1/\epsilon}\leq\rho(t)$ for all $t\in\mathbb{R}_{\geq 0}$.

######  Lemma 7.15.

The map $H$ is a coarse map.

###### Proof..

It is easy to show that $H$ is metrically proper, so we will show that $H$ is
bornologous.

Let $(x,t),(y,t^{\prime})\in Z^{\text{Vis}}$. For $x$, $y$, we choose
$\gamma_{x},\gamma_{y}\in{{}_{X}\mathcal{L}}_{a}^{\infty}$,
$T_{x},T_{y}\in\mathbb{R}_{\geq 0}$,
$\eta_{x},\eta_{y}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that

$\displaystyle x=\gamma_{x}(T_{x}),$
$\displaystyle\quad\log^{\epsilon}(x)=T_{x}^{\epsilon}[\gamma_{x}],\quad\partial
F([\gamma_{x}])=[\eta_{x}],\quad\exp_{\epsilon}(s[\eta_{x}])=\eta_{x}(s^{1/\epsilon})\quad(\forall
s),$ $\displaystyle y=\gamma_{(}T_{y}),$
$\displaystyle\quad\log^{\epsilon}(y)=T_{y}^{\epsilon}[\gamma_{y}],\quad\partial
F([\gamma_{y}])=[\eta_{y}],\quad\exp_{\epsilon}(s[\eta_{y}])=\eta_{y}(s^{1/\epsilon})\quad(\forall
s).$

Set $\delta:=\sup\\{u\in\mathbb{R}_{\geq 0}:r(u^{\epsilon})\leq 1\\}$. By
Lemma 7.5, we have

$\displaystyle\left\lvert
r(t^{\epsilon})^{1/\epsilon}-r(t^{\prime\epsilon})^{1/\epsilon}\right\rvert\leq\left\lvert
t-t^{\prime}\right\rvert+2\delta.$

Let $\tau$ be a constant such that $\rho$ is a $\tau$-rough contraction. Then

$\displaystyle\left\lvert\rho(T_{x}-t)-\rho(T_{y}-t^{\prime})\right\rvert\leq\left\lvert
T_{x}-T_{y}\right\rvert+\left\lvert
t-t^{\prime}\right\rvert+\tau\leq\left\lvert
t-t^{\prime}\right\rvert+\tilde{\theta}(\,\overline{x,y}\,)+\tau.$

Set $s:=\rho(T_{x}-t)+r(t^{\epsilon})^{1/\epsilon}$ and
$s^{\prime}:=\rho(T_{y}-t^{\prime})+r(t^{\prime\epsilon})^{1/\epsilon}$. Then
we have

$\displaystyle\left\lvert s-s^{\prime}\right\rvert$
$\displaystyle\leq\left\lvert\rho(T_{x}-t)-\rho(T_{y}-t^{\prime})\right\rvert+\left\lvert
r(t^{\epsilon})^{1/\epsilon}-r(t^{\prime\epsilon})^{1/\epsilon}\right\rvert$
$\displaystyle\leq\tilde{\theta}(\,\overline{x,y}\,)+2\left\lvert
t-t^{\prime}\right\rvert+2\delta+\tau.$

Now set $q:=\min\\{s,s^{\prime}\\}$. Then by Lemma 2.4, we have

(15) $\displaystyle\,\overline{H(x,t),H(y,t^{\prime})}\,=$
$\displaystyle\,\overline{\eta_{x}(s),\eta_{y}(s^{\prime})}\,$
$\displaystyle=$
$\displaystyle\,\overline{\eta_{x}(q),\eta_{y}(q)}\,+\lambda\left\lvert
s-s^{\prime}\right\rvert+k_{1}$ $\displaystyle\leq$ $\displaystyle
E^{\prime}\left(\,\overline{\eta_{x}(\rho(T_{x})),\eta_{y}(\rho(T_{y}))}\,+\lambda^{\prime}\tilde{\theta}(\,\overline{\eta_{x}(\rho(T_{x})),\eta_{y}(\rho(T_{y}))}\,\right)$
$\displaystyle+D^{\prime}+\lambda(\tilde{\theta}(\,\overline{x,y}\,)+2\left\lvert
t-t^{\prime}\right\rvert+2\delta+\tau)+k^{\prime}_{1}.$

Since $F$ is
$\rho$-$T$-${{}_{X}\mathcal{L}}_{a}^{\infty}$-${{}_{Y}\mathcal{L}}_{b}^{\infty}$-equivariant,
there exists $\xi_{x},\xi_{y}\in{{}_{Y}\mathcal{L}}_{b}^{\infty}$ such that

(16) $\displaystyle\sup_{u\in\mathbb{R}_{\geq
0}}\max\left\\{\,\overline{\xi_{x}(\rho(u)),F(\gamma_{x}(u))}\,,\,\overline{\xi_{y}(\rho(u)),F(\gamma_{y}(u))}\,\right\\}<T.$

By Lemma 3.15, $[\eta_{x}]=\partial F([\gamma_{x}])=[\xi_{x}]$ and
$[\eta_{y}]=\partial F([\gamma_{y}])=[\xi_{y}]$. So by Lemma 2.5,

(17) $\displaystyle\sup_{u\in\mathbb{R}_{\geq
0}}\max\left\\{\,\overline{\eta_{x}(\rho(u)),\xi_{x}(\rho(u)))}\,,\,\overline{\eta_{y}(\rho(u)),\xi_{y}(\rho(u))}\,\right\\}\leq
D.$

Then we have

(18)
$\displaystyle\,\overline{\eta_{x}(\rho(T_{x})),\eta_{y}(\rho(T_{y}))}\,\leq$
$\displaystyle\,\overline{\eta_{x}(\rho(T_{x})),F(\gamma_{x}(\rho(T_{x})))}\,+\,\overline{F(\gamma_{x}(\rho(T_{x}))),F(\gamma_{y}(\rho(T_{y})))}\,$
$\displaystyle+\,\overline{F(\gamma_{y}(\rho(T_{y}))),\eta_{y}(\rho(T_{y}))}\,$
$\displaystyle\leq$
$\displaystyle\,\overline{F(\gamma_{x}(\rho(T_{x}))),F(\gamma_{y}(\rho(T_{y})))}\,+2(T+D).$

Set $T:=\min\\{T_{x},T_{y}\\}$. Then by Lemma 2.4

(19)
$\displaystyle\,\overline{\gamma_{x}(\rho(T_{x})),\gamma_{y}(\rho(T_{y}))}\,\leq$
$\displaystyle\,\overline{\gamma_{x}(\rho(T)),\gamma_{y}(\rho(T))}\,+\lambda\left\lvert\rho(T_{x})-\rho(T_{y})\right\rvert+k_{1}$
$\displaystyle\leq$ $\displaystyle
E(\,\overline{x,y}\,+\lambda\tilde{\theta}(\,\overline{x,y}\,)+k_{1})+D+\lambda(\tilde{\theta}(\,\overline{x,y}\,)+\tau)+k_{1}.$

The estimates (15), (18), (19) and the assumption that $F$ is a large scale
Lipschitz map imply $H$ is bornologous. ∎

######  Lemma 7.16.

The map $\mathop{\mathrm{rad^{\prime}}}\nolimits\partial F$ is coarsely
homotopic to the restriction $F|_{X^{\text{Vis}}}$.

###### Proof..

By Lemma 7.15, the map $H$ is a coarse map. For $(x,T_{x})\in Z^{\text{Vis}}$,
we have
$H(x,T_{x})=\eta_{x}(r(T_{x}^{\epsilon})^{1/\epsilon})=\mathop{\mathrm{rad^{\prime}}}\nolimits\partial
F$. It is enough to show that $H(-,0)$ is close to $F|_{X^{\text{Vis}}}$, that
is, $\sup_{x\in X^{\text{Vis}}}\,\overline{H(x,0),F(x)}\,<\infty$.

Let $x\in X^{\text{Vis}}$. Then $H(x,0)=\eta_{x}(\rho(T_{x}))$. Then by (16)
and (17) in the proof of Lemma 7.15, we have
$\,\overline{\eta_{x}(\rho(T_{x})),F(\gamma_{x}(T_{x}))}\,\leq T+D$. Therefore
$H(-,0)$ is close to $F|_{X^{\text{Vis}}}$. ∎

######  Proposition 7.17.

The map $\mathop{\mathrm{rad}}\nolimits\partial F$ is coarsely homotopic to
$F$.

###### Proof..

By Lemma 7.16, $\mathop{\mathrm{rad}}\nolimits\partial
F=\mathop{\mathrm{rad^{\prime}}}\nolimits\partial F\circ\varphi$ is coarsely
homotopic to the composite $F\circ\varphi$. In [7, Section 5.5], it is proved
that $\varphi$ is coarsely homotopic to the identity. Therefore
$\mathop{\mathrm{rad}}\nolimits\partial F$ is coarsely homotopic to $F$. ∎

## References

  * [1] Kyle Austin and Žiga Virk, _Higson compactification and dimension raising_ , Topology Appl. 215 (2017), 45–57. MR 3576439
  * [2] JÃ©rÃ©mie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, and Damian Osajda, _Helly groups_ , 2020, arXiv:2002.06895.
  * [3] Dominic Descombes and Urs Lang, _Convex geodesic bicombings and hyperbolicity_ , Geom. Dedicata 177 (2015), 367–384. MR 3370039
  * [4] J. Dydak and Ž. Virk, _Inducing maps between Gromov boundaries_ , Mediterr. J. Math. 13 (2016), no. 5, 2733–2752. MR 3554273
  * [5] Ryszard Engelking, _Dimension theory_ , North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978, Translated from the Polish and revised by the author, North-Holland Mathematical Library, 19. MR 0482697
  * [6] Tomohiro Fukaya, _Coarse dynamics and fixed-point theorem_ , Nagoya Math. J. 202 (2011), 1–13. MR 2804542 (2012h:55001)
  * [7] Tomohiro Fukaya and Shin-ichi Oguni, _A coarse Cartan–Hadamard theorem with application to the coarse Baum–Connes conjecture_ , J. Topol. Anal. 12 (2020), no. 3, 857–895. MR 4139217
  * [8] Thomas Haettel, Nima Hoda, and Harry Petyt, _The coarse helly property, hierarchical hyperbolicity, and semihyperbolicity_ , arXiv:2009.14053.
  * [9] Nigel Higson and John Roe, _On the coarse Baum-Connes conjecture_ , Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, Cambridge, 1995, pp. 227–254. MR 1388312 (97f:58127)
  * [10] Takahisa Miyata and Žiga Virk, _Dimension-raising maps in a large scale_ , Fund. Math. 223 (2013), no. 1, 83–97. MR 3125134
  * [11] Damian Osajda and Jingyin Huang, _Large-type Artin groups are systolic_ , Preprint.
  * [12] Damian Osajda and Piotr Przytycki, _Boundaries of systolic groups_ , Geom. Topol. 13 (2009), no. 5, 2807–2880. MR 2546621
  * [13] John Roe, _Lectures on coarse geometry_ , University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003. MR 2007488 (2004g:53050)
  * [14] Rufus Willett, _Band-dominated operators and the stable higson corona_ , PhD thesis, Penn State (2009).

Yuuhei Ezawa

Department of Mathematical Science, Tokyo Metropolitan University, Minami-
osawa Hachioji, Tokyo, 192-0397, Japan

E-mail address:

E-mail address<EMAIL_ADDRESS>

Tomohiro Fukaya

Department of Mathematical Science, Tokyo Metropolitan University, Minami-
osawa Hachioji, Tokyo, 192-0397, Japan

E-mail address:

E-mail address<EMAIL_ADDRESS>