Horoboundaries of coarsely convex spaces

Ikkei Sato Ikkei Sato Department of Mathematical Sciences Tokyo Metropolitan University Minami-osawa Hachioji Tokyo 192-0397 Japan sato-ikkei@tmu.ac.jp

Abstract.

A horoboundary is one of the attempts to compactify metric spaces, and is constructed using continuous functions on metric spaces. It is a concept that includes global information of metric spaces, and its correspondence with an ideal boundary constructed using geodesics has been studied in nonpositive curvature spaces such as CAT(0) spaces and geodesic Gromov hyperbolic spaces. We will introduce a certain correspondence between the horoboundary and the ideal boundary of coarsely convex spaces, which can be regarded as a generalization of spaces of nonpositive curvature.

Key words and phrases:

coarse geometry, coarsely convex space, horoboundary

1. Introduction

1.1. Introduction

Coarse geometry is the field of studying invariant properties of metric spaces under quasi-isometry. The geodesic Gromov hyperbolic spaces introduced by Gromov are invariant under quasi-isometry and have been studied as the coarse geometric analogues of Riemannian manifolds with negative sectional curvature. Subsequently, the construction of analogues in the metric geometry of Riemannian manifolds with nonpositive sectional curvature was advanced, and CAT(0) spaces and Busemann spaces were introduced. However, these spaces are not invariant under quasi-isometry. Coarsely convex spaces introduced by Fukaya and Oguni in [FO20] are coarse geometric analogues of simply connected Riemannian manifolds of nonpositive sectional curvature. They are a class that includes not only CAT(0) spaces and Busemann spaces, but also geodesic hyperbolic spaces, finite-dimensional systolic complexes, proper injective spaces. Furthermore, in coarsely convex spaces, as in other non-positive curvature spaces, ideal boundaries are defined by using (quasi-)geodesic rays, the coarse Cartan-Hadamard theorem, which states that the open cones defined on the ideal boundary and the coarsely convex spaces themselves are coarse homotopy equivalence, and coarsely convex spaces satisfy various properties that allow them to be regarded as generalizations of nonpositive curvature spaces.

The horoboundary was introduced by Gromov in [Gro81], as a way to compactify proper metric spaces. In particular, he constructed it as a concept that includes Busemann functions constructed using geodesic rays. The elements of the horoboundary are called horofunctions, and in CAT(0) spaces all horofunctions are Busemann functions. In particular, for the CAT(0) space, there exists a bijection between the set of Busemann functions and that of geodesic rays. That is, the horoboundary is homeomorphic to the ideal boundary.

So what about in other non-positive curvature spaces? This is not viable. For example, in geodesic hyperbolic spaces, by Webster and Winchester [WW03], it was shown that there exists a surjective continuous map from the horoboundary to the ideal boundary. However, Arosio, Fiacchi, Gontard and Guerini [AFGG22] gave an example of Gromov hyperbolic space whose horoboundary differs from the ideal boundary. Other similar results were shown in the Busemann spaces by Andreev [And08]. However, in [And18], Andreev introduced the cone metric $d_{c}$ for Busemann spaces $(X,d)$ , and he showed that the horoboundary of $(X,d_{c})$ is homeomorphic to the ideal boundary of $(X,d)$ .

We note that the cone metric defined by Andreev is similar to the Euclidean cone metric for open cones, and show the following results on coarsely convex spaces.

Theorem 1.1.

Let $X$ be a proper coarsely convex space and $o\in X$ be a base point of $X$ . Let $\partial_{o}X$ be the ideal boundary of $X$ with respect to the base point $o$ and let $\mathcal{O}\partial_{o}X$ be an open cone over $\partial_{o}X$ . Then, the horoboundary $\partial_{h}\mathcal{O}\partial_{o}X$ of the open cone $\mathcal{O}\partial_{o}X$ is homeomorphic to the ideal boundary $\partial_{o}X$ of $X$ .

We construct the cone metric $d_{c}$ on a coarsely convex space $X$ . Due to the coarse geometric nature of coarsely convex spaces, we need to modify the formulation of the horoboundary $\partial^{c}_{h}X$ of $X$ with the cone metric $d_{c}$ . When $X$ is a Busemann space, $d_{c}$ coincide with the one defined by Andreev in [And18], and $\partial^{c}_{h}X$ coincide with the one studied in [And18].

We compare the horoboundary with the ideal boundary. Since the ideal boundary of the coarsely convex space is the set of asymptotic classes of quasi-geodesic rays, we need to take a quotient by bounded functions. The quotient is called the reduced horoboundary denoted by $\partial^{c}_{h}X/\sim$ . For more detail, see Definition 6.9. We show that the ideal boundary coincides with the reduced horoboundary.

Theorem 1.2.

Let $(X,d)$ be a coarsely convex space and let $o\in X$ be a base point of $X$ . Let $d_{c}$ be the cone metric of $(X,d)$ . Let $\partial_{o}X$ be the ideal boundary of $X$ with respect to the base point $o$ and $\partial^{c}_{h}X/\sim$ be the reduced horoboundary of $(X,d_{c})$ . Then $\partial_{o}X$ and $\partial^{c}_{h}X/\sim$ are homeomorphic.

In geodesic Gromov hyperbolic spaces, by [WW03], the ideal boundary coincides with the reduced horoboundary in the usual sense. Theorem 1.2 generalize this fact to geodesic coarsely convex spaces in a sense.

Furthermore, in geodesic $(1,0)$ -coarsely convex spaces such as Busemann spaces, the reduced horoboundary with the cone metric and the horoboundary with the cone metric coincide, so Theorem 1.2 generalize a result in [And18] by Andreev.

1.2. Outline

In Section 2, we introduce coarsely convex spaces and give some examples according to [FO20, FM24].

In Section 3, we describe the construction of the ideal boundaries for coarsely convex spaces, and we discuss their properties.

In Section 4, we summarize some known facts on horoboundary

In Section 5, we define open cones, and we study the horoboundary of open cones. We give a proof of Theorem 1.1.

In Section 6, we define the cone metric for coarsely convex spaces, and we analyze the horoboundary of the coarsely convex space associated with the cone metric. We complete the proof of Theorem 1.2.

2. Coarsely convex space

Let $(X,d)$ be a metric space. A bicombing on $X$ is a map $\Gamma\colon X\times X\times[0,1]\rightarrow X$ such that for $x,y\in X$ , we have $\Gamma(x,y,0)=x$ , $\Gamma(x,y,1)=y$ .

Let $\lambda\geq 1$ and $k\geq 0$ be constants. A bicombing $\Gamma$ is a $(\lambda,k)$ -quasi-geodesic bicombing if

\frac{1}{\lambda}|t-s|d(x,y)-k\leq d(\Gamma(x,y,t),\Gamma(x,y,s))\leq\lambda|t-s|d(x,y)+k

holds for $t,s\in[0,1]$ . In particular, $\Gamma\colon X\times X\times[0,1]\rightarrow X$ is a geodesic-bicombing if $\lambda=1$ , $k=0$ is satisfied, that is,

d(\Gamma(x,y,t),\Gamma(x,y,s))=|t-s|d(x,y)

holds for $t,s\in[0,1]$ .

Definition 2.1.

Let $(X,d)$ be a metric space, and let $\lambda\geq 1$ , $k\geq 0$ , $E\geq 1$ , $C\geq 0$ be constants. Let $\theta\colon\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ be a non-decreasing function. A $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing on a metric space $(X,d)$ is a $(\lambda,k)$ -quasi-geodesic bicombing $\Gamma\colon X\times X\times[0,1]\rightarrow X$ with the following items (i) and (ii):

(i)

Let $x_{1},x_{2},y_{1},y_{2}\in X$ and let $a,b\in[0,1]$ . Let $y^{\prime}_{1}\coloneqq\Gamma(x_{1},y_{1},a)$ and $y^{\prime}_{2}\coloneqq\Gamma(x_{2},y_{2},b)$ . Then, for $c\in[0,1]$ , we have

d(\Gamma(x_{1},y_{1},ca),\Gamma(x_{2},y_{2},cb))\leq(1-c)Ed(x_{1},x_{2})+cEd(y^{\prime}_{1},y^{\prime}_{2})+C.

(ii)

Let $x_{1},x_{2},y_{1},y_{2}\in X$ . Then for $t,s\in[0,1]$ we have

|td(x_{1},y_{1})-sd(x_{2},y_{2})|\leq\theta(d(x_{1},x_{2})+d(\Gamma(x_{1},y_{1},t),\Gamma(x_{2},y_{2},s))).

A geodesic $(E,C)$ -coarsely convex bicombing is a geodesic bicombing $\Gamma$ satisfying item (i) in the above.

Remark 2.2.

If $\Gamma$ is a geodesic bicombing, then $\Gamma$ satisfies item (ii) in Definition 2.1 due to the triangle inequality.

Definition 2.3.

We say that $X$ is a coarsely convex space if there are constants $\lambda\geq 1$ , $k\geq 0$ , $E\geq 1$ , $C\geq 0$ and non-decreasing function $\theta\colon\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ such that $X$ admits a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing $\Gamma$ . In particular, $X$ is a geodesic coarsely $(E,C)$ -coarsely convex space if $X$ admits a geodesic $(E,C)$ -coarsely convex bicombing $\Gamma$ .

Example 2.4.

Let $V$ be a normed vector space. The bicombing of $V$ is given by affine lines. So $V$ is a geodesic $(1,0)$ -coarsely convex space.

Example 2.5.

We say that a geodesic space $(X,d)$ is a Busemann space if any two geodesics $\gamma_{1}\colon[0,a_{1}]\rightarrow X$ and $\gamma_{2}\colon[0,a_{2}]\rightarrow X$ satisfy the following inequality

d(\gamma_{1}(ta_{1}),\gamma_{2}(ta_{2}))\leq(1-t)d(\gamma_{1}(0),\gamma_{2}(0))+td(\gamma_{1}(a_{1}),\gamma_{2}(a_{2}))

for any $t\in[0,1]$ . It follows that a Busemann space $(X,d)$ is a unique geodesic space. So $X$ admits the canonical $(1,0)$ -coarsely convex bicombing. Then the Busemann space $(X,d)$ is a geodesic $(1,0)$ -coarsely convex space.

The following reparametrization is used to construct ideal boundaries in Section 3 and cone metrics in Section 6.

Definition 2.6.

Let $(X,d)$ be a metric space and let $\Gamma\colon X\times X\times[0,1]\rightarrow X$ be a $(\lambda,k)$ -quasi-geodesic bicombing on $X$ . A reparametrised bicombing of $\Gamma$ is a map $\texttt{rp}\Gamma\colon X\times X\times\mathbb{R}_{\geq 0}\rightarrow X$ defined by

\texttt{rp}\Gamma(x,y,t)\coloneqq\begin{cases}\Gamma\left(x,y,\frac{t}{d(x,y)}\right)&\text{if}\>t\leq d(x,y)\\ y&\text{if}\>t>d(x,y).\end{cases}

In particular, for any $x,y\in X$ , $\texttt{rp}\Gamma(x,y,-)|_{[0,d(x,y)]}\colon[0,d(x,y)]\rightarrow X$ is a $(\lambda,k)$ -quasi-geodesic connecting $x$ and $y$ . If $\Gamma$ is a geodesic bicombing, rp $\Gamma(x,y,-)|_{[0,d(x,y)]}$ is a geodesic connecting $x$ and $y$ .

3. Ideal boundary

For more details on the proof of the statements in this section, see [FM24, Section 4] and [FO20, Section 4].

Let $(X,d)$ be a coarsely convex space with a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing $\Gamma$ , and let $\texttt{rp}\Gamma$ be a reparametrised bicombing of $\Gamma$ . We fix $o\in X$ as the base point of $X$ .

Definition 3.1.

Set $k_{1}=\lambda+k$ , $D=2(1+E)k_{1}+C$ , and $D_{1}=2D+2$ . We define a product $(-\>|\>-)_{o}\colon X\times X\rightarrow\mathbb{R}_{\geq 0}$ by

(x\>|\>y)_{o}\coloneqq\min\{d(o,x),d(o,y),\sup\{t\in\mathbb{R}_{\geq 0}\>|\>d(\texttt{rp}\Gamma(o,x,t),\texttt{rp}\Gamma(o,y,t))\leq D_{1}\}\}.

Lemma 3.2.

Set $D_{2}\coloneqq E(D_{1}+2k)$ . For $x,y,z\in X$ , we have

(x\>|\>z)_{o}\geq\frac{1}{D_{2}}\min\{(x\>|\>y)_{o},(y\>|\>z)_{o}\}.

Following [FOY22], we construct the ideal boundary of coarsely convex space $X$ by a set of sequences in $X$ tending to infinity.

We let

S^{\infty}_{o}X\coloneqq\{\{x_{n}\}_{n\in\mathbb{N}}\>|\>\{x_{n}\}_{n\in\mathbb{N}}\text{ is a sequence such that }(x_{n}\>|\>x_{m})_{o}\rightarrow\infty\text{ as }n,m\rightarrow\infty\},

and we define the relation $\sim$ on $S^{\infty}_{o}X$ as follows. For every $\{x_{n}\}_{n\in\mathbb{N}},\{y_{n}\}_{n\in\mathbb{N}}\in S^{\infty}_{o}X$ , we define $\{x_{n}\}_{n\in\mathbb{N}}\sim\{y_{n}\}_{n\in\mathbb{N}}$ if and only if

(x_{n}\>|\>y_{n})_{o}\rightarrow\infty\text{ as }n\rightarrow\infty.

Then the relation $\sim$ is an equivalence relation on $S^{\infty}_{o}X$ .

Definition 3.3.

The ideal boundary of a coarsely convex space $X$ , denote by $\partial_{o}X$ is the quotient of the set $S^{\infty}_{o}X$ by the equivalence relation $\sim$ . We denote by $\bar{X}$ the disjoint union of $X$ and $\partial_{o}X$ . Namely,

\partial_{o}X\coloneqq S^{\infty}_{o}X/\sim,\quad\bar{X}\coloneqq X\cup\partial_{o}X.

For $x\in X$ and a sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in $X$ , we write $\{x_{n}\}_{n\in\mathbb{N}}\in x$ if $x_{n}=x$ for every $n\in\mathbb{N}$ . We extend $(-\>|\>-)_{o}\colon X\times X\times\rightarrow\mathbb{R}_{\geq 0}$ to the function $(-\>|\>-)_{o}\colon\bar{X}\times\bar{X}\rightarrow\mathbb{R}_{\geq 0}$ as follows

(x\>|\>y)_{o}\coloneqq\sup\{\liminf_{n\rightarrow\infty}(x_{n}\>|\>y_{n})_{o}\>|\>\{x_{n}\}_{n\in\mathbb{N}}\in x,\{y_{n}\}_{n\in\mathbb{N}}\in y\}

for any $x,y\in\bar{X}$ .

For $n\in\mathbb{N}$ , let

V_{n}\coloneqq\{(x,y)\in\bar{X}\times\bar{X}\>|\>(x\>|\>y)_{o}>n\}\cup\left\{(x,y)\in X\times X\>|\>d(x,y)<\frac{1}{n}\right\}.

Then $\{V_{n}\}_{n\in\mathbb{N}}$ is a base of a metrizable uniformly topology on $\bar{X}$ . For any $x\in\bar{X}$ , we define $V_{n}[x]\subset\bar{X}$ as follows

V_{n}[x]\coloneqq\{y\in\bar{X}\>|\>(x,y)\in V_{n}\}.

Then the family $\{V_{n}[x]\}_{n\in\mathbb{N}}$ is a fundamental system of neighborhoods of $x$ .

Proposition 3.4 ([FO20, Proposition 4.19.]).

For sufficiently small $\epsilon>0$ , there exists a constant $K\geq 1$ depending on $D,\theta(0)$ and $\epsilon$ , and there exists a metric $d_{\epsilon}$ on $\partial_{o}X$ such that, for all $x,y\in\partial_{o}X$ ,

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

Especially, $d_{\epsilon}$ is compatible with the topology of $\partial_{o}X$ , and the diameter of $(\partial_{o}X,d_{\epsilon})$ is less than or equal to 1.

We extend the reparametrised bicombing $\texttt{rp}\Gamma\colon X\times X\times\mathbb{R}_{\geq 0}\rightarrow X$ to $\texttt{rp}\bar{\Gamma}\colon X\times\bar{X}\times\mathbb{R}_{\geq 0}\rightarrow X$ .

Lemma 3.5 ([FM24, Lemma 4.8.]).

Suppose that $X$ is proper. Then there exists a map $\texttt{rp}\bar{\Gamma}\colon X\times\bar{X}\times\mathbb{R}_{\geq 0}\to X$ satisfying the following.

(i)

For $x,y\in X$ , we have $\texttt{rp}\bar{\Gamma}(x,y,-)=\texttt{rp}\Gamma(x,y,-)$ .
(ii)

For $(o,x)\in X\times\partial_{o}X$ , there exists a sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in $X$ such that the sequence of maps $\{\texttt{rp}\Gamma(o,x_{n},-)|_{\mathbb{N}}\colon\mathbb{N}\rightarrow X\}_{n\in\mathbb{N}}$ converges to $\texttt{rp}\bar{\Gamma}(o,x,-)|_{\mathbb{N}}$ pointwise. In particular, if $\Gamma$ is a geodesic bicombing, the sequence of maps $\{\texttt{rp}\Gamma(o,x_{n},-)\colon\mathbb{R}_{\geq 0}\rightarrow X\}_{n\in\mathbb{N}}$ converges to $\texttt{rp}\bar{\Gamma}(o,x,-)$ uniformly on compact sets.
(iii)

For $(o,x)\in X\times\partial_{o}X$ , we have

$(\text{rp}\bar{\Gamma}(o,x,t)\>|\>x)_{o}\rightarrow\infty\>(t\rightarrow\infty).$
(iv)

For $(o,x)\in X\times\partial_{o}X$ , the map $\texttt{rp}\bar{\Gamma}(o,x,-)\colon\mathbb{R}_{\geq 0}\rightarrow X$ is a $(\lambda,k_{1})$ -quasi geodesic, where $k_{1}\coloneqq\lambda+k$ . In particular, if $\Gamma$ is a geodesic bicombing, $\texttt{rp}\bar{\Gamma}(o,x,-)\colon\mathbb{R}_{\geq 0}\rightarrow X$ is a geodesic.

Definition 3.6.

We call the map $\texttt{rp}\bar{\Gamma}$ given in Lemma 3.4 an extended bicombing on $X\times\bar{X}$ corresponding to $\Gamma$ . For $(o,x)\in X\times\partial_{o}X$ , we abbreviate $\texttt{rp}\bar{\Gamma}(o,x,-)$ by $\gamma^{x}_{o}(-)$ .

Lemma 3.7 ([FM24, Lemma 4.10.]).

The extended bicombing $\texttt{rp}\bar{\Gamma}$ on $X\times\bar{X}$ satisfies the following

(i)

Let $(o_{1},x_{1}),(o_{2},x_{2})\in X\times\partial_{o}X$ and let $a,b\in\mathbb{R}_{\geq 0}$ . Set ${x_{1}}^{\prime}\coloneqq\texttt{rp}\bar{\Gamma}(o_{1},x_{1},a)$ and ${x_{2}}^{\prime}\coloneqq\texttt{rp}\bar{\Gamma}(o_{2},x_{2},b)$ . Then, for $c\in[0,1]$ , we have

d(\texttt{rp}\bar{\Gamma}(o_{1},x_{1},ca),\texttt{rp}\bar{\Gamma}(o_{2},x_{2},cb))\leq(1-c)Ed(o_{1},o_{2})+cEd({x_{1}}^{\prime},{x_{2}}^{\prime})+D

(ii)

We define a non-decreasing function $\tilde{\theta}(t)\coloneqq\theta(t+1)+1$ . Let $(o_{1},x_{1}),(o_{2},x_{2})\in X\times\partial_{o}X$ . Then for $t,s\in\mathbb{R}_{\geq 0}$ , we have

|t-s|\leq\tilde{\theta}(d(o_{1},o_{2})+d(\texttt{rp}\bar{\Gamma}(o_{1},x_{1},t),\texttt{rp}\bar{\Gamma}(o_{2},x_{2},s))).

Definition 3.8.

For $x,y\in\partial_{o}X$ , we define

(\gamma^{x}_{o}\>|\>\gamma^{y}_{o})_{o}\coloneqq\sup\{t\in\mathbb{R}_{\geq 0}\>|\>d(\gamma^{x}_{o}(t),\gamma^{y}_{o}(t))\leq D_{1}\}.

For $x\in\partial_{o}X$ and $p\in X$ , we define

(\gamma^{x}_{o}\>|\>p)_{o}\coloneqq\min\{d(o,p),\sup\{t\in\mathbb{R}_{\geq 0}\>|\>d(\Gamma(o,p,t),\gamma^{x}_{o}(t))\leq D_{1}\}\}.

Lemma 3.9 ([FM24, Lemma 4.12.]).

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

(1)

For $x,y\in\partial_{o}X$ , we have

$(\gamma^{x}_{o}\>|\>\gamma^{y}_{o})_{o}\leq(x\>|\>y)_{o}\leq\Omega(\gamma^{x}_{o}\>|\>\gamma^{y}_{o})_{o}.$

(2)

For $x,y,z\in\partial_{o}X$ , we have

(\gamma^{x}_{o}\>|\>\gamma^{y}_{o})_{o}\geq\Omega^{-1}\min\{(\gamma^{x}_{o}\>|\>\gamma^{y}_{o})_{o},(\gamma^{y}_{o}\>|\>\gamma^{z}_{o})_{o}\}.

(3)

For $x,y,z\in\bar{X}$ , we have

$(x\>|\>z)_{o}\geq\Omega^{-1}\min\{(x\>|\>y)_{o},(y\>|\>z)_{o}\}.$
(4)

Let $x,y\in\partial_{o}X$ . For all $t\in\mathbb{R}_{\geq 0}$ with $t\leq(\gamma^{x}_{o}\>|\>\gamma^{y}_{o})$ , we have

$d(\gamma^{x}_{o}(t),\gamma^{y}_{o}(t))\leq\Omega.$
(5)

Let $o\in X$ and let $x,y\in\bar{X}$ . If $\texttt{rp}\bar{\Gamma}(o,x,a)=\texttt{rp}\bar{\Gamma}(o,y,b)$ for some $a,b\in\mathbb{R}_{\geq 0}$ , then for all $t\in[0,\max\{a,b\}]$ we have

$d(\texttt{rp}\bar{\Gamma}(o,x,t),\texttt{rp}\bar{\Gamma}(o,y,t))\leq\Omega.$
(6)

Let $o\in X$ and $x\in\partial_{o}X$ . For $v\in X$ and $t\in[0,1]$ , we have

$(x\>|\>\Gamma(o,v,t))_{o}\geq\Omega^{-1}\min\{(x\>|\>v)_{o},td(o,v)\}.$

4. Horoboundary

4.1. Horoboundary

Let $(X,d)$ be a proper metric space and fix $o\in X$ as the base point. Also, let $C(X)$ be the set of all $\mathbb{R}$ -valued continuous functions on $X$ , and equip with the topology of uniform convergence on compact sets. It is a standard fact that the space $C(X)$ is Hausdorff.

We define $\phi\colon X\rightarrow C(X)$ as follows. For $x\in X$ , let

x\mapsto\phi_{x}\coloneqq d(-,x)-d(o,x).

Proposition 4.1.

The map $\phi$ is injective and continuous.

Proof..

The triangle inequality implies that $\phi_{x}(u)-\phi_{y}(u)\leq 2d(x,y)$ , for all $u,x,y\in X$ . The continuity of $\phi$ follows.

Let $x$ and $y$ be distinct points in $X$ such that $d(o,x)\geq d(o,y)$ . We have

	$\displaystyle\phi_{y}(x)-\phi_{x}(x)$	$\displaystyle=d(x,y)-d(o,y)-d(x,x)+d(o,x)$
		$\displaystyle\geq d(x,y),$

which shows that $\phi_{x}$ and $\phi_{y}$ are distinct. ∎

Definition 4.2.

Let $(X,d)$ be a proper metric space and fix $o\in X$ as the base point. Let $\phi\colon X\rightarrow C(X)$ be the map defined above. We define the horoboundary $\partial_{h}X$ of $(X,d)$ as follows

\partial_{h}X\coloneqq\operatorname{cl}\phi(X)\setminus\phi(X).

Here $\operatorname{cl}\phi(X)$ denotes the closure of $\phi(X)$ in $C(X)$ . Since $C(X)$ with the topology of uniform convergence on compact sets is Hausdorff, the horoboundary $\partial_{h}X$ is Hausdorff.

Lemma 4.3.

Let $(X,d)$ be a proper metric space. If the sequence of maps $\{\phi_{x_{n}}\}_{n\in\mathbb{N}}$ converges to $\xi\in\partial_{h}X$ , then only finitely many of the points of $\{x_{n}\}_{n\in\mathbb{N}}$ lie in any bounded subset of $X$ .

Proof..

If there is an infinite number of points of $\{x_{n}\}_{n\in\mathbb{N}}$ contained some closed ball $\bar{B}(o,R)$ , we can take a subsequence $\{x_{n(k)}\}_{k\in\mathbb{N}}$ of $\{x_{n}\}_{n\in\mathbb{N}}$ such that $\{x_{n(k)}\}_{k\in\mathbb{N}}$ converges to some point $\tilde{x}\in\bar{B}(o,R)$ . By hypothesis, a sequence of maps $\{\phi_{x_{n(k)}}\}_{k\in\mathbb{N}}$ converges to $\xi\in\partial_{h}X$ . On the other hand, by Proposition 4.1, $\{\phi_{x_{n(k)}}\}_{k\in\mathbb{N}}$ converges to $\phi_{\tilde{x}}$ . Therefore, we have $\xi=\phi_{\tilde{x}}$ . This contradicts to $\xi\notin\phi(X)$ .

∎

Proposition 4.4.

The horoboundary does not depend on the choice of the base point.

Proof..

We take $o,o^{\prime}\in X$ arbitrarily. We define $\phi\colon X\rightarrow C(X)$ and $\phi^{\prime}\colon X\rightarrow C(X)$ by

x\mapsto\phi_{x}(-)\coloneqq d(-,x)-d(o,x)\text{$\quad$ and $\quad$}x\mapsto\phi^{\prime}_{x}(-)\coloneqq d(-,x)-d(o^{\prime},x).

Then, we have $\phi^{\prime}_{x}=\phi_{x}-\phi_{x}(o^{\prime})\text{$\quad$ and $\quad$}\phi_{x}=\phi^{\prime}_{x}-\phi^{\prime}_{x}(o)$ for any $x\in X$ .

Let $\partial_{h}X\coloneqq\mathrm{cl}\phi(X)\setminus\phi(X)$ and $\partial^{\prime}_{h}X\coloneqq\mathrm{cl}\phi^{\prime}(X)\setminus\phi^{\prime}(X)$ . We define $F\colon\partial_{h}X\rightarrow C(X)$ by

\xi\mapsto\xi-\xi(o^{\prime}).

We show $F(\xi)\in\partial^{\prime}_{h}X$ for any $\xi\in\partial_{h}X$ . Let $\{x_{n}\}_{n\in\mathbb{N}}\subset X$ be a sequence such that $d(o,x_{n})\rightarrow\infty$ as $n\rightarrow\infty$ and a sequence of maps $\{\phi_{x_{n}}\}_{n\in\mathbb{N}}$ converges uniformly to $\xi$ on compact sets. For any $R>0$ , we have

	$\displaystyle\sup_{u\in\bar{B}(o,R)}\|F(\xi)(u)-\phi^{\prime}_{x_{n}}(u)\|$	$\displaystyle=\sup_{u\in\bar{B}(o,R)}\|\{\xi(u)-\xi(o^{\prime})\}-\{\phi_{x_{n}}(u)-\phi_{x_{n}}(o^{\prime})\}\|$
		$\displaystyle\leq\sup_{u\in\bar{B}(o,R)}\|\xi(u)-\phi_{x_{n}}(u)\|+\|\xi(o^{\prime})-\phi_{x_{n}}(o^{\prime})\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

So we have $F(\xi)\in\operatorname{cl}\phi^{\prime}(X)$ . We assume $F(\xi)\in\phi^{\prime}(X)$ . Then there is some $z\in X$ such that

F(\xi)=\xi-\xi(o^{\prime})=\phi^{\prime}_{z}=\phi_{z}-\phi_{z}(o^{\prime}).

Since $\xi(o)=\phi_{z}(o)=0$ , we have $\xi(o^{\prime})=\phi_{z}(o^{\prime})$ . Thus, we get $\xi=\phi_{z}$ , but this contradicts to $\xi\notin\phi(X)$ . Therefore, $\xi^{\prime}\notin\phi^{\prime}(X)$ and $\xi^{\prime}=F(\xi)\in\partial^{\prime}_{h}X$ . Thus, the map $F\colon\partial_{h}X\to\partial^{\prime}_{h}X$ is well-defined.

Similarly, we define $\tilde{F}\colon\partial^{\prime}_{h}X\to\partial_{h}X$ by

\xi^{\prime}\mapsto\xi^{\prime}-\xi(o).

We show that $\tilde{F}$ and $F$ are the inverse of each other. Since $\xi(o)=\xi^{\prime}(o^{\prime})=0$ , we have

	$\displaystyle\tilde{F}\circ F(\xi)=\tilde{F}(\xi-\xi(o^{\prime}))=(\xi-\xi(o^{\prime}))-(\xi(o)-\xi^{\prime}(o))=\xi$
	$\displaystyle F\circ\tilde{F}(\xi^{\prime})=F(\xi^{\prime}-\xi^{\prime}(o))=(\xi^{\prime}-\xi(o^{\prime}))-(\xi^{\prime}(o^{\prime})-\xi(o^{\prime}))=\xi^{\prime}.$

We show $F\colon\partial_{h}X\rightarrow\partial^{\prime}_{h}X$ is continuous. Let $\{\xi_{m}\}_{m\in\mathbb{N}}\subset\partial_{h}X$ be a sequence of maps such that $\xi_{m}$ converges uniformly to some $\xi\in\partial_{h}X$ on compact sets. For any $L>0$ , we have

	$\displaystyle\sup_{u\in\bar{B}(o,L)}\|F(\xi_{m})(u)-F(\xi)(u)\|$	$\displaystyle=\sup_{u\in\bar{B}(o,L)}\|\{\xi_{m}(u)-\xi_{m}(o^{\prime})\}-\{\xi(u)-\xi(o^{\prime})\}\|$
		$\displaystyle\leq\sup_{u\in\bar{B}(o,L)}\|\xi_{m}(u)-\xi(u)\|+\|\xi_{m}(o^{\prime})-\xi(o^{\prime})\|$
		$\displaystyle\rightarrow 0\text{ as }m\rightarrow\infty.$

Thus, a sequence of maps $\{F(\xi_{m})\}_{m\in\mathbb{N}}$ converges uniformly to $F(\xi)$ on compact sets. Therefore, $F\colon\partial_{h}X\rightarrow\partial^{\prime}_{h}X$ is continuous. In the same way, we can show $\tilde{F}\colon\partial^{\prime}_{h}X\rightarrow\partial_{h}X$ is continuous. Hence, $\partial_{h}X$ and $\partial^{\prime}_{h}X$ are homeomorphic. ∎

Example 4.5.

Let $(\mathbb{R}^{2},l^{1})$ be the space $\mathbb{R}^{2}$ with $l^{1}$ metric. Here, $l^{1}$ is defined as follows. For $(x_{1},y_{1}),\>(x_{2},y_{2})\in\mathbb{R}^{2}$

l^{1}((x_{1},y_{1}),(x_{2},y_{2}))=|x_{1}-x_{2}|+|y_{1}-y_{2}|.

Then horoboundary $\partial_{h}(\mathbb{R}^{2},l^{1})$ is homeomorphic to $\partial[-\infty,\infty]^{2}$ .

Proposition 4.6.

If $X$ is a geodesic space, then $\phi\colon X\rightarrow C(X)$ is a topological embedding.

Proof..

First, we show the following claim.

Claim.

Let $\{x_{n}\}_{n\in\mathbb{N}}$ be a sequence on $X$ such that $d(o,x_{n})\rightarrow\infty$ and $\phi_{x_{n}}$ converges to $\xi\in\operatorname{cl}\phi(X)$ uniformly on compact sets. Then we show that there is no $y\in X$ such that $\xi=\phi_{y}$ .

We prove this by contradiction. That is, suppose there is some $y\in X$ and $\phi_{x_{n}}$ converges uniformly on a compact set to $\phi_{y}$ .

Since $d(y,x_{n})\geq d(o,x_{n})-d(o,y)$ from the triangle inequality, $d(y,x_{n})$ diverges to infinity. We let $\gamma_{n}\colon[0,d(y,x_{n})]\rightarrow X$ be a geodesic connecting $y$ and $x_{n}$ . That is, $\gamma_{n}(0)=y$ and $\gamma_{n}(d(y,x_{n}))=x_{n}$ . We take $x^{\prime}_{n}$ to be a point on the image $\gamma_{n}([0,d(y,x_{n})])$ of the geodesic $\gamma_{n}$ , satisfying $d(y,x^{\prime}_{n})=d(o,y)+1$ . For any $n\in\mathbb{N}$ , $x^{\prime}_{n}$ is contained in $\bar{B}(y,d(o,y)+1)$ . Since $\phi_{x_{n}}$ converges uniformly to $\phi_{y}$ on compact sets, the following holds.

	$\displaystyle\|\phi_{y}(x^{\prime}_{n})-\phi_{x_{n}}(x^{\prime}_{n})\|$	$\displaystyle\leq\sup_{u\in\bar{B}(y,d(o,y)+1)}\|\phi_{y}(u)-\phi_{x_{n}}(u)\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

For any $n\in\mathbb{N}$ , since $x_{n},x^{\prime}_{n},y\in X$ lie on the geodesic $\gamma_{n}$ , we have

	$\displaystyle\phi_{y}(x^{\prime}_{n})-\phi_{x_{n}}(x^{\prime}_{n})$	$\displaystyle=\{d(x^{\prime}_{n},y)-d(o,y)\}-\{d(x^{\prime}_{n},x_{n})-d(o,x_{n})\}$
		$\displaystyle=1-d(x^{\prime}_{n},x_{n})+d(o,x_{n})$
		$\displaystyle\geq 1-d(x^{\prime}_{n},x_{n})+d(x_{n},y)-d(o,y)$
		$\displaystyle=1+d(y,x^{\prime}_{n})-d(o,y)$
		$\displaystyle=2.$

This contradicts the results above, so the claim holds. Now we show that if a sequence of maps $\{\phi_{x_{n}}\}_{n\in\mathbb{N}}$ converges uniformly to $\phi_{x}$ $(x\in X)$ on compact sets, a sequence $\{x_{n}\}_{n\in\mathbb{N}}$ converges to $x$ .

By the claim, the set $\{x_{n}\}$ is bounded. Set

R\coloneqq\max\{d(o,x),\,\sup_{n\in\mathbb{N}}d(o,x_{n})\}.

Since $\phi_{x_{n}}$ converges pointwise to $\phi_{x}$ , we have

	$\displaystyle\|\phi_{x_{n}}(x)-\phi_{x}(x)\|$	$\displaystyle=\|(d(x,x_{n})-d(o,x_{n}))-(d(x,x)-d(o,x))\|$
		$\displaystyle=\|d(x,x_{n})-d(o,x_{n})+d(o,x)\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

On the other hand, $\phi_{x_{n}}$ converges uniformly to $\phi_{x}$ on compact sets, so we obtain

	$\displaystyle\|\phi_{x_{n}}(x_{n})-\phi_{x}(x_{n})\|$	$\displaystyle\leq\sup_{u\in\bar{B}(o,R)}\|\phi_{x_{n}}(u)-\phi_{x}(u)\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

Thus, we obtain the following

	$\displaystyle\|\phi_{x_{n}}(x_{n})-\phi_{x}(x_{n})\|$	$\displaystyle=\|d(x_{n},x_{n})-d(o,x_{n})-d(x_{n},x)+d(o,x)\|$
		$\displaystyle=\|-d(o,x_{n})-d(x_{n},x)+d(o,x)\|$
		$\displaystyle=\|d(o,x_{n})+d(x_{n},x)-d(o,x)\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

By triangle inequality, we have

	$\displaystyle 2d(x_{n},x)$	$\displaystyle\leq\|d(x,x_{n})-d(o,x_{n})+d(o,x)\|$
		$\displaystyle\phantom{+\quad}+\|d(o,x_{n})+d(x_{n},x)-d(o,x)\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

∎

4.2. Reduced horoboundary

Definition 4.7.

Let $(X,d)$ be a proper metric space and let $\partial_{h}X$ be the horoboundary. We define that two functions $\xi$ and $\eta$ in $\partial_{h}X$ are equivalent, denoted by $\xi\sim\eta$ , if

\sup_{x\in X}|\xi(x)-\eta(x)|<\infty.

This determines an equivalence relation on $\partial_{h}X$ . A reduced horoboundary of $X$ , denoted by $\partial_{h}X/\sim$ , is the quotient of $\partial_{h}X$ by the equivalence relation $\sim$ with the quotient topology.

In general, a geodesic Gromov hyperbolic space does not necessarily coincide with its horoboundary and its ideal boundary.

On the other hand, in geodesic Gromov hyperbolic spaces, Webster and Winchester [WW03] showed the following.

Theorem 4.8 ([WW03, Theorem 4.5.]).

Let $(X,d)$ be a proper geodesic Gromov hyperbolic space with the base point $o\in X$ . Let $\partial_{o}X$ be an ideal boundary with respect to the base point $o$ and let $\partial_{h}X$ be a horoboundary of $X$ . Then there is a natural continuous quotient map from $\partial_{h}X$ onto $\partial_{o}X$ .

It is well known that the reduced horoboundary coincides with the ideal boundary in proper geodesic Gromov hyperbolic spaces as a corollary of Theorem 4.8.

The same result does not hold for coarsely convex spaces. For example, we have seen that all normed spaces are geodesic coarsely convex spaces in Example 2.4. There are examples such as the following. Here, we remark that an ideal boundary of a proper coarsely convex space is compact and metrizable

Example 4.9.

The reduced horoboundary of $(\mathbb{R}^{2},l^{1})$ is not Hausdorff.

5. Open cone and horoboundary

Definition 5.1.

Let $(Y,d_{Y})$ be a compact metric space with diameter less than or equal to 1. Let the open cone over $Y$ be the following quotient topological space.

\mathcal{O}Y=([0,\infty)\times Y)/(\{0\}\times Y)

For $(t,y)\in[0,\infty]\times Y$ , we denote by $tx$ the point in $\mathcal{O}Y$ represented by $(t,y)$ . We define a metric $d_{\mathcal{O}Y}$ on $\mathcal{O}Y$ by

d_{\mathcal{O}Y}(tx,sy)\coloneqq|t-s|+\min\{t,s\}d_{Y}(x,y).

Set $o\coloneqq 0y\in\mathcal{O}Y$ as the base point of $\mathcal{O}Y$ .

Proposition 5.2.

Let $(Y,d_{Y})$ be a compact metric space with diameter less than or equal to 1 and $\mathcal{O}Y$ be an open cone over $Y$ . Then $\mathcal{O}Y$ is a proper metric space.

Proof..

Let $K\subset\mathcal{O}Y$ be a closed bounded subset. There exists $R\geq 0$ such that $K$ is contained in $[0,R]\times Y/(\{0\}\times Y)$ . Since $[0,R]\times Y$ is compact, the quotient $[0,R]\times Y/(\{0\}\times Y)$ is compact. Therefore, $K$ is compact. ∎

As in the following example, $\mathcal{O}Y$ is not necessarily a geodesic space.

Example 5.3.

We consider a two-points set $Y=\{a,b\}$ and define $d\colon Y\times Y\rightarrow\mathbb{R}_{\geq 0}$ to satisfy $d(a,a)=0,d(b,b)=0,d(a,b)=d(b,a)=1$ . Then $d$ is a metric on $Y$ , and $(Y,d)$ is a compact metric space with diameter less than or equal to 1.

Then $\mathcal{O}Y$ is not a geodesic space. We prove this by contradiction.

We assume there exists some geodesic $\gamma\colon[0,1]\rightarrow\mathcal{O}Y$ such that $\gamma(0)=1a$ and $\gamma(1)=1b$ . In this case, there is a map $f\colon[0,1]\rightarrow\mathbb{R}_{\geq 0}$ and a map $g\colon[0,1]\rightarrow Y$ such that $\gamma(t)=f(t)g(t)$ for any $t\in[0,1]$ .

We show $f\colon[0,1]\rightarrow\mathbb{R}_{\geq 0}$ is a continuous function. By the continuity of $\gamma\colon[0,1]\rightarrow\mathcal{O}Y$ , for any $t_{1}\in[0,1]$ and any $\epsilon>0$ , there exists some $\delta>0$ and for any $t\in[0,1]$ with $|t-t_{1}|<\delta$ , we have

d_{\mathcal{O}Y}(\gamma(t),\gamma(t_{1}))=|f(t)-f(t_{1})|+\min\{f(t),f(t_{1})\}d(g(t),g(t_{1}))<\epsilon.

So we have $|f(t)-f(t_{1})|<\epsilon$ .

By the compactness of $[0,1]$ and the continuity of $f$ , there exists minimum value $m\coloneqq\min_{t\in[0,1]}f(t)$ . Then, the minimum value $m$ is equal to 0. We prove this by contradiction. Since $f(t)\geq 0$ for all $t\in[0,1]$ , we may assume that $m>0$ . From the continuity of $\gamma\colon[0,1]\rightarrow\mathcal{O}Y$ , for any ${t_{1}}^{\prime}\in[0,1]$ and any $\epsilon^{\prime}>0$ , there exists some $\delta^{\prime}>0$ and for any $t^{\prime}\in[0,1]$ with $|t^{\prime}-{t_{1}}^{\prime}|<\delta$ , we have

d_{\mathcal{O}Y}(\gamma(t^{\prime}),\gamma({t_{1}}^{\prime}))=|f(t^{\prime})-f({t_{1}}^{\prime})|+\min\{f(t^{\prime}),f({t_{1}}^{\prime})\}d_{Y}(g(t^{\prime}),g({t_{1}}^{\prime}))<m\epsilon.

So we have

md_{Y}(g(t^{\prime}),g({t_{1}}^{\prime}))<m\epsilon.

By the hypothesis of $m>0$ , we get

d_{Y}(g(t^{\prime}),g({t_{1}}^{\prime}))<\epsilon.

Therefore, $g\colon[0,1]\rightarrow Y$ is a continuous map. Since $[0,1]$ is connected and $Y$ is discrete, the map $g\colon[0,1]\rightarrow Y$ is a constant map. We assume that $g(t)=1a$ for any $t\in[0,1]$ . Then, we have

	$\displaystyle 0=d_{\mathcal{O}Y}(\gamma(1),\gamma(1))$	$\displaystyle=d_{\mathcal{O}Y}(1b,f(1)g(1))=d_{\mathcal{O}Y}(1b,f(1)a)$
		$\displaystyle=\|1-f(1)\|+\min\{1,f(1)\}d_{Y}(a,b)$
		$\displaystyle\geq\min\{1,m\}>0.$

This is contradiction. So we have $m=0$ . In particular, there exists some $t_{0}\in[0,1]$ such that $f(t_{0})=0$ . It follows that $\gamma(t_{0})=f(t_{0})g(t_{0})=o$ . Since $1b\neq o$ , we have $t_{0}\neq 1$ . On the other hand, since $\gamma\colon[0,1]\rightarrow\mathcal{O}Y$ is a geodesic, we have $t_{0}=d_{\mathcal{O}Y}(\gamma(0),\gamma(t_{0}))=d_{\mathcal{O}Y}(1a,o)=1$ . This is a contradiction.

We will show that $\mathcal{O}Y$ can be topologically embedded in $C(\mathcal{O}Y)$ . As stated in the above, $\mathcal{O}Y$ is not necessarily geodesic spaces. So, we cannot apply directly Proposition 4.6.

Proposition 5.4.

Let $(Y,d_{Y})$ be a compact metric space with diameter less than or equal to 1 and $\mathcal{O}Y$ be an open cone over $Y$ . Let $C(\mathcal{O}Y)$ be the space of continuous functions on $\mathcal{O}Y$ equipped with the topology of uniform convergence on compact sets. For each $sx\in\mathcal{O}Y$ , we consider the function $\phi_{sx}\colon\mathcal{O}Y\rightarrow\mathbb{R}$ as following,

	$\displaystyle\phi_{sx}(ty)$	$\displaystyle\coloneqq d_{\mathcal{O}Y}(ty,sx)-d_{\mathcal{O}Y}(o,sx)$
		$\displaystyle=d_{\mathcal{O}Y}(ty,sx)-s$

Then the map $\phi\colon\mathcal{O}Y\rightarrow C(\mathcal{O}Y),sx\mapsto\phi_{sx}$ is a topological embedding.

Proof..

Let $\{z_{n}\}_{n\in\mathbb{N}}=\{t_{n}y_{n}\}_{n\in\mathbb{N}},(t_{n}\in\mathbb{R}_{\geq 0},\>y_{n}\in Y)$ be a sequence in $\mathcal{O}Y$ escaping to infinity, that is $t_{n}\rightarrow\infty$ .

We claim that no subsequence of $\phi_{z_{n}}$ converges to a function $\phi_{z}$ with $z\coloneqq ty\in\mathcal{O}Y$ . Suppose contrarily, there exists a subsequence of $\phi_{z_{n}}$ which converges to $\phi_{z}$ with $z=ty\in\mathcal{O}Y$ . By replacing the subsequence, we suppose that $\phi_{z_{n}}$ converges to $\phi_{z}$ .

First we consider the case $t=0$ .

We have $\phi_{z}(1y)=\phi_{o}(1y)=1$ . On the other hand, for any $n\in\mathbb{N}$ with $t_{n}\geq 1$ , we have $\phi_{z_{n}}(1y)=-1+d_{Y}(y_{n},y)\leq 0$ . This contradicts that $\phi_{z_{n}}$ converges pointwise to $\phi_{o}$ .

Next we consider the case $t\neq 0$ .

Let $R>0$ be a constant that satisfies $R>t+1$ . Set $x=(t+1)y\in\mathcal{O}Y$ .

Since the topology of $C(\mathcal{O}Y)$ is given by the uniform convergence on compact sets, $\phi_{z_{n}}$ converges uniformly on $\bar{B}(0,R)$ to $\phi_{z}$ . By taking $n\in\mathbb{N}$ large enough, we have

	$\displaystyle\|\phi_{z_{n}}(z)-\phi_{z}(z)\|$	$\displaystyle=\|(d_{\mathcal{O}Y}(z_{n},z)-d_{\mathcal{O}Y}(0,z_{n}))-(d_{\mathcal{O}Y}(z,z)-d_{\mathcal{O}Y}(0,z))\|$
		$\displaystyle=\|(\|t_{n}-t\|+\min\{t_{n},t\}d_{Y}(y_{n},y)-t_{n})-(-t)\|$
		$\displaystyle=td_{Y}(y_{n},y)\rightarrow 0\text{ as }n\rightarrow\infty.$

Since $t\neq 0$ , we have $d_{Y}(y_{n},y)\rightarrow 0\text{ as }n\rightarrow\infty$ . On the other hand, for $n\in\mathbb{N}$ with $t_{n}\geq t+1$ ,

	$\displaystyle\|\phi_{z_{n}}(x)-\phi_{z}(x)\|$
	$\displaystyle=\|(\|t_{n}-(t+1)\|+\min\{t_{n},t\}d_{Y}(y_{n},y)-t_{n})-(1+td_{Y}(y,y)-t)\|$
	$\displaystyle=\|-2+td_{Y}(y_{n},y)\|\rightarrow 2\neq 0\text{ as }n\rightarrow\infty.$

This contradicts the fact that $\phi_{z_{n}}$ converges pointwise to $\phi_{z}$ . This completes the proof of the claim.

The rest of the proof can be shown as in Proposition 4.6.

∎

Proposition 5.5.

Let $(Y,d_{Y})$ be a compact metric space with diameter less than or equal to 1, and let $\mathcal{O}Y$ be an open cone over $Y$ . Let $\{t_{n}y_{n}\}_{n\in\mathbb{N}}$ be a sequence satisfying $t_{n}\rightarrow\infty$ on $\mathcal{O}Y$ . If $\phi_{t_{n}y_{n}}\in C(\mathcal{O}Y)$ converges uniformly on compact sets to $\xi\in\mathrm{cl}\{\phi_{s}x\>|\>sx\in\mathcal{O}Y\}$ , then $y_{n}$ converges to some point on $Y$ .

Proof..

Since $Y$ is compact, it is enough to show that $\{y_{n}\}_{n\in\mathbb{N}}$ is a Cauchy sequence.

Let $m$ and $n$ be arbitrary natural numbers. Since $\phi_{t_{n}y_{n}}$ converges uniformly to $\xi\in\partial_{h}\mathcal{O}Y$ on compact sets, we have

	$\displaystyle\|\phi_{t_{m}y_{m}}(1y_{n})-\phi_{t_{n}y_{n}}(1y_{n})\|$	$\displaystyle\leq\|\phi_{t_{m}y_{m}}(1y_{n})-\xi(1y_{n})\|+\|\phi_{t_{n}y_{n}}(1y_{n})-\xi(1y_{n})\|$
		$\displaystyle\leq\sup_{u\in\bar{B}(o,1)}\|\phi_{t_{m}y_{m}}(u)-\xi(u)\|+\sup_{u\in\bar{B}(o,1)}\|\phi_{t_{n}y_{n}}(u)-\xi(u)\|$
		$\displaystyle\rightarrow 0\text{ as }m,n\rightarrow\infty.$

On the other hand, by taking $m$ and $n$ large enough, we have

	$\displaystyle\|\phi_{t_{m}y_{m}}(1y_{n})-\phi_{t_{n}y_{n}}(1y_{n})\|$	$\displaystyle=\|d_{\mathcal{O}Y}(1y_{n},t_{m}y_{m})-d_{\mathcal{O}Y}(o,t_{m}y_{m})-d_{\mathcal{O}Y}(1y_{n},t_{n}y_{n})+d_{\mathcal{O}Y}(o,t_{n}y_{n})\|$
		$\displaystyle=\|(t_{m}-1)+d_{Y}(y_{m},y_{n})-t_{m}-(t_{n}-1)-d_{Y}(y_{n},y_{n})+t_{n}\|$
		$\displaystyle=d_{Y}(y_{m},y_{n}).$

So we have $d_{Y}(y_{m},y_{n})\rightarrow 0\text{ as }m,n\rightarrow\infty$ . ∎

Proposition 5.6.

Let $(Y,d_{Y})$ be a compact metric space with diameter less than or equal to 1. Let $(\mathcal{O}Y,d_{\mathcal{O}Y})$ be an open cone over $Y$ . Then $Y$ and $\partial_{h}\mathcal{O}Y$ are homeomorphic.

Proof..

We define $F\colon Y\rightarrow C(\mathcal{O}Y)$ as follows,

y\mapsto(F(y)(sx)\coloneqq-s+sd(x,y)).

For any $y\in Y$ , we show $F(y)\in\partial_{h}\mathcal{O}Y$ .

We observe that a sequence $\{\phi_{ny}\}_{n\in\mathbb{N}}$ converges to $F(y)$ . We fix $R>0$ . For all $n>R$ , we have

	$\displaystyle\sup_{sx\in\bar{B}(o,R)}\|\phi_{ny}(sx)-F(y)(sx)\|$
	$\displaystyle=\sup_{sx\in\bar{B}(o,R)}\|\{d_{\mathcal{O}Y}(sx,ny)-d_{\mathcal{O}Y}(o,ny)\}-\{-s+sd_{Y}(x,y)\}\|$
	$\displaystyle=\sup_{sx\in\bar{B}(o,R)}\|\{n-s+sd_{Y}(x,y)-n\}-\{-s+sd_{Y}(x,y)\}\|$
	$\displaystyle=0.$

Thus, the sequence of maps $\{\phi_{ny}\}_{n\in\mathbb{N}}$ converges uniformly to $F(y)$ on compact sets. Since the sequence $\{ny\}_{n\in\mathbb{N}}$ is unbounded, we have $F(y)=\lim_{n\rightarrow\infty}\phi_{ny}\in\partial_{h}\mathcal{O}Y$ by Proposition 5.4.

We show $F\colon Y\rightarrow\partial_{h}\mathcal{O}Y$ is continuous.

We take a sequence $\{y_{n}\}_{n\in\mathbb{N}}\subset Y$ such that $y_{n}$ converges to some $y\in Y$ . We take $L>0$ arbitrarily. Then we have

	$\displaystyle\sup_{sx\in\bar{B}(o,L)}\|F(y_{n})(sx)-F(y)(sx)\|$
	$\displaystyle=\sup_{sx\in\bar{B}(o,L)}\|\{-s+sd_{Y}(x,y_{n})\}-\{-s+sd_{Y}(x,y)\}\|$
	$\displaystyle\leq\sup_{sx\in\bar{B}(o,L)}sd_{Y}(y_{n},y)$
	$\displaystyle\leq Ld_{Y}(y_{n},y)\rightarrow 0\text{ as }n\rightarrow\infty.$

Thus, a sequence of maps $\{F(y_{n})\}_{n\in\mathbb{N}}$ converges uniformly to $F(y)$ on compact sets, so $F\colon Y\rightarrow\partial_{h}\mathcal{O}Y$ is continuous.

We show $F\colon Y\rightarrow\partial_{h}\mathcal{O}Y$ is injective. For any $y,z\in Y$ , we assume $F(y)=F(z)$ . Then we have

F(y)(1z)=-1+d_{Y}(y,z)=-1=F(z)(1z).

Therefore, we have $d_{Y}(y,z)=0$ , that is, $F\colon Y\rightarrow\partial_{h}\mathcal{O}Y$ is injective.

We show $F\colon Y\rightarrow\partial_{h}\mathcal{O}Y$ is surjective. We take $\xi\in\partial_{h}\mathcal{O}Y$ arbitrarily. Then, there is a sequence $\{t_{n}y_{n}\}_{n\in\mathbb{N}}\subset\mathcal{O}Y$ such that $t_{n}\rightarrow\infty$ as $n\rightarrow\infty$ and a sequence of maps $\{\phi_{t_{n}y_{n}}\}_{n\in\mathbb{N}}$ converges uniformly to $\xi$ on compact sets. By Proposition 5.5, the sequence $\{y_{n}\}_{n\in\mathbb{N}}$ converges to some point $y\in Y$ . For any $sx\in\mathcal{O}Y$ , we have

	$\displaystyle\xi(sx)$	$\displaystyle=\lim_{n\rightarrow\infty}\phi_{t_{n}y_{n}}(sx)$
		$\displaystyle=\lim_{n\rightarrow\infty}d_{\mathcal{O}Y}(sx,t_{n}y_{n})-d_{\mathcal{O}Y}(o,t_{n}y_{n})$
		$\displaystyle=\lim_{n\rightarrow\infty}t_{n}-s+sd_{Y}(x,y_{n})-t_{n}$
		$\displaystyle=-s+sd(x,y)$
		$\displaystyle=F(y)(sx).$

Therefore, we have $\xi=F(y)$ , that is, $F\colon Y\rightarrow\partial_{h}\mathcal{O}Y$ is surjective. Since $Y$ is compact and $\partial_{h}\mathcal{O}Y$ is Hausdorff, $F\colon Y\rightarrow\partial_{h}\mathcal{O}Y$ is a homeomorphism. ∎

By Proposition 3.4, an ideal boundary of a proper coarsely convex space is a compact metric space with its diameter if less than or equal to 1. Applying Proposition 5.6, we obtain Theorem 1.1.

6. Cone metric and horoboundary

Andreev defined a cone metric for Busemann spaces [And18]. In this section, we construct a cone metric for coarsely convex spaces. Then we construct the horoboundary using the cone metric.

6.1. Cone metric

For $t\in\mathbb{R}$ , we denote by $\lfloor t\rfloor$ the greatest integer less than or equal to $t$ .

Definition 6.1.

Let $(X,d)$ be a $(\lambda,k,E,C,\theta)$ -coarsely convex space with a base point $o\in X$ . Let $\Gamma$ be a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing on $X$ . For given points $x,y\in X$ . The cone metric $d_{c}(x,y)$ is defined by

	$\displaystyle d_{c}(x,y)\coloneqq$	$\displaystyle\|d(o,x)-d(o,y)\|$
		$\displaystyle\>+d(\texttt{rp}\Gamma(o,x,\lfloor\min\{d(o,x),d(o,y)\}\rfloor),\texttt{rp}\Gamma(o,y,\lfloor\min\{d(o,x),d(o,y)\}\rfloor)).$

By the definition of the cone metric, for any $x\in X$ , we have $d_{c}(o,x)=d(o,x)$ . Also, the cone metric is non-negative, symmetric.

Remark 6.2.

If $(X,d)$ is a geodesic $(E,C)$ -coarsely convex space with a geodesic $(E,C)$ -coarsely convex bicombing $\Gamma$ and a base point $o\in X$ , we define cone metric as follows

	$\displaystyle d_{c}(x,y)\coloneqq$	$\displaystyle\|d(o,x)-d(o,y)\|$
		$\displaystyle\>+d(\texttt{rp}\Gamma(o,x,\min\{d(o,x),d(o,y)\}),\texttt{rp}\Gamma(o,y,\min\{d(o,x),d(o,y)\}))$

for any $x,y\in X$ . From Example 2.5, a Busemann space is a geodesic $(1,0)$ -coarsely convex space. Then, the cone metric defined above coincides with the definition given by Andreev in [And18]. Furthermore, Definition 6.1 is an extension of the cone metric defined above to quasi-geodesic spaces.

Lemma 6.3.

Let $(X,d)$ be a $(\lambda,k,E,C,\theta)$ -coarsely convex space with $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing $\Gamma$ . For any three points $x,y,z\in X$ , we have

d_{c}(x,z)\leq\lambda Ed_{c}(x,y)+\lambda Ed_{c}(y,z)+4\lambda+2k+C.

Proof..

We consider three points $x,y,z\in X$ satisfy $d(o,x)\leq d(o,y)\leq d(o,z)$ . Since $\Gamma$ is a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing, we have

	$\displaystyle d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))$
	$\displaystyle=d\left(\Gamma\left(o,y,\frac{\lfloor d(o,x)\rfloor}{d(o,y)}\right),\Gamma\left(o,z,\frac{\lfloor d(o,x)\rfloor}{d(o,z)}\right)\right)$
	$\displaystyle\leq\frac{\lfloor d(o,x)\rfloor}{\lfloor d(o,y)\rfloor}Ed\left(\Gamma\left(o,y,\frac{\lfloor d(o,y)\rfloor}{d(o,y)}\right),\Gamma\left(o,z,\frac{\lfloor d(o,y)\rfloor}{d(o,z)}\right)\right)+C$
	$\displaystyle\leq Ed(\texttt{rp}\Gamma(o,y,\lfloor d(o,y)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,y)\rfloor))+C.$

Thus, we have the following

	$\displaystyle d_{c}(x,y)$	$\displaystyle=d(o,y)-d(o,x)+d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor))$
		$\displaystyle\leq d(o,z)-d(o,x)+d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))$
		$\displaystyle=d_{c}(x,z)+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))$
		$\displaystyle\leq d_{c}(x,z)+d(o,z)-d(o,y)+Ed(\texttt{rp}\Gamma(o,y,\lfloor d(o,y)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,y)\rfloor))+C$
		$\displaystyle\leq Ed_{c}(x,z)+Ed_{c}(y,z)+C.$

Similarly, we have

	$\displaystyle d_{c}(x,z)$	$\displaystyle=d(o,z)-d(o,x)+d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))$
		$\displaystyle\leq d(o,z)-d(o,x)+d(o,y)-d(o,y)+d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor))$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))$
		$\displaystyle=d_{c}(x,y)+d(o,z)-d(o,y)+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))$
		$\displaystyle\leq d_{c}(x,y)+d(o,z)-d(o,y)+Ed(\texttt{rp}\Gamma(o,y,\lfloor d(o,y)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,y)\rfloor))+C$
		$\displaystyle\leq Ed_{c}(x,y)+Ed_{c}(y,z)+C.$

Since $\Gamma$ is a $(\lambda,k)$ -quasi geodesic bicombing, we have

	$\displaystyle d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,y,\lfloor d(o,y)\rfloor))$	$\displaystyle\leq\lambda\left(\frac{\lfloor d(o,y)\rfloor}{d(o,y)}-\frac{\lfloor d(o,x)\rfloor}{d(o,y)}\right)d(o,y)+k$
		$\displaystyle\leq\lambda(d(o,y)-d(o,x))+2\lambda+k.$

Similarly, we have

	$\displaystyle d(\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,y)\rfloor))$	$\displaystyle\leq\lambda\left(\frac{\lfloor d(o,y)\rfloor}{d(o,z)}-\frac{\lfloor d(o,x)\rfloor}{d(o,z)}\right)d(o,z)+k$
		$\displaystyle\leq\lambda(d(o,y)-d(o,x))+2\lambda+k.$

So we have

	$\displaystyle d_{c}(y,z)$	$\displaystyle=d(o,z)-d(o,y)+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,y)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,y)\rfloor))$
		$\displaystyle\leq d(o,z)-d(o,y)+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,y,\lfloor d(o,y)\rfloor))$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor))$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,y)\rfloor))$
		$\displaystyle\leq d(o,z)-d(o,y)+\lambda(d(o,y)-d(o,x))+2\lambda+k$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor))$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))+\lambda(d(o,y)-d(o,x))+2\lambda+k$
		$\displaystyle\leq\lambda(d(o,z)-d(o,y))+\lambda(d(o,y)-d(o,x))+2\lambda+k$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor))$
		$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,z,\lfloor d(o,x)\rfloor))+\lambda(d(o,y)-d(o,x))+2\lambda+k$
		$\displaystyle\leq\lambda d_{c}(x,y)+\lambda d_{c}(x,z)+4\lambda+2k.$

∎

Lemma 6.4.

Let $(X,d)$ be a $(\lambda,k,E,C,\theta)$ -coarsely convex space and let $d_{c}$ be a cone metric on $X$ . For any $x,y\in X$ , we have

d_{c}(x,y)\leq(\lambda+2)d(x,y)+2\lambda+2k.

Proof..

We do not lose generality by assuming $d(o,x)\leq d(o,y)$ . Since $\Gamma$ is a $(\lambda,k)$ -quasi geodesic bicombing, we have

	$\displaystyle d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor))$
	$\displaystyle\leq d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),x)+d(x,y)+d(y,\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor))$
	$\displaystyle=d\left(\Gamma\left(o,x,\frac{\lfloor d(o,x)\rfloor}{d(o,x)}\right),\Gamma(o,x,1)\right)+d(x,y)+d\left(\Gamma(o,y,1),\Gamma\left(o,y,\frac{\lfloor d(o,x)\rfloor}{d(o,y)}\right)\right)$
	$\displaystyle\leq\lambda+k+d(x,y)+\lambda(d(o,y)-d(o,x)+1)+k$
	$\displaystyle\leq(\lambda+1)d(x,y)+2\lambda+2k.$

So we have

	$\displaystyle d_{c}(x,y)$	$\displaystyle=d(o,y)-d(o,x)+d(\texttt{rp}\Gamma(o,x,\lfloor d(o,x)\rfloor),\texttt{rp}\Gamma(o,y,\lfloor d(o,x)\rfloor))$
		$\displaystyle\leq(\lambda+2)d(x,y)+2\lambda+2k.$

∎

6.2. Horoboundary

Now, we consider horoboundary for the cone metric, and study the relation with the ideal boundary. Let $B(X)$ be the set of all $\mathbb{R}$ -valued functions on $X$ , and equip with the topology of uniform convergence on compact sets. We define $\psi\colon X\rightarrow B(X)$ as follows. For $x\in X$ , let

	$\displaystyle x\mapsto\psi_{x}$	$\displaystyle\coloneqq d_{c}(-,x)-d_{c}(o,x)$
		$\displaystyle\coloneqq d_{c}(-,x)-d(o,x).$

Definition 6.5.

Let $(X,d)$ be a proper $(\lambda,k,E,C,\theta)$ -coarsely convex space and let $B_{b}(X)$ be the set of all $\mathbb{R}$ -valued bounded functions on $X$ . Let $\psi\colon X\rightarrow B(X)$ be the map defined above. Let

B_{b,\lambda,E}(X)\coloneqq\left\{f\in B(X)\mathrel{}\middle|\mathrel{}\frac{1}{\lambda E}\psi_{x}-\beta\leq f\leq\lambda E\psi_{x}+\beta,\>\exists x\in X,\exists\beta\in B_{b}(X)\right\}.

We define the horoboundary of $X$ with cone metric, denoted by $\partial^{c}_{h}X$ as follows

\partial^{c}_{h}X\coloneqq\operatorname{cl}\psi(X)\setminus B_{b,\lambda,E}(X).

We also say that $\partial^{c}_{h}X$ is a horoboundary of $(X,d_{c})$ . Here $\operatorname{cl}\psi(X)$ denotes the closure of $\psi(X)$ in $B(X)$ .

We call a function which belongs to $\partial^{c}_{h}X$ a horofunction with cone metric, or we simply call it horofunction if there is no risk of misunderstanding.

Lemma 6.6.

Let $(X,d)$ be a proper $(\lambda,k,E,C,\theta)$ -coarsely convex space. If the sequence $\psi_{x_{n}}$ converges to $\xi\in\partial^{c}_{h}X$ , then only finitely many of the points $x_{n}$ lie in any bounded subset of $X$ .

Proof..

We assume that infinitely many $x_{n}$ contained in some ball $\bar{B}(o,R)$ where $R>0$ .

Since $X$ is proper, $\bar{B}(o,R)$ is compact. By taking a subsequence, we can assume that $\{x_{n}\}_{n\in\mathbb{N}}$ converges to $\tilde{x}\in\bar{B}(o,R)$ as $n\rightarrow\infty$ . By hypothesis, $\psi_{x_{n}}$ converges pointwise to $\xi$ on $X$ . By Lemma 6.3 and Lemma 6.4, we have

	$\displaystyle\psi_{x_{n}}(x)-\lambda E\psi_{\tilde{x}}(x)$	$\displaystyle=\{d_{c}(x,x_{n})-d_{c}(o,x_{n})\}-\lambda E\{d_{c}(x,\tilde{x})-d_{c}(o,\tilde{x})\}$
		$\displaystyle=\{d_{c}(x,x_{n})-\lambda Ed_{c}(x,\tilde{x})\}+\{\lambda Ed_{c}(o,\tilde{x})-d_{c}(o,x_{n})\}$
		$\displaystyle\leq\lambda Ed_{c}(x_{n},\tilde{x})+4\lambda+2k+C+\{\lambda Ed(o,\tilde{x})-d(o,x_{n})\}$
		$\displaystyle\leq\lambda E\{(\lambda+2)d(x_{n},\tilde{x})+2\lambda+2k\}+4\lambda+2k+C$
		$\displaystyle\phantom{\leq\quad}+\{\lambda Ed(o,\tilde{x})-d(o,x_{n})\}$

for all $x\in X$ . Since $d(x_{n},\tilde{x})\rightarrow 0$ and $\psi_{x_{n}}(x)\rightarrow\xi(x)$ for all $x\in X$ , we get

	$\displaystyle\xi(x)-\lambda E\psi_{\tilde{x}}(x)$	$\displaystyle=\lim_{n\rightarrow\infty}\psi_{x_{n}}(x)-\lambda E\psi_{\tilde{x}}(x)$
		$\displaystyle\leq\lim_{n\rightarrow\infty}\lambda E\{(\lambda+2)d(x_{n},\tilde{x})+2\lambda+2k\}+4\lambda+2k+C$
		$\displaystyle\phantom{\leq\quad\quad\quad}+\{\lambda Ed(o,\tilde{x})-d(o,x_{n})\}$
		$\displaystyle=(\lambda E-1)d(o,\tilde{x})+2\lambda^{2}E+2\lambda kE+4\lambda+2k+C$
		$\displaystyle\leq(\lambda E-1)R+2\lambda^{2}E+2\lambda kE+4\lambda+2k+C.$

So we have

\xi(x)\leq\lambda E\psi_{\tilde{x}}(x)+(\lambda E-1)R+2\lambda^{2}E+2\lambda kE+4\lambda+2k+C

for any $x\in X$ .

Similarly, we have

\psi_{\tilde{x}}(x)-\lambda E\xi(x)\leq(\lambda E-1)R+2\lambda^{2}E+2\lambda kE+4\lambda+2k+C

for any $x\in X$ , that is,

\frac{1}{\lambda E}\psi_{\tilde{x}}(x)-\frac{1}{\lambda E}\{(\lambda E-1)R+2\lambda^{2}E+2\lambda kE+4\lambda+2k+C\}\leq\xi(x)

for any $x\in X$ . This contradicts to $\xi\in\partial^{c}_{h}X$ . ∎

In the rest of this section, let $(X,d)$ be a proper metric space with a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing $\Gamma$ , fix the base point $o\in X$ and let $\partial_{o}X$ be the ideal boundary of $X$ with respect to the base point $o$ . We construct a continuous map from $\partial^{c}_{h}X$ to $\partial_{o}X$ .

Lemma 6.7.

Let $\xi\in\partial^{c}_{h}X$ and let $\{x_{n}\}_{n\in\mathbb{N}}$ on $X$ be a sequence such that $\psi_{x_{n}}$ converges uniformly to $\xi$ on compact sets.

(1)

The sequence $\{x_{n}\}_{n\in\mathbb{N}}$ belongs to $S^{\infty}_{o}X$ . That is, $(x_{n}\>|\>x_{m})_{o}\rightarrow\infty$ as $n,m\rightarrow\infty$ .
(2)

Let $\{y_{n}\}_{n\in\mathbb{N}}$ be another sequence on $X$ such that $\psi_{y_{n}}$ converges uniformly to $\xi$ on compact sets. Then we have $(x_{n}\>|\>y_{n})_{o}\rightarrow\infty$ as $n\rightarrow\infty$ .

Proof..

We take $R>0$ arbitrarily. Set $\tilde{x}_{n}=\texttt{rp}\Gamma(o,x_{n},R)$ . By the definition of $\Gamma$ , for any $n\in\mathbb{N}$ , we have

d(o,\tilde{x}_{n})=d\left(o,\texttt{rp}\Gamma(o,x_{n},R)\right)\leq\lambda R+k.

So we have $\tilde{x}_{n}\in\bar{B}(o,\lambda R+k)$ for any $n\in\mathbb{N}$ . Since $X$ is proper, $\bar{B}(o,\lambda R+k)$ is compact.

Since $\psi_{x_{n}}$ converges uniformly to $\xi$ on $\bar{B}(o,\lambda R+k)$ , we have

	$\displaystyle\|\psi_{x_{n}}(\tilde{x}_{n})-\psi_{x_{m}}(\tilde{x}_{n})\|$	$\displaystyle\leq\|\psi_{x_{n}}(\tilde{x}_{n})-\xi(\tilde{x}_{n})\|+\|\psi_{x_{m}}(\tilde{x}_{n})-\xi(\tilde{x}_{n})\|$
		$\displaystyle\leq\sup_{u\in\bar{B}(o,\lambda R+k)}\|\psi_{x_{n}}(u)-\xi(u)\|+\sup_{u\in\bar{B}(o,\lambda R+k)}\|\psi_{x_{m}}(u)-\xi(u)\|$
		$\displaystyle\rightarrow 0\text{ as }n,m\rightarrow\infty.$

On the other hand, for any $n,m\in\mathbb{N}$ with $d(o,x_{n}),d(o,x_{m})>R$ , we have

(6.1)		$\displaystyle\|\psi_{x_{n}}(\tilde{x}_{n})-\psi_{x_{m}}(\tilde{x}_{n})\|$
	$\displaystyle=\|\{d_{c}(\tilde{x}_{n},x_{n})-d(o,x_{n})\}-\{d_{c}(\tilde{x}_{n},x_{m})-d(o,x_{m})\}\|$
	$\displaystyle=\|\{d(o,x_{n})-d(o,\tilde{x}_{n})+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))-d(o,x_{n})\}$
	$\displaystyle\phantom{{=}\quad}-\{d(o,x_{m})-d(o,\tilde{x}_{n})+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{m},\lfloor d(o,\tilde{x}_{n})\rfloor))-d(o,x_{m})\}\|$
	$\displaystyle=\|d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{{=}\quad}-d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{m},\lfloor d(o,\tilde{x}_{n})\rfloor))\|.$

Since $\Gamma$ is a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing, we have

	$\displaystyle\|R-d(o,\tilde{x}_{n})\|$	$\displaystyle=\left\|\frac{R}{d(o,x_{n})}d(o,x_{n})-d(o,\tilde{x}_{n})\right\|$
		$\displaystyle\leq\theta\left(d\left(\Gamma\left(o,x_{n},\frac{R}{d(o,x_{n})}\right),\Gamma(o,\tilde{x}_{n},1)\right)\right)$
		$\displaystyle=\theta(0).$

So we have

(6.2)		$\displaystyle d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\leq d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},d(o,\tilde{x}_{n})))+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},d(o,\tilde{x}_{n})),\texttt{rp}\Gamma(o,x_{n},R))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x_{n},R),\texttt{rp}\Gamma(o,x_{n},d(o,\tilde{x}_{n})))+d(\texttt{rp}\Gamma(o,x_{n},d(o,\tilde{x}_{n})),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\leq\lambda+k+\lambda\|R-d(o,\tilde{x}_{n})\|+k+\lambda+k$
	$\displaystyle\leq\lambda(\theta(0)+2)+3k.$

By eq. 6.1 and eq. 6.2, there is some $N\in\mathbb{N}$ such that we have

	$\displaystyle d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{m},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\leq d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))+1$
	$\displaystyle\leq\lambda(\theta(0)+2)+3k+1$

for any $n,m\in\mathbb{N}$ with $n,m\geq N$ . Therefore, we have

	$\displaystyle d(\texttt{rp}\Gamma(o,x_{n},R),\texttt{rp}\Gamma(o,x_{m},R))$
	$\displaystyle\leq d(\texttt{rp}\Gamma(o,x_{n},R),\texttt{rp}\Gamma(o,x_{n},d(o,\tilde{x}_{n})))+d(\texttt{rp}\Gamma(o,x_{n},d(o,\tilde{x}_{n})),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{m},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x_{m},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{m},d(o,\tilde{x}_{n})))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x_{m},d(o,\tilde{x}_{n})),\texttt{rp}\Gamma(o,x_{m},R))$
	$\displaystyle\leq\lambda\theta(0)+k+\lambda+k+\lambda(\theta(0)+2)+3k+\lambda(\theta(0)+2)+3k+1+\lambda+k+\lambda\theta(0)+k$
	$\displaystyle=\lambda(4\theta(0)+6)+10k+1$

for any $n,m\in\mathbb{N}$ with $n,m\geq N$ .

Set $L\coloneq\lambda(4\theta(0)+6)+10k+1$ . Note that $L>1$ . Then, for any $R>0$ , there is some $N\in\mathbb{N}$ such that for any $n,m\in\mathbb{N}$ with $n,m\geq N$ , we have

d(\texttt{rp}\Gamma(o,x_{n},R),\texttt{rp}\Gamma(o,x_{m},R))\leq L.

We show, for any $s\in\mathbb{R}_{\geq 0}$ , there is some $N^{\prime}\in\mathbb{N}$ such that for any $n,m\in\mathbb{N}$ with $n,m\geq N^{\prime}$ , we have $(x_{n}\>|\>x_{m})_{o}\geq s$ . From above, there is some $N^{\prime}\in\mathbb{N}$ such that for any $n,m\in\mathbb{N}$ with $n,m\geq N^{\prime}$ , we have

d(\texttt{rp}\Gamma(o,x_{n},ELs),\texttt{rp}\Gamma(o,x_{m},ELs))\leq L.

Since $\Gamma$ is a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing, we have

	$\displaystyle d(\texttt{rp}\Gamma(o,x_{n},s),\texttt{rp}\Gamma(o,x_{m},s))$	$\displaystyle=d\left(\texttt{rp}\Gamma\left(o,x_{n},\frac{1}{EL}ELs\right),\texttt{rp}\Gamma\left(o,x_{m},\frac{1}{EL}ELs\right)\right)$
		$\displaystyle\leq\frac{1}{EL}Ed(\texttt{rp}\Gamma(o,x_{n},ELs),\texttt{rp}\Gamma(o,x_{m},ELs))+C$
		$\displaystyle\leq 1+C\leq D_{1}.$

So we have

(x_{n}\>|\>x_{m})_{o}=\sup\{t\in\mathbb{R}_{\geq 0}\>|\>d(\texttt{rp}\Gamma(o,x_{n},t),\texttt{rp}\Gamma(o,x_{m},t))\leq D_{1}\}\geq s

for any $n,m\in\mathbb{N}$ with $n,m\geq N^{\prime}$ . This completes the proof of item (1).

Now we will show item (2). Since both $\psi_{x_{n}}$ and $\psi_{y_{n}}$ converge uniformly to $\xi$ on $\bar{B}(o,\lambda R+k)$ , we have

	$\displaystyle\|\psi_{x_{n}}(\tilde{x}_{n})-\psi_{y_{n}}(\tilde{x}_{n})\|$	$\displaystyle\leq\|\psi_{x_{n}}(\tilde{x}_{n})-\xi(\tilde{x}_{n})\|+\|\psi_{y_{n}}(\tilde{x}_{n})-\xi(\tilde{x}_{n})\|$
		$\displaystyle\leq\sup_{u\in\bar{B}(o,\lambda R+k)}\|\psi_{x_{n}}(u)-\xi(u)\|+\sup_{u\in\bar{B}(o,\lambda R+k)}\|\psi_{y_{n}}(u)-\xi(u)\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

On the other hand, for any $n\in\mathbb{N}$ with $d(o,x_{n}),d(o,y_{n})>R$ , we have

(6.3)		$\displaystyle\|\psi_{x_{n}}(\tilde{x}_{n})-\psi_{y_{n}}(\tilde{x}_{n})\|$
	$\displaystyle=\|\{d_{c}(\tilde{x}_{n},x_{n})-d(o,x_{n})\}-\{d_{c}(\tilde{x}_{n},y_{n})-d(o,y_{n})\}\|$
	$\displaystyle=\|\{d(o,x_{n})-d(o,\tilde{x}_{n})+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))-d(o,x_{n})\}$
	$\displaystyle\phantom{{=}\quad}-\{d(o,y_{n})-d(o,\tilde{x}_{n})+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,y_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))-d(o,y_{n})\}\|$
	$\displaystyle=\|d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{{=}\quad}-d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,y_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))\|.$

By eq. 6.2 and eq. 6.3, there is some $N\in\mathbb{N}$ such that we have

	$\displaystyle d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,y_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\leq d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))+1$
	$\displaystyle\leq\lambda(\theta(0)+2)+3k+1$

for any $n\in\mathbb{N}$ with $n\geq N$ . Therefore, we have

	$\displaystyle d(\texttt{rp}\Gamma(o,x_{n},R),\texttt{rp}\Gamma(o,y_{n},R))$
	$\displaystyle\leq d(\texttt{rp}\Gamma(o,x_{n},R),\texttt{rp}\Gamma(o,x_{n},d(o,\tilde{x}_{n})))+d(\texttt{rp}\Gamma(o,x_{n},d(o,\tilde{x}_{n})),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,y_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,y_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,y_{n},d(o,\tilde{x}_{n})))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,y_{n},d(o,\tilde{x}_{n})),\texttt{rp}\Gamma(o,y_{n},R))$
	$\displaystyle\leq\lambda\theta(0)+k+\lambda+k+\lambda(\theta(0)+2)+3k+\lambda(\theta(0)+2)+3k+1+\lambda+k+\lambda\theta(0)+k$
	$\displaystyle=\lambda(4\theta(0)+6)+10k+1=L$

for any $n\in\mathbb{N}$ with $n\geq N$ . The rest of the proof of item (2) can be shown as in item (1). ∎

By Lemma 6.7, we define $\mathrm{Pr}\colon\partial^{c}_{h}X\rightarrow\partial_{o}X$ as follows. For any $\xi\in\partial^{c}_{h}X$ , there is a sequence $\{x_{n}\}_{n\in\mathbb{N}}\in S^{\infty}_{o}X$ such that $\psi_{x_{n}}$ converges uniformly to $\xi$ on compact sets. Let $x\in\partial_{o}X$ be the equivalence class of the sequence $\{x_{n}\}_{n\in\mathbb{N}}$ . Then we define $\mathrm{Pr}(\xi)\coloneqq x$ . By item (2) of Lemma 6.7, $\mathrm{Pr}(\xi)$ does not depend on the choice of a sequence $\{x_{n}\}_{n\in\mathbb{N}}$ such that $\xi=\lim_{n\rightarrow\infty}\psi_{x_{n}}$ .

Proposition 6.8.

$\mathrm{Pr}\colon\partial^{c}_{h}X\rightarrow\partial_{o}X$ is continuous.

Proof..

We take $\xi\in\partial^{c}_{h}X$ arbitrarily and assume that the sequence of maps $\{\xi_{m}\}_{m\in\mathbb{N}}$ in $\partial^{c}_{h}X$ converges uniformly to $\xi$ on compact sets.

We take $\{x_{n}\}_{n\in\mathbb{N}}$ to be a sequence on $X$ such that $\{\psi_{x_{n}}\}_{n\in\mathbb{N}}$ converges uniformly to $\xi$ on compact sets. By Lemma 6.7, the sequence $\{x_{n}\}_{n\in\mathbb{N}}$ belongs to $S^{\infty}_{o}X$ , and we denote by $x$ the equivalence class of $\{x_{n}\}_{n\in\mathbb{N}}$ on $\partial_{o}X$ . By the definition of the map $\mathrm{Pr}$ , we get $\mathrm{Pr}(\xi)=x$ .

Similarly, we take $\{y_{m,l}\}_{l\in\mathbb{N}}$ to be a sequence on $X$ such that $\{\psi_{y_{m,l}}\}_{l\in\mathbb{N}}$ converges uniformly to $\xi_{m}$ on compact sets for each $m\in\mathbb{N}$ . Lemma 6.7, the sequence $\{y_{m,l}\}_{l\in\mathbb{N}}$ belongs to $S^{\infty}_{o}X$ each $l\in\mathbb{N}$ , and we denote by $y_{m}$ the equivalence class of $\{y_{m,l}\}_{l\in\mathbb{N}}$ on $\partial_{o}X$ . By the definition of the map $\mathrm{Pr}$ , we get $\mathrm{Pr}(\xi_{m})=y_{m}$ for any $m\in\mathbb{N}$ .

Here, we show $\lim_{m\rightarrow\infty}(x\>|\>y_{m})_{o}=\infty$ . Since we have $\lim_{n\rightarrow\infty}(x\>|\>x_{n})_{o}=\infty$ and $\lim_{l\rightarrow\infty}(y_{m,l}\>|\>y_{m})_{o}=\infty$ , it is enough to show $\lim_{m\rightarrow\infty}\liminf_{n,l\rightarrow\infty}(x_{n}\>|\>y_{m,l})_{o}=\infty$ .

We take $R>0$ arbitrarily. Set $\tilde{x}_{n}\coloneqq\texttt{rp}\Gamma(o,x_{n},R)$ . We have $\tilde{x}_{n}\in\bar{B}(o,\lambda R+k)$ for any $n\in\mathbb{N}$ . Since $\psi_{x_{n}}$ converges uniformly to $\xi$ on compact sets and $\psi_{y_{m,l}}$ converges uniformly to $\xi_{m}$ on compact sets for each $m\in\mathbb{N}$ , we have

	$\displaystyle\liminf_{n,l\rightarrow\infty}\|\psi_{x_{n}}(\tilde{x}_{n})-\psi_{y_{m,l}}(\tilde{x}_{n})\|$
	$\displaystyle\leq\liminf_{n,l\rightarrow\infty}\|\psi_{x_{n}}(\tilde{x}_{n})-\xi(\tilde{x}_{n})\|+\|\xi(\tilde{x}_{n})-\xi_{m}(\tilde{x}_{n})\|+\|\xi_{m}(\tilde{x}_{n})-\psi_{y_{m,l}}(\tilde{x}_{n})\|$
	$\displaystyle\leq\lim_{n\rightarrow\infty}\sup_{u\in\bar{B}(o,\lambda R+k)}\|\psi_{x_{n}}(u)-\xi(u)\|+\sup_{u\in\bar{B}(o,\lambda R+k)}\|\xi(u)-\xi_{m}(u)\|$
	$\displaystyle\phantom{\lim_{n\rightarrow\infty}\quad}+\lim_{l\rightarrow\infty}\sup_{u\in\bar{B}(o,\lambda R+k)}\|\xi_{m}(u)-\psi_{y_{m,l}}(u)\|$
	$\displaystyle=\sup_{u\in\bar{B}(o,\lambda R+k)}\|\xi(u)-\xi_{m}(u)\|.$

Since $\xi_{m}$ converges uniformly to $\xi$ on compact sets, for any $R>0$ , there is some $M>0$ such that for any $m\in\mathbb{N}$ with $m\geq M$ and for a sufficiently small $\epsilon$ , we have

\sup_{u\in\bar{B}(o,\lambda R+k)}|\xi(u)-\xi_{m}(u)|\leq 1-\epsilon.

We fix $m\in\mathbb{N}$ with $m\geq M$ . Then, we have

\liminf_{n,l\rightarrow\infty}|\psi_{x_{n}}(\tilde{x}_{n})-\psi_{y_{m,l}}(\tilde{x}_{n})|\leq 1-\epsilon.

There is some $N_{m}\in\mathbb{N}$ such that for any $i\in\mathbb{N}$ with $i\geq N_{m}$ , there are some $n,l\in\mathbb{N}$ such that $n,l\geq i$ and we have $|\psi_{x_{n}}(\tilde{x}_{n})-\psi_{y_{m,l}}(\tilde{x}_{n})|<1$ .

On the other hand, by Lemma 6.6, for large enough $n\in\mathbb{N}$ , we have $d(o,x_{n})>\lambda R+k$ . Similarly, for large enough $l\in\mathbb{N}$ , we have $d(o,y_{m,l})>\lambda R+k$ . So we have

	$\displaystyle\|\psi_{x_{n}}(\tilde{x}_{n})-\psi_{y_{m,l}}(\tilde{x}_{n})\|$
	$\displaystyle=\|\{d(o,x_{n})-d(o,\tilde{x}_{n})+d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))-d(o,x_{n})\}$
	$\displaystyle\phantom{{=}\quad}-\{d(o,y_{m,l})-d(o,\tilde{x}_{n})+d(\texttt{rp}\Gamma(o,y_{m,l},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor)-d(o,y_{m,l})\}\|$
	$\displaystyle=\|d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{{=}\quad}-d(\texttt{rp}\Gamma(o,y_{m,l},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))\|$

for large enough $n,l\in\mathbb{N}$ . By the proof of Lemma 6.7, we have

d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))\leq\lambda(\theta(0)+2)+3k

for each $n\in\mathbb{N}$ . Thus, for any $i\in\mathbb{N}$ with $i\geq N_{m}$ , there is some $n,l\in\mathbb{N}$ such that $n,l\geq i$ and we have

d(\texttt{rp}\Gamma(o,y_{m,l},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))\leq\lambda(\theta(0)+2)+3k+1.

By the proof of Lemma 6.7, we have $|R-d(o,\tilde{x}_{n})|\leq\theta(0)$ . Therefore, for any $i\in\mathbb{N}$ with $i\geq N_{m}$ , there is some $n,l\in\mathbb{N}$ such that $n,l\geq i$ and we have

	$\displaystyle d(\texttt{rp}\Gamma(o,y_{m,l},R),\texttt{rp}\Gamma(o,x_{n},R))$
	$\displaystyle\leq d(\texttt{rp}\Gamma(o,y_{m,l},R),\texttt{rp}\Gamma(o,y_{m,l},d(o,\tilde{x}_{n})))+d(\texttt{rp}\Gamma(o,y_{m,l},d(o,\tilde{x}_{n})),\texttt{rp}\Gamma(o,y_{m,l},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,y_{m,l},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor)$
	$\displaystyle\phantom{\leq\quad}+d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,x_{n},d(o,\tilde{x}_{n})))+d(\texttt{rp}\Gamma(o,x_{n},d(o,\tilde{x}_{n})),\texttt{rp}\Gamma(o,x_{n},R))$
	$\displaystyle\leq\lambda(4\theta(0)+6)+10k+1.$

Set $L\coloneqq\lambda(4\theta(0)+6)+10k+1$ . Note $L>1$ . For any $R>0$ , there is some $M\in\mathbb{N}$ such that for any $m\in\mathbb{N}$ with $m\geq M$ , there is some $N_{m}\in\mathbb{N}$ such that for any $i\in\mathbb{N}$ with $i\geq N_{m}$ , there are some $n,l\in\mathbb{N}$ such that $n,l\geq i$ and we have

d(\texttt{rp}\Gamma(o,y_{m,l},R),\texttt{rp}\Gamma(o,x_{n},R))\leq L.

We show, for any $s\in\mathbb{R}_{\geq 0}$ , there is some $M^{\prime}\in\mathbb{N}$ such that for any $m\in\mathbb{N}$ with $m\geq M^{\prime}$ , we have $\liminf_{n,l\rightarrow\infty}(x_{n}\>|\>y_{m,l})_{o}\geq s$ . From above, there is some $M^{\prime}\in\mathbb{N}$ such that for any $m\in\mathbb{N}$ with $m\geq M^{\prime}$ , there is some $N_{m}\in\mathbb{N}$ such that for any $i\in\mathbb{N}$ with $i\geq N_{m}$ , there are some $n,l\in\mathbb{N}$ such that $n,l\geq i$ and we have

d(\texttt{rp}\Gamma(o,y_{m,l},ELs),\texttt{rp}\Gamma(o,x_{n},ELs))\leq L.

Since $\Gamma$ is a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing, we have

	$\displaystyle d(\texttt{rp}\Gamma(o,y_{m,l},s),\texttt{rp}\Gamma(o,x_{n},s))$	$\displaystyle=d\left(\texttt{rp}\Gamma\left(o,y_{m,l},\frac{1}{EL}ELs\right),\texttt{rp}\Gamma\left(o,x_{n},\frac{1}{EL}ELs\right)\right)$
		$\displaystyle\leq\frac{1}{EL}Ed(\texttt{rp}\Gamma(o,y_{m,l},ELs),\texttt{rp}\Gamma(o,x_{n},ELs))+C$
		$\displaystyle\leq 1+C\leq D_{1}.$

This means that for any $s\in\mathbb{R}_{\geq 0}$ , there is some $M^{\prime}\in\mathbb{N}$ such that for any $m\in\mathbb{N}$ with $m\geq M^{\prime}$ , we have $\liminf_{n,l\rightarrow\infty}(x_{n}\>|\>y_{m,l})_{o}\geq s$ . Therefore, we have

\lim_{m\rightarrow\infty}\liminf_{n,l\rightarrow\infty}(x_{n}\>|\>y_{m,l})_{o}=\infty.

∎

6.3. Reduced horoboundary

Definition 6.9.

Let $(X,d)$ be a proper $(\lambda,k,E,C,\theta)$ -coarsely convex space and let $\partial^{c}_{h}X$ be the horoboundary defined above, using the cone metric.

We define that two functions $\xi$ and $\eta$ in $\partial^{c}_{h}X$ are equivalent, denoted by $\xi\sim\eta$ , if

\sup_{u\in X}|\xi(u)-\eta(u)|<\infty.

This determines an equivalence relation on $\partial^{c}_{h}X$ . A reduced horoboundary of $X$ with cone metric, denoted by $\partial^{c}_{h}X/\sim$ , is the quotient of $\partial^{c}_{h}X$ by the equivalence relation $\sim$ with the quotient topology. We also say that $\partial^{c}_{h}X/\sim$ is a reduced horoboundary of $(X,d_{c})$ .

Let $\pi\colon\partial^{c}_{h}X\rightarrow\partial^{c}_{h}X/\sim$ be the natural projection. If $\xi\sim\eta$ , we have $\mathrm{Pr}(\xi)=\mathrm{Pr}(\eta)$ . So there uniquely exists a continuous map $f\colon\partial^{c}_{h}X/\sim\,\rightarrow\partial_{o}X$ such that $\mathrm{Pr}=f\circ\pi$ . We construct the inverse of $f$ .

Definition 6.10.

For any $x\in\partial_{o}X$ , we define a $(\lambda,k_{1})$ -quasi geodesic ray $\gamma_{x}\colon\mathbb{R}_{\geq 0}\rightarrow X$ as follows

\gamma_{x}(-)\coloneqq\texttt{rp}\bar{\Gamma}(o,x,-)

where $k_{1}\coloneqq\lambda+k$ . Then we define $b_{x}\colon X\rightarrow\mathbb{R}$ as the Busemann function respect to the cone metric by

u\mapsto-d(o,u)+d(\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor),\gamma_{x}(\lfloor d(o,u)\rfloor)).

Proposition 6.11.

The Busemann function respect to the cone metric is a horofunction.

Proof..

Let $x\in\partial_{o}X$ and define $\gamma_{x}\colon\mathbb{R}_{\geq 0}\rightarrow X$ by $\gamma_{x}(-)=\texttt{rp}\bar{\Gamma}(o,x,-)$ . Then, there is a sequence $\{x_{n}\}_{n\in\mathbb{N}}\in S^{\infty}_{o}X$ such that we have $(x_{n}\>|\>x)_{o}\rightarrow\infty$ as $n\rightarrow\infty$ and the sequence of maps $\{\texttt{rp}\Gamma(o,x_{n},-)|_{\mathbb{N}}\colon\mathbb{N}\rightarrow X\}_{n\in\mathbb{N}}$ converges pointwise to $\gamma_{x}(-)=\texttt{rp}\bar{\Gamma}(o,x,-)$ .

First, we show $b_{x}\in\operatorname{cl}\psi(X)$ . For any $R>0$ , by taking $n\in\mathbb{N}$ large enough, we have

	$\displaystyle\sup_{u\in\bar{B}(o,R)}\|\psi_{x_{n}}(u)-b_{x}(u)\|$
	$\displaystyle=\sup_{u\in\bar{B}(o,R)}\|\{d(o,x_{n})-d(o,u)+d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,u)\rfloor),\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor))-d(o,x_{n})\}$
	$\displaystyle\phantom{\sup_{u\in\bar{B}(o,R)}\quad\quad}-\{-d(o,u)+d(\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor),\gamma_{x}(\lfloor d(o,u)\rfloor))\}$
	$\displaystyle=\sup_{u\in\bar{B}(o,R)}\|d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,u)\rfloor),\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor)-d(\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor,\gamma_{x}(\lfloor d(o,u)\rfloor))\|$
	$\displaystyle\leq\sup_{u\in\bar{B}(o,R)}d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,u)\rfloor),\gamma_{x}(\lfloor d(o,u)\rfloor))$
	$\displaystyle=\max_{t\in\{1,2,\cdots,\lfloor R\rfloor\}}d(\texttt{rp}\Gamma(o,x_{n},t),\gamma_{x}(t))\rightarrow 0\text{ as }n\rightarrow\infty.$

Second, we show $b_{x}\notin B_{b,\lambda,E}(X)$ by contradiction. That is, there are some $y\in X$ and $\beta\in B_{b}(X)$ such that we have

(6.4)

\displaystyle\frac{1}{\lambda E}\psi_{y}-b_{x}\leq\beta.

We take $R>0$ with

d(o,y)\leq\frac{1}{\lambda}R-k.

Set $\tilde{x}_{n}\coloneqq\texttt{rp}\Gamma(o,x_{n},R)$ . Then, for any $n\in\mathbb{N}$ , we have

d(o,y)\leq\frac{1}{\lambda}R-k\leq d(o,\tilde{x}_{n})\leq\lambda R+k.

For large enough $n\in\mathbb{N}$ , we have $d(o,x_{n})>\lambda R+k$ .

By the proof of Lemma 6.7, we have

d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))\leq\lambda(\theta(0)+2)+3k.

Thus, we have

	$\displaystyle\frac{1}{\lambda E}\psi_{y}(\tilde{x}_{n})-\psi_{x_{n}}(\tilde{x}_{n})$
	$\displaystyle=\frac{1}{\lambda E}\{d(o,\tilde{x}_{n})-d(o,y)+d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,y)\rfloor),\texttt{rp}\Gamma(o,y,\lfloor d(o,y)\rfloor))-d(o,y)\}$
	$\displaystyle\phantom{{=}\quad}-\{d(o,x_{n})-d(o,\tilde{x}_{n})+d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))-d(o,x_{n})\}$
	$\displaystyle=\left(\frac{1}{\lambda E}+1\right)d(o,\tilde{x}_{n})-\frac{2}{\lambda E}d(o,y)+\frac{1}{\lambda E}d(\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,y)\rfloor),\texttt{rp}\Gamma(o,y,\lfloor d(o,y)\rfloor))$
	$\displaystyle\phantom{{=}\quad}-d(\texttt{rp}\Gamma(o,x_{n},\lfloor d(o,\tilde{x}_{n})\rfloor),\texttt{rp}\Gamma(o,\tilde{x}_{n},\lfloor d(o,\tilde{x}_{n})\rfloor))$
	$\displaystyle\geq\left(\frac{1}{\lambda E}+1\right)\left(\frac{1}{\lambda}R-k\right)-\frac{2}{\lambda E}d(o,y)-\lambda(\theta(0)+2)-3k.$

Therefore, we have

\displaystyle\sup_{u\in\bar{B}(o,\lambda R+k)}\frac{1}{\lambda E}\psi_{y}(u)-\psi_{x_{n}}(u)\geq\left(\frac{1}{\lambda E}+1\right)\left(\frac{1}{\lambda}R-k\right)-\frac{2}{\lambda E}d(o,y)-\lambda(\theta(0)+2)-3k

for large enough $n\in\mathbb{N}$ .

On the other hand, since $\psi_{x_{n}}$ converges uniformly to $b_{x}$ on compact sets, we have

\lim_{n\rightarrow\infty}\sup_{u\in\bar{B}(o,\lambda R+k)}\frac{1}{\lambda E}\psi_{y}(u)-\psi_{x_{n}}(u)=\sup_{u\in\bar{B}(o,\lambda R+k)}\frac{1}{\lambda E}\psi_{y}(u)-b_{x}(u).

So we have

\sup_{u\in\bar{B}(o,\lambda R+k)}\frac{1}{\lambda E}\psi_{y}(u)-b_{x}(u)\geq\left(\frac{1}{\lambda E}+1\right)\left(\frac{1}{\lambda}R-k\right)-\frac{2}{\lambda E}d(o,y)-\lambda(\theta(0)+2)-3k.

Since the right-hand side of the above equation contains $R>0$ which can be arbitrary large, this contradicts to eq. 6.4. ∎

For $x\in\partial_{o}X$ , we define $g\colon\partial_{o}X\rightarrow\partial^{c}_{h}X/\sim$ as $g(x)\coloneqq\pi(b_{x})$ .

Proposition 6.12.

The map $g$ is the inverse of $f$ .

Proof..

First, we show $f\circ g=\mathrm{id}_{\partial_{o}X}$ . We take $x\in\partial_{o}X$ arbitrarily. Then, we have $g(x)=\pi(b_{x})$ . By Proposition 6.11, there is a sequence $\{x_{n}\}_{n\in\mathbb{N}}\in S^{\infty}_{o}X$ such that we have $(x_{n}\>|\>x)_{o}\rightarrow\infty$ as $n\rightarrow\infty$ and $\lim_{n\rightarrow\infty}\psi_{x_{n}}=b_{x}$ .

Thus, we obtain

f\circ g(x)=f(\pi(b_{x}))=\mathrm{Pr}(b_{x})=x.

Second, we show $g\circ f=\mathrm{id}_{\partial^{c}_{h}X/\sim}$ . We take $\xi\in\partial^{c}_{h}X$ arbitrarily. By Lemma 6.7, there is a sequence of maps $\{\psi_{y_{n}}\}_{n\in\mathbb{N}}$ converges uniformly to $\xi$ on compact sets, and we have $\{y_{n}\}_{n\in\mathbb{N}}\in S^{\infty}_{o}X$ . Let $y\in\partial_{o}X$ be an equivalence class of a sequence $\{y_{n}\}_{n\in\mathbb{N}}\in S^{\infty}_{o}X$ . By the definition of $\mathrm{Pr}$ , we have $\mathrm{Pr}(\xi)=y$ .

There is a sequence $\{z_{n}\}_{n\in\mathbb{N}}\in S^{\infty}_{o}X$ such that we have $(z_{n}\>|\>y)_{o}\rightarrow\infty$ as $n\rightarrow\infty$ and the sequence of maps $\{\psi_{z_{n}}\}_{n\in\mathbb{N}}$ converges uniformly to $b_{y}$ on compact sets. By the definition of $f$ and $g$ , we have

g\circ f(\pi(\xi))=g(\mathrm{Pr}(\xi))=g(y)=\pi(b_{y}).

Thus, it is enough to show

\sup_{u\in X}|\xi(u)-b_{y}(u)|=\sup_{u\in X}\lim_{n\rightarrow\infty}|\psi_{y_{n}}(u)-\psi_{z_{n}}(u)|<\infty.

We take $u\in X$ arbitrarily. For large enough $n\in\mathbb{N}$ , we have

	$\displaystyle\|\psi_{y_{n}}(u)-\psi_{z_{n}}(u)\|$
	$\displaystyle=\|\{d(o,y_{n})-d(o,u)+d(\texttt{rp}\Gamma(o,y_{n},\lfloor d(o,u)\rfloor),\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor))-d(o,y_{n})\}$
	$\displaystyle\phantom{=\quad}-\{d(o,z_{n})-d(o,u)+d(\texttt{rp}\Gamma(o,z_{n},\lfloor d(o,u)\rfloor),\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor))-d(o,z_{n})\}\|$
	$\displaystyle\leq d(\texttt{rp}\Gamma(o,y_{n},\lfloor d(o,u)\rfloor),\texttt{rp}\Gamma(o,z_{n},\lfloor d(o,u)\rfloor))$

By $(y_{n}\>|\>y)_{o}\rightarrow\infty$ and $(z_{n}\>|\>y)_{o}\rightarrow\infty$ as $n\rightarrow\infty$ , we have

\liminf_{n\rightarrow\infty}(y_{n}\>|\>z_{n})_{o}=\infty.

Thus, there is some $N\in\mathbb{N}$ such that for any $n\in\mathbb{N}$ with $n\geq N$ , there is some $n^{\prime}\in\mathbb{N}$ with $n^{\prime}\geq n$ such that we have

\sup\{t\in\mathbb{R}_{\geq 0}\>|\>d(\texttt{rp}\Gamma(o,y_{n^{\prime}},t),\texttt{rp}\Gamma(o,z_{n^{\prime}},t))\leq D_{1}\}\geq\lfloor d(o,u)\rfloor.

So, there is some $s\geq\lfloor d(o,u)\rfloor$ such that we have

d(\texttt{rp}\Gamma(o,y_{n^{\prime}},s),\texttt{rp}\Gamma(o,z_{n^{\prime}},s))\leq D_{1}.

Therefore, there is some $N\in\mathbb{N}$ such that for any $n\in\mathbb{N}$ with $n\geq N$ , there is some $n^{\prime}\in\mathbb{N}$ with $n^{\prime}\geq n$ such that we have

	$\displaystyle d(\texttt{rp}\Gamma(o,y_{n^{\prime}},\lfloor d(o,u)\rfloor),\texttt{rp}\Gamma(o,z_{n^{\prime}},\lfloor d(o,u)\rfloor)$
	$\displaystyle=d\left(\texttt{rp}\Gamma\left(o,y_{n^{\prime}},\frac{\lfloor d(o,u)\rfloor}{s}s\right),\texttt{rp}\Gamma\left(o,z_{n^{\prime}},\frac{\lfloor d(o,u)\rfloor}{s}s\right)\right)$
	$\displaystyle\leq\frac{\lfloor d(o,u)\rfloor}{s}Ed(\texttt{rp}\Gamma(o,y_{n^{\prime}},s),\texttt{rp}\Gamma(o,z_{n^{\prime}},s))+C$
	$\displaystyle\leq ED_{1}+C.$

From the above, we obtain

	$\displaystyle\|\xi(u)-b_{y}(u)\|$	$\displaystyle=\lim_{n\rightarrow\infty}\|\psi_{y_{n}}(u)-\psi_{z_{n}}(u)\|$
		$\displaystyle=\liminf_{n\rightarrow\infty}\|\psi_{y_{n}}(u)-\psi_{z_{n}}(u)\|$
		$\displaystyle\leq\liminf_{n\rightarrow\infty}d(\texttt{rp}\Gamma(o,y_{n},\lfloor d(o,u)\rfloor),\texttt{rp}\Gamma(o,z_{n},\lfloor d(o,u)\rfloor))$
		$\displaystyle\leq ED_{1}+C$

for any $u\in X$ . So, we have

\sup_{u\in X}|\xi(u)-b_{y}(u)|\leq ED_{1}+C.

∎

We show that $g\colon\partial_{o}X\rightarrow\partial^{c}_{h}X/\sim$ is continuous. We recall the following lemma from general topology.

Lemma 6.13.

Let $(Z,\mathcal{O})$ be a topological space and let $\alpha\in Z$ be a point on $Z$ . Let $\{a_{n}\}_{n\in\mathbb{N}}$ be a sequence on $Z$ such that for any subsequence $\{a_{n(m)}\}_{m\in\mathbb{N}}$ has a subsequence $\{a_{n(m)(k)}\}_{k\in\mathbb{N}}$ such that $a_{n(m)(k)}$ converges to $\alpha$ . Then $a_{n}$ converges to $\alpha$ .

Proof..

We prove this by contradiction. That is, there is an open set $U\in\mathcal{O}$ such that $\alpha\in U$ and for any $N\in\mathbb{N}$ , there is some $n\in\mathbb{N}$ , $n\geq N$ and $a_{n}\notin U$ . We construct a subsequence $\{a_{n(m)}\}_{m\in\mathbb{N}}$ of $\{a_{n}\}_{n\in\mathbb{N}}$ by induction as follows.

By the assumption, there is $n(1)\in\mathbb{N}$ such that $n(1)\geq 1$ and $a_{n(1)}\notin U$ . Also, there is $n(2)\in\mathbb{N}$ such that $n(2)>n(1)$ and $a_{n(2)}\notin U$ . In the same way, for any $m\in\mathbb{N}$ , there is $n(m+1)\in\mathbb{N}$ such that $n(m+1)>n(m)$ and $a_{n(m+1)}\notin U$ . Thus, we obtain a subsequence $\{a_{n(m)}\}_{m\in\mathbb{N}}$ that does not intersect with $U$ . By the construction of $\{a_{n(m)}\}_{m\in\mathbb{N}}$ , there is no subsequence of $\{a_{n(m)}\}_{m\in\mathbb{N}}$ that converges to $\alpha$ . This is a contradiction. ∎

Proposition 6.14.

$g\colon\partial_{o}X\rightarrow\partial^{c}_{h}X/\sim$ is continuous.

Proof..

We take $x\in\partial_{o}X$ arbitrarily, and we take a sequence $\{x_{n}\}_{n\in\mathbb{N}}\subset\partial_{o}X$ arbitrarily such that $\liminf_{n\rightarrow\infty}(x_{n}\>|\>x)_{o}=\infty$ . By Proposition 6.11, we have $b_{x}\in\partial^{c}_{h}X$ and $b_{x_{n}}\in\partial^{c}_{h}X$ for any $n\in\mathbb{N}$ . Then we show $\lim_{n\rightarrow\infty}\pi(b_{x_{n}})=\pi(b_{x})$ .

Set $\gamma_{x}(-)\coloneqq\text{rp}\bar{\Gamma}(o,x,-)$ and $\gamma_{x_{n}}(-)\coloneqq\text{rp}\bar{\Gamma}(o,x_{n},-)$ for any $n\in\mathbb{N}$ . By Lemma 3.9, we have $\liminf_{n\rightarrow\infty}(\gamma_{x_{n}}\>|\>\gamma_{x})_{o}=\infty$ .

We take any subsequence $\{\pi(b_{x_{n(m)}})\}_{m\in\mathbb{N}}$ of $\{\pi(b_{x_{n}})\}_{n\in\mathbb{N}}$ . Correspondingly, we take a sequence of $(\lambda,k_{1})$ -quasi geodesic rays $\{\gamma_{x_{n(m)}}\}_{m\in\mathbb{N}}$ where $k_{1}=\lambda+k$ . By the properness of $X$ and induction, we can take a subsequence $\{\gamma_{x_{n(m)(l)}}\}_{l\in\mathbb{N}}$ of $\{\gamma_{x_{n(m)}}\}_{m\in\mathbb{N}}$ such that $\{\gamma_{x_{n(m)(l)}}\}_{l\in\mathbb{N}}$ converges uniformly to some $(\lambda,k_{1})$ -quasi geodesic ray $\tilde{\gamma}$ on compact sets. For details, see [FO20, Proposition 4.17]. We define $b_{\tilde{\gamma}}\colon X\rightarrow\mathbb{R}$ as follows

u\mapsto-d(o,u)+d(\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor),\tilde{\gamma}(\lfloor d(o,u)\rfloor))

for any $u\in X$ . Then, for any $R>0$ , we have

	$\displaystyle\sup_{u\in\bar{B}(o,R)}\|b_{x_{n(m)(l)}}(u)-b_{\tilde{\gamma}}(u)\|$
	$\displaystyle=\sup_{u\in\bar{B}(o,R)}\|\{-d(o,u)+d(\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor),\gamma_{x_{n(m)(l)}}(\lfloor d(o,u)\rfloor))\}$
	$\displaystyle\phantom{=\quad}-\{-d(o,u)+d(\texttt{rp}\Gamma(o,u,\lfloor d(o,u)\rfloor),\tilde{\gamma}(\lfloor d(o,u)\rfloor)\}\|$
	$\displaystyle\leq\sup_{u\in\bar{B}(o,R)}d(\gamma_{x_{n(m)(l)}}(\lfloor d(o,u)\rfloor),\tilde{\gamma}(\lfloor d(o,u)\rfloor))$
	$\displaystyle\leq\max_{t\in\{1,\cdots,\lfloor R\rfloor\}}d(\gamma_{x_{n(m)(l)}}(t),\tilde{\gamma}(t))\rightarrow 0\text{ as }l\rightarrow\infty.$

Thus, $\{b_{x_{n(m)(l)}}\}_{l\in\mathbb{N}}$ converges uniformly to $b_{\tilde{\gamma}}$ on compact sets. Since $b_{x_{n(m)(l)}}\in\partial^{c}_{h}X\subset\operatorname{cl}\psi(X)$ for any $l\in\mathbb{N}$ , we have $\lim_{l\rightarrow\infty}b_{x_{n(m)(l)}}=b_{\tilde{\gamma}}\in\operatorname{cl}\psi(X)$ . Also, we have $b_{\tilde{\gamma}}\notin B_{b,\lambda,E}$ . We take $R>0$ and $y\in X$ arbitrarily. By the proof of Proposition 6.11, we have

\sup_{u\in\bar{B}(o,\lambda R+k)}\frac{1}{\lambda E}\psi_{y}(u)-b_{x_{n(m)(l)}}(u)\geq\left(\frac{1}{\lambda E}+1\right)\left(\frac{1}{\lambda}R-k\right)-\frac{2}{\lambda E}d(o,y)-\lambda(\theta(0)+2)-3k

for any $l\in\mathbb{N}$ . Since $b_{x_{n(m)(l)}}$ converges uniformly to $b_{\tilde{\gamma}}$ on compact sets, we have

\sup_{u\in\bar{B}(o,\lambda R+k)}\frac{1}{\lambda E}\psi_{y}(u)-b_{\tilde{\gamma}}(u)\geq\left(\frac{1}{\lambda E}+1\right)\left(\frac{1}{\lambda}R-k\right)-\frac{2}{\lambda E}d(o,y)-\lambda(\theta(0)+2)-3k

for any $R>0$ and $y\in X$ . This means $b_{\tilde{\gamma}}\notin B_{b,\lambda,E}(X)$ and we have $b_{\tilde{\gamma}}\in\partial^{c}_{h}X$ .

We show $\pi(b_{\tilde{\gamma}})=\pi(b_{x})$ , that is $\sup_{u\in X}|b_{\tilde{\gamma}}(u)-b_{x}(u)|<\infty$ . We prove this by contradiction. We assume that

\sup_{u\in X}|b_{\tilde{\gamma}}(u)-b_{x}(u)|=\sup_{u\in X}\lim_{l\rightarrow\infty}|b_{x_{n(m)(l)}}(x)-b_{x}(u)|=\infty.

For any $M>D_{1}+C$ , there is $x_{M}\in X$ such that for any $L\in\mathbb{N}$ , there is some $l\in\mathbb{N}$ , $l\geq L$ such that

|b_{\tilde{\gamma}}(x_{M})-b_{x_{n(m)(l)}}(x_{M})|=|d(x_{M},\tilde{\gamma}(d(o,x_{M})))-d(x_{M},\gamma_{x_{n(m)(l)}}(d(o,x_{M})))|>M.

By the triangle inequality, for any $L\in\mathbb{N}$ there is some $l\in\mathbb{N}$ such that $l\geq L$ and we have

d(\tilde{\gamma}(d(o,x_{M})),\gamma_{x_{n(m)(l)}}(d(o,x_{M})))>M>D_{1}+C.

On the other hand, since $\{\gamma_{x_{n(m)(l)}}\}_{l\in\mathbb{N}}$ is a subsequence of $\{\gamma_{x_{n}}\}_{n\in\mathbb{N}}$ , we have

\liminf_{l\rightarrow\infty}(\gamma_{x_{n(m)(l)}}\>|\>\gamma_{x})_{o}=\infty.

Thus, there is some $\tilde{L}\in\mathbb{N}$ such that for any $\tilde{l}\in\mathbb{N}$ , $\tilde{l}\geq\tilde{L}$ , there is $t_{M}>d(o,x_{M})$ such that $d(\gamma_{x_{n(m)(\tilde{l})}}(t_{M}),\gamma_{x}(t_{M}))\leq D_{1}$ . Thus, we have

	$\displaystyle d(\gamma_{x_{n(m)(\tilde{l})}}(d(o,x_{M})),\gamma_{x}(d(o,x_{M}))$	$\displaystyle=d(\texttt{rp}\bar{\Gamma}(o,x_{n(m)(\tilde{l})},d(o,x_{M})),\texttt{rp}\bar{\Gamma}(o,x,d(o,x_{M})))$
		$\displaystyle\leq\frac{d(o,x_{M})}{t_{M}}d(\texttt{rp}\bar{\Gamma}(o,x_{n(m)(\tilde{l})},t_{M}),\texttt{rp}\bar{\Gamma}(o,x,t_{M}))+C$
		$\displaystyle=\frac{d(o,x_{M})}{t_{M}}d(\gamma_{x_{n(m)(\tilde{l})}}(t_{M}),\gamma_{x}(t_{M}))+C$
		$\displaystyle\leq D_{1}+C<M.$

This is a contradiction. So we have

\sup_{u\in X}|b_{\tilde{\gamma}}(u)-b_{x}(u)|<\infty.

In particular, we have

\lim_{l\rightarrow\infty}\pi(b_{x_{n(m)(l)}})=\pi(\lim_{l\rightarrow\infty}b_{x_{n(m)(l)}})=\pi(b_{\tilde{\gamma}})=\pi(b_{x}).

From above, any subsequence $\{\pi(b_{x_{n(m)}})\}_{m\in\mathbb{N}}$ of $\{\pi(b_{x_{n}})\}_{n\in\mathbb{N}}$ has a subsequence which converges to $\pi(b_{x})$ . By Lemma 6.13, we have $\lim_{n\rightarrow\infty}\pi(b_{x_{n}})=\pi(b_{x})$ . ∎

Combining Propositions 6.8, 6.12 and 6.14, we complete the proof of Theorem 1.2.

Acknowledgement

I would like to thank Professor Tomohiro Fukaya for giving me tremendous opportunities for growth through weekly instruction and off-campus placements over the past four years.

I would like to thank Dr. Kenshiro Tashiro for providing us with very useful information on horoboundary and for his very careful guidance.

I would also like to thank Dr. Yoshito Ishiki, Dr. Yuya Kodama, and Dr. Takumi Matsuka for their constant encouragement.

Finally, I would like to thank my parents for allowing me to study for a total of six years at the university and graduate school.

References

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[And18] Pavel Andreev. The cone metric of a busemann space. Journal of Geometry, 109(1):25, 2018.
[FM24] Tomohiro Fukaya and Takumi Matsuka. Boundary of free products of metric spaces. arXiv:2402.06862, 2024.
[FO20] Tomohiro Fukaya and Shin-ichi Oguni. A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture. J. Topol. Anal., 12(3):857–895, 2020.
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	$\displaystyle\sup_{u\in\bar{B}(o,R)}\|F(\xi)(u)-\phi^{\prime}_{x_{n}}(u)\|$	$\displaystyle=\sup_{u\in\bar{B}(o,R)}\|\{\xi(u)-\xi(o^{\prime})\}-\{\phi_{x_{n}}(u)-\phi_{x_{n}}(o^{\prime})\}\|$
		$\displaystyle\leq\sup_{u\in\bar{B}(o,R)}\|\xi(u)-\phi_{x_{n}}(u)\|+\|\xi(o^{\prime})-\phi_{x_{n}}(o^{\prime})\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

	$\displaystyle\sup_{u\in\bar{B}(o,L)}\|F(\xi_{m})(u)-F(\xi)(u)\|$	$\displaystyle=\sup_{u\in\bar{B}(o,L)}\|\{\xi_{m}(u)-\xi_{m}(o^{\prime})\}-\{\xi(u)-\xi(o^{\prime})\}\|$
		$\displaystyle\leq\sup_{u\in\bar{B}(o,L)}\|\xi_{m}(u)-\xi(u)\|+\|\xi_{m}(o^{\prime})-\xi(o^{\prime})\|$
		$\displaystyle\rightarrow 0\text{ as }m\rightarrow\infty.$

	$\displaystyle\|\phi_{x_{n}}(x)-\phi_{x}(x)\|$	$\displaystyle=\|(d(x,x_{n})-d(o,x_{n}))-(d(x,x)-d(o,x))\|$
		$\displaystyle=\|d(x,x_{n})-d(o,x_{n})+d(o,x)\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

	$\displaystyle\|\phi_{x_{n}}(x_{n})-\phi_{x}(x_{n})\|$	$\displaystyle=\|d(x_{n},x_{n})-d(o,x_{n})-d(x_{n},x)+d(o,x)\|$
		$\displaystyle=\|-d(o,x_{n})-d(x_{n},x)+d(o,x)\|$
		$\displaystyle=\|d(o,x_{n})+d(x_{n},x)-d(o,x)\|$
		$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty.$

	$\displaystyle\|\phi_{z_{n}}(z)-\phi_{z}(z)\|$	$\displaystyle=\|(d_{\mathcal{O}Y}(z_{n},z)-d_{\mathcal{O}Y}(0,z_{n}))-(d_{\mathcal{O}Y}(z,z)-d_{\mathcal{O}Y}(0,z))\|$
		$\displaystyle=\|(\|t_{n}-t\|+\min\{t_{n},t\}d_{Y}(y_{n},y)-t_{n})-(-t)\|$
		$\displaystyle=td_{Y}(y_{n},y)\rightarrow 0\text{ as }n\rightarrow\infty.$