⁰⁰02020 Mathematics Subject Classification. Primary 51F30; Secondary 20F65, 58B34.

Boundary of free products of metric spaces

Tomohiro Fukaya^∗ and Takumi Matsuka Tomohiro Fukaya Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-osawa Hachioji, Tokyo, 192-0397, Japan tmhr@tmu.ac.jp Takumi Matsuka Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-osawa Hachioji, Tokyo, 192-0397, Japan takumi.matsuka1@gmail.com

Abstract.

In this paper, we compute (co)homologies of ideal boundaries of free products of geodesic coarsely convex spaces in terms of those of each of the components. The (co)homology theories we consider are, $K$ -theory, Alexander-Spanier cohomology, $K$ -homology, and Steenrod homology. These computations led to the computation of $K$ -theory of the Roe algebra of free products of geodesic coarsely convex spaces via the coarse Baum–Connes conjecture.

Key words and phrases:

coarsely convex spaces, ideal boundary, cohomology,

The first author was supported by JSPS KAKENHI Grant number JP19K03471. The second author was supported by JST, the establishment of university fellowships towards the creation of science technology innovations, Grant number JPMJFS2139.

1. Introduction

Coarsely convex spaces are coarse geometric analogues of simply connected Riemannian manifolds of nonpositive sectional curvature introduced by Oguni and the first author. In [11], ideal boundaries of coarsely convex spaces are constructed, and it was shown that coarsely convex spaces are coarsely homotopy equivalent to the open cones over their ideal boundaries. Combining with Higson-Roe’s work on the coarse Baum-Connes conjecture for open cones [16], it was shown that the coarse Baum-Connes conjecture holds for proper coarsely convex spaces [11].

The class of coarsely convex spaces includes geodesic Gromov hyperbolic spaces, CAT(0) spaces, Busemann spaces, and systolic complexes [20]. Descombes and Lang [7] showed that proper injective metric spaces admit geodesic convex bicombings. This implies that proper injective metric spaces are coarsely convex. Recently, Chalopin, Chepoi, Genevois, Hirai, and Osajda [6] showed that Helly graphs are coarsely dense in their injective hulls, and these injective hulls are proper. It follows that Helly graphs are coarsely convex.

In [9], we constructed free products of metric spaces, which generalizes the construction by Bridson-Haefliger [3, Theorem II.11.18]. We showed that free products of geodesic coarsely convex spaces are geodesic coarsely convex. In [10], the authors introduced trees of spaces, which are generalizations of free products of metric spaces. They showed that if each of the components of a tree of spaces is geodesic coarsely convex, then the tree of spaces is geodesic coarsely convex. This generalized the main result of [9] for trees of spaces.

In this paper, we study the topology of ideal boundaries of trees of spaces whose components are coarsely convex. For this purpose, we introduce augmented spaces of trees of spaces. We determine the topological type of the ideal boundaries of augmented spaces. Then we compute $K$ -theory, Alexander-Spanier cohomology, and $K$ -homology of the ideal boundaries of the trees of spaces in terms of ideal boundaries of augmented spaces, and ideal boundaries of the components of the trees of spaces.

By applying Theorems 9.3 and 9.5 for free products, we have the following.

Theorem 1.1.

Let $X$ and $Y$ be proper geodesic coarsely convex spaces with nets. Suppose that the net of $X$ and that of $Y$ are coarsely dense in $X$ and $Y$ , respectively, and, both $X$ and $Y$ are unbounded. Let $X*Y$ be the free product of $X$ and $Y$ . Let $\partial X$ , $\partial Y$ , and $\partial(X*Y),$ denote the ideal boundaries of $X$ , $Y$ , and $X*Y$ , respectively. Let $\mathcal{C}$ denote the Cantor space.

Let $M^{*}=(M^{n})_{n\in\mathbb{N}}$ be the $K$ -theory or the Alexander-Spanier cohomology. Then,

\displaystyle\tilde{M}^{p}(\partial(X*Y))

\displaystyle\cong\tilde{M}^{p}(\mathcal{C})\oplus\bigoplus_{\mathbb{N}}\tilde{M}^{p}(\partial X)\oplus\bigoplus_{\mathbb{N}}\tilde{M}^{p}(\partial Y)

Let $M_{*}=(M_{n})_{n\in\mathbb{N}}$ be the $K$ -homology or the Steenrod homology. Then,

\displaystyle\tilde{M}_{p}(\partial(X*Y))

\displaystyle\cong\tilde{M}_{p}(\mathcal{C})\times\prod_{\mathbb{N}}\tilde{M}_{p}(\partial X)\times\prod_{\mathbb{N}}\tilde{M}_{p}(\partial Y).

Here $\tilde{M}_{p}$ and $\tilde{M}^{p}$ denote the reduced (co)homology corresponding to $M_{p}$ and $M^{p}$ .

Let $Y$ be a metric space. We denote by $C^{*}(Y)$ the Roe algebra of $Y$ . The coarse Baum-Connes conjecture states that the $K$ -theory of the Roe algebra $C^{*}(Y)$ is isomorphic to the coarse $K$ -homology $KX_{*}(Y)$ of $Y$ . As mentioned above, it is proved in [11] that the coarse Baum-Connes conjecture holds for proper coarsely convex spaces. It is also proved that the coarse $K$ -homology of a coarsely convex space $Y$ is isomorphic to the reduced $K$ -homology of the ideal boundary $\partial Y$ with sifting the degree by one [11, Theorem 6.7.]. Therefore we can compute the $K$ -theory of the Roe algebra of a free product coarsely convex spaces as follows.

For a $C^{*}$ -algebra $A$ , we denote by $K_{p}(A)$ the operator $K$ -theory of $A$ . We denote by $\tilde{K}_{p}(-)$ the reduced $K$ -homology.

Theorem 1.2.

\displaystyle K_{p}(C^{*}(X*Y))

\displaystyle\cong\tilde{K}_{p-1}(\mathcal{C})\times\prod_{\mathbb{N}}\tilde{K}_{p-1}(\partial X)\times\prod_{\mathbb{N}}\tilde{K}_{p-1}(\partial Y).

Corollary 1.3.

Let $X$ and $Y$ be proper geodesic coarsely convex spaces with nets satisfying the condition of Theorem 1.2. We have

\displaystyle K_{p}(C^{*}(X*Y))

\displaystyle\cong\tilde{K}_{p-1}(\mathcal{C})\times\prod_{\mathbb{N}}K_{p}(C^{*}(X))\times\prod_{\mathbb{N}}K_{p}(C^{*}(Y)).

It might be interesting to compare Corollary 1.3 with a formula for $K$ -theories of the group $C^{*}$ algebras of free products of groups [2, 10.11.11 (f) to (h)].

We also study the topological dimensions of the ideal boundaries of trees of spaces. Applying Theorem 12.4 for free products of metric spaces, we obtain a formula of the dimension.

Theorem 1.4.

Let $X$ and $Y$ be geodesic coarsely convex spaces with nets.

\displaystyle\mathop{\mathrm{dim}}\nolimits(\partial(X*Y))=\max\{\mathop{\mathrm{dim}}\nolimits\partial X,\mathop{\mathrm{dim}}\nolimits\partial Y\}.

We remark that Theorem 1.4 is an analogue of the formula for the cohomological dimensions of the free products of groups. Namely, in the setting of Theorem 1.4, if we assume that both $X$ and $Y$ admits geometric action by group $G$ and $H$ , respectively, and both $G$ and $H$ admits a finite model for the classifying space $BG$ and $BH$ , respectively, then the above formula follows from a well-known formula on the cohomological dimensions of free products of groups. See Section 12.2 for details.

1.1. Outline

In Section 2, based on the work of Groves and Manning, we introduce combinatorial horoballs for metric spaces with lattices, and study the shape of geodesics in them.

In Sections 3, 4 and 5, we review the coarsely convex bicombings and the construction of ideal boundaries. In this paper, we only deal with geodesic coarsely convex space, so in Section 5, we restrict results in previous sections to geodesic coarsely convex bicombings, and we improve some of them.

In Section 6, we prepare some technical lemmata to show the continuity of certain retractions on ideal boundaries, which play key roles in the computations of (co)homologies.

In Sections 7 and 8, we introduce trees of spaces and augmented spaces. We construct geodesic coarsely convex bicombings on them assuming that each component admits such bicombings.

In Section 9, we compute (co)homologies of ideal boundaries of trees of geodesic coarsely convex spaces by using Mayer-Vietoris exact sequences. Key tools are retractions on ideal boundaries. We give proofs of Theorems 1.1, 1.2 and 9.7.

In Section 10, we construct the retractions used in Section 9 and show that they are continuous. Here we use the results in Section 6.

In Section 11, we study the topological type of the ideal boundary of the augmented spaces. We show (Theorem 11.4) that under certain conditions, the ideal boundaries of the augmented spaces are homeomorphic to the Cantor space.

In Section 12, we show a formula for the topological dimensions of the ideal boundaries of trees of geodesic coarsely convex spaces in terms of the dimensions of each component. In Section 12.2, we compare these results with a well known formula for the cohomological dimensions of the free products of groups.

2. Combinatorial and metric horoballs

All materials in this section are slight modifications of the work of Groves and Manning [15] on the combinatorial horoballs.

In the rest of this section, we assume that $X$ is a geodesic metric space.

Definition 2.1.

Let $(X,d_{X})$ be a metric space and $A\subset X$ . Let $C\geq 0$ .

(1)

The subset $A$ is $C$ -discrete if for all $a,a^{\prime}\in A$ with $a\neq a^{\prime}$ , we have $d_{Z}(a,a^{\prime})\geq C$ .
(2)

The subset $A$ is $C$ -dense if for all $x\in X$ , there exists $a\in A$ such that $d_{Z}(x,a)\leq C$ .

Remark 2.2.

A maximal $C$ -discrete subset $A\subset X$ is $2C$ -dense.

Definition 2.3.

A metric space with a lattice is a pair $(X,X^{(0)})$ of metric space $(X,d_{X})$ and 1-discrete 2-dense subset $X^{(0)}\subset X$ . We call $X^{(0)}$ the lattice of $X$ .

Groves and Manning defined an augmented space and introduced an equivalent definition of relatively hyperbolic groups [15]. The augmented space is constructed by gluing a ”combinatorial horoballs” based on each $P\in\mathcal{P}$ onto the Cayley graph of $G$ . They defined that $G$ is hyperbolic relative to $\mathcal{P}$ if the augmented space is hyperbolic.

The construction presented here is closely modeled on Groves–Manning’s combinatorial horoballs. They constructed horoballs based on an arbitrary connected graph, and showed that such a horoball is always $\delta$ –hyperbolic. A minor modification of the Groves–Manning construction produces connected horoballs based on any metric space.

Definition 2.4.

Let $(X^{(0)},d_{X})$ be a 1-discrete metric space. A combinatorial horoball $\mathcal{H}_{\mathsf{comb}}(X^{(0)})$ is a graph with a set of vertices $\mathcal{H}_{\mathsf{comb}}(X^{(0)})^{(0)}$ and a set of edges $\mathcal{H}_{\mathsf{comb}}(X^{(0)})^{(1)}$ defined as follows:

(1)

$\mathcal{H}_{\mathsf{comb}}(X^{(0)})^{(0)}\coloneqq X^{(0)}\times(\mathbb{N}\cup\{0\})$ .
(2)
$\mathcal{H}_{\mathsf{comb}}(X^{(0)})^{(1)}$ contains the following two types of edges:
1. (a)
  
  For each $x\in X^{(0)}$ and $l\in\mathbb{N}\cup\{0\}$ , there exists a vertical edge connecting $(x,l)$ and $(x,l+1)$ .
2. (b)
  
  For each $x,y\in X^{(0)}$ and $l\in\mathbb{N}$ , if $0<d_{X}(x,y)\leq 2^{l}$ holds for some $l\in\mathbb{N}$ , there exists a horizontal edge connecting $(x,l)$ and $(y,l)$ .

We endow $(X^{(0)},d_{X})$ with a graph metric such that each edge has length one, and we consider $(X^{(0)},d_{X})$ as a geodesic space.

Let $\mathcal{H}_{\mathsf{comb}}(X^{(0)})$ be a combinatorial horoball. A horizontal segment is a path consisting only of horizontal edges. A vertical segment is a path consisting only of vertical edges. Each $x\in\mathcal{H}_{\mathsf{comb}}(X^{(0)})^{(0)}$ can be identified with $x=(x_{0},l)$ , where $x_{0}\in X^{(0)}$ and $l\in\mathbb{N}\cup\{0\}$ . We say that the depth of $x$ is $l$ , denoted by $D(x)=l$ . Moreover, let $p$ be a horizontal segment. When the depth of a point on the horizontal segment $p$ is $l$ , we say that the depth of the horizontal segment $p$ is $l$ , denoted by $D(p)=l$ .

Definition 2.5.

Let $(X,X^{(0)})$ be a metric space with a lattice. An metric horoball $\mathcal{H}(X,X^{(0)})$ is a quotient space of $X\sqcup\mathcal{H}_{\mathsf{comb}}(X^{(0)})$ by the equivalent relation generated by

\displaystyle\mathcal{H}_{\mathsf{comb}}(X^{(0)})^{(0)}\ni(x_{0},0)\sim x_{0}\in X^{(0)}\subset X

for any $x_{0}\in X^{(0)}$ .

Geodesics in combinatorial horoballs are particularly easy to understand.

Proposition 2.6 ([15], Lemma 3.10).

Let $x,y\in\mathcal{H}(X,X^{(0)})$ . There exists a geodesic segment on $\mathcal{H}(X,X^{(0)})$ from $x$ to $y$ which consists of at most two vertical segments and a single horizontal segment of length at most $5$ . We say that this geodesic segment is a normal geodesic segment, denoted by $\mathtt{n}\gamma(x,y)$ . Moreover, for each geodesic segment from $x$ to $y$ , denoted by $\gamma(x,y)$ , we have

\displaystyle d_{H}(\mathrm{Im}(\gamma(x,y)),\mathrm{Im}(\mathtt{n}\gamma(x,y)))\leq 4

,where $d_{H}$ is the Hausdorff metric on $X$ .

Proof..

Let $\gamma(x,y)$ be a geodesic segment from $x$ to $y$ . We can put

\displaystyle\gamma(x,y)=p_{1}\ast p_{2}\ast\cdots\ast p_{n}

such that for all $i$ , we have that if $p_{i}$ is a horizontal segment, then $p_{i+1}$ is a vertical segment, and if $p_{i}$ is a vertical segment, then $p_{i+1}$ is a horizontal segment. We can assume that $p_{1}$ and $p_{n}$ are vertical segments. Let $s(p_{i})$ be an initial point of $p_{i}$ and let $t(p_{1})$ be a terminal point of $p_{i}$ . Note that $t(p_{i})=s(p_{i+1})$ , $s(p_{1})=x$ , and $t(p_{n})=y$ .

First, it is easily shown that there does not exist a vertical segment $p$ such that $D(s(p))>D(t(p))$ . We will show that as a consequence of lemmas.

Claim 2.7.

Let $D_{M}$ be the maximum depth of horizontal segments which compose $\gamma(x,y)$ . If there exists a horizontal segment $p$ which composes $\gamma(x,y)$ such that $D(p)<D_{M}$ , then the length of $p$ is $1$ .

Let $p_{i}$ be a horizontal segment with $D(p_{i})<D_{M}$ . Suppose the length of $p_{i}$ is $2$ . Let $s(p_{i})=(x_{i},D(p_{i}))$ and $t(p_{i})=(y_{i},D(p_{i}))$ . Note that $p_{i+1}$ is a vertical segment and $t(p_{i})=s(p_{i+1})$ . We denote the depth of $t(p_{i+1})$ by $D_{1}$ . Since $D(p_{i})<D_{1}$ , there exists a horizontal edge from $(x_{i},D_{1})$ to $(y_{i},D_{1})$ , denoted by $q_{2}$ . Let $q_{1}$ be the vertical segment from $s(p_{i})=(x_{i},D(p_{i}))$ to $(x_{i},D_{1})$ . The path $q_{1}\ast q_{2}$ is a geodesic segment from $s(p_{i})$ to $t(p_{i+1})$ . The length of $q_{1}\ast q_{2}$ is shorter than $p_{i}\ast p_{i+1}$ . This is a contradiction.

Claim 2.8.

There does not exist a subgeodesic $h_{1}\ast v_{1}\ast h_{2}\ast v_{2}$ such that $h_{i}$ are horizontal segments and $v_{i}$ are vertical segments.

Suppose that there exists a subgeodesic $h_{1}\ast v_{1}\ast h_{2}\ast v_{2}$ such that $h_{i}$ are horizontal segments and $v_{i}$ are vertical segments. Let $D(h_{1})=D_{1}$ , $D(h_{2})=D_{2}$ , and $D(t(v_{2}))=D$ . By Claim 2.7, the length of $h_{1}$ and $h_{2}$ are $1$ . We can put $s(h_{1})=(x_{1},D_{1})$ , $s(h_{2})=(x_{2},D_{2})$ , and $t(v_{2})=(x_{3},D)$

	$\displaystyle d_{X}(x_{1},x_{3})$	$\displaystyle\leq d_{X}(x_{1},x_{2})+d_{X}(x_{2},x_{3})$
		$\displaystyle\leq 2^{D_{1}}+2^{D_{2}}$
		$\displaystyle\leq 2^{D_{2}+1}\leq 2^{D}.$

Therefore, there exits a horizontal edge $q_{2}$ from $(x_{1},D)$ to $t(v_{2})$ . Now, we define a path $q_{1}$ to be a vertical segment from $s(h_{1})=(x_{1},D_{1})$ to $(x_{1},D)$ . The length of the path $q_{1}\ast q_{2}$ is smaller than $h_{1}\ast v_{1}\ast h_{2}\ast v_{2}$ . This is a contradiction.

Therefore, a geodesic segment $\gamma(x,y)$ has the following form:

\displaystyle\gamma(x,y)=v_{1}\ast h_{1}\ast v_{2}\ast h_{2}\ast v_{3}\ast h_{3}\ast v_{4},

where $v_{i}$ are vertical segments and $h_{i}$ are horizontal segments. Note that $h_{2}$ is the highest horizontal segment. By Claim 2.7 and Claim 2.8, we have that the length of $h_{i}$ is $1$ for $i\in\{1,3\}$ . Put $s(h_{1})=(x,D(h_{1}))$ and $t(h_{3})=(y,D(h_{3}))$ . We denote by $v^{\prime}_{1}$ , a vertical segment from $(x,D(h_{1}))$ to $(x,D_{M})$ . and denote by $v^{\prime}_{2}$ , a vertical segment from $(y,D_{M})$ and $t(h_{3})=(y,D(h_{3}))$ . There exists a geodesic segment from $(x,D_{M})$ to $(y,D_{M})$ , denoted by $h_{M}$ . We replace $h_{1}\ast v_{2}\ast h_{2}\ast v_{3}\ast h_{3}$ with $v^{\prime}_{1}\ast h_{M}\ast v^{\prime}_{2}$ . The replaced path is also a geodesic segment from $x$ to $y$ . Therefore, we obtain a normal geodesic segment from $x$ to $y$ .

For $x,y\in\mathcal{H}(X,X^{(0)})$ , let $\mathtt{n}\gamma(x,y)=u\ast h\ast v$ , where $u$ and $v$ are vertical segments and $h$ is a horizontal segment. Finally, we will show that the length of $h$ is at most $5$ . Suppose that the length of $h$ is $6$ . Let $s(h)=(p,D(h))$ and $t(h)=(q,D(h))$ . We can connect $(p,D(h)+1)$ and $(q,D(h)+1)$ with three horizontal edges. Then, there exists a path of length $5$ connecting $s(h)$ and $t(h)$ . This is the contradiction.

Moreover, for each geodesic segment from $x$ to $y$ , denoted by $\gamma(x,y)$ , we have

\displaystyle d_{H}(\mathrm{Im}(\gamma(x,y)),\mathrm{Im}(\mathtt{n}\gamma(x,y)))\leq 4.

This completes the proof. ∎

Proposition 2.9 ([15]).

$\mathcal{H}(X,X^{(0)})$ is a Gromov hyperbolic space.

Proof..

Let $x,y,z\in X^{(0)}$ . We denote by $[x,y]$ , an image of a geodesic segment on $\mathcal{H}(X,X^{(0)})$ from $x$ to $y$ . Let $p_{1}\in[x,y]$ , $p_{2}\in[y,z]$ , and $p_{3}\in[z,x]$ . We define

\displaystyle\mathrm{diam}\{p_{1},p_{2},p_{3}\}\coloneqq\max\{d_{X}(p_{i},p_{j})\colon i\neq j\}.

One of the equivalent definitions of Gromov hyperbolicity is that the following quantity

\displaystyle\min\{\mathrm{diam}\{p_{1},p_{2},p_{3}\}\colon p_{1}\in[x,y],p_{2}\in[y,z],p_{3}\in[z,x]\}

is bounded by a constant independent of choices of geodesic segments and geodesic triangles. See [13, 21.- Proposition] for details. We say that this quantity is the minimum diameter of geodesic triangle $[x,y]\cup[y,z]\cup[z,x]$ .

By Proposition 2.6, there exists a normal geodesic segment connecting each two points, that is,

•

$\mathtt{n}\gamma(x,y)=u_{1}\ast h_{1}\ast v_{1}$ ,
•

$\mathtt{n}\gamma(y,z)=u_{2}\ast h_{2}\ast v_{2}$ ,
•

$\mathtt{n}\gamma(z,x)=u_{3}\ast h_{3}\ast v_{3}$ ,

where $u_{i}$ and $v_{i}$ are vertical segments and each $h_{i}$ is a horizontal segment. Suppose $D(h_{1})\leq D(h_{2})\leq D(h_{3})$ .

It is easily shown that $\mathop{\mathrm{Im}}\nolimits(u_{1})\subset\mathop{\mathrm{Im}}\nolimits(v_{3})$ and $\mathop{\mathrm{Im}}\nolimits(v_{1})\subset\mathop{\mathrm{Im}}\nolimits(u_{2})$ . Then, we have

\displaystyle\mathop{\mathrm{Im}}\nolimits(h_{1})\subset N(\mathop{\mathrm{Im}}\nolimits(v_{3}),3)\cup N(\mathop{\mathrm{Im}}\nolimits(u_{2}),3).

Therefore,

\displaystyle\min\{\mathrm{diam}\{p_{1},p_{2},p_{3}\}\colon p_{1}\in\mathop{\mathrm{Im}}\nolimits(\mathtt{n}\gamma(x,y)),p_{2}\in\mathop{\mathrm{Im}}\nolimits(\mathtt{n}\gamma(y,z)),p_{3}\in\mathop{\mathrm{Im}}\nolimits(\mathtt{n}\gamma(z,x))\}

is bounded by $5$ . By Proposition 2.6, for each geodesic segment from $x$ to $y$ , denoted by $\gamma(x,y)$ , we have

\displaystyle d_{H}(\mathrm{Im}(\gamma(x,y)),\mathrm{Im}(\mathtt{n}\gamma(x,y)))\leq 4.

For any $x,y,z\in\mathcal{H}(X,X^{(0)})$ and any geodesic triangle $\Delta(x,y,z)$ , the minimal diameter is bounded by $9$ . Therefore, $\mathcal{H}(X,X^{(0)})$ is a Gromov hyperbolic space. {comment} It is easily shown that $\mathop{\mathrm{Im}}\nolimits(u_{1})\subset\mathop{\mathrm{Im}}\nolimits(v_{3})$ and $\mathop{\mathrm{Im}}\nolimits(v_{1})\subset\mathop{\mathrm{Im}}\nolimits(u_{2})$ . Then, we have

\displaystyle\mathop{\mathrm{Im}}\nolimits(h_{1})\subset N(\mathop{\mathrm{Im}}\nolimits(v_{3}),3)\cup N(\mathop{\mathrm{Im}}\nolimits(u_{2}),3).

Next, we will show that $H(h_{3})-H(h_{2})$ is bounded. Let $s(h_{3})=(z,H(h_{3}))$ , $t(h_{3})=(x,H(h_{3}))$ , and $s(h_{2})=(y,H(h_{2}))$ . Since the length of $h_{i}$ is at most $4$ , we have

	$\displaystyle d(x,z)$	$\displaystyle\leq d(x,y)+d(y,z)$
		$\displaystyle\leq 4\cdot 2^{H(h_{1})}+4\cdot 2^{H(h_{2})}$
		$\displaystyle\leq 8\cdot 2^{H(h_{2})}\leq 2^{H(h_{2})+4}$

Therefore, we have $H(h_{3})\leq H(h_{2})+4$ . This implies that

	$\displaystyle\mathop{\mathrm{Im}}\nolimits(h_{3})$	$\displaystyle\subset N(\mathop{\mathrm{Im}}\nolimits(h_{2}),8),$
	$\displaystyle\mathop{\mathrm{Im}}\nolimits(h_{2})$	$\displaystyle\subset N(\mathop{\mathrm{Im}}\nolimits(h_{3}),8).$

Finally,

\displaystyle\mathop{\mathrm{Im}}\nolimits(v_{3})\setminus\mathop{\mathrm{Im}}\nolimits(u_{1})

\displaystyle\subset

∎

3. coarsely convex bicombing

Let $(X,d_{X})$ be a metric space. For $\lambda\geq 1$ and $k\geq 0$ , a $(\lambda,k)$ -quasi-geodesic bicombing on $X$ is a map $\gamma\colon X\times X\times[0,1]\to X$ such that for $x,y\in X$ , we have $\gamma(x,y,0)=x$ , $\gamma(x,y,1)=y$ , and

\displaystyle\lambda^{-1}\left\lvert t-s\right\rvert d_{X}(x,y)-k\leq d_{X}(\gamma(x,y,t),\gamma(x,y,s))\leq\lambda\left\lvert t-s\right\rvert d_{X}(x,y)+k\ \ \ \ (t,s\in[0,1]).

If the space $X$ admits a $(\lambda,k)$ -quasi-geodesic bicombing for some $\lambda\geq 1$ and $k>0$ , we say that $X$ admits a quasi-geodesic bicombing. If a group $G$ acts on $X$ by isometries, a geodesic bicombing $\gamma:X\times X\times[0,1]\to X$ is $G$ -equivariant if

\displaystyle g\cdot\gamma(x,y)(t)=\gamma(gx,gy)(t)

holds for any $g\in G$ , $x,y\in X$ , and $t\in[0,1]$ .

Definition 3.1.

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.

A $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing on a metric space $(X,d_{X})$ is a $(\lambda,k)$ -quasi-geodesic bicombing $\gamma\colon X\times X\times[0,1]\to X$ with the following:

(i)

Let $x_{1},x_{2},y_{1},y_{2}\in X$ and let $a,b\in[0,1]$ . Set $y_{1}^{\prime}:=\gamma(x_{1},y_{1},a)$ and $y_{2}^{\prime}:=\gamma(x_{2},y_{2},b)$ . Then, for $c\in[0,1]$ , we have

\displaystyle d_{X}(\gamma(x_{1},y_{1},ca),\gamma(x_{2},y_{2},cb))\leq(1-c)Ed_{X}(x_{1},x_{2})+cEd_{X}(y_{1}^{\prime},y_{2}^{\prime})+C.

(ii)

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

\displaystyle\left\lvert td_{X}(x_{1},y_{1})-sd_{X}(x_{2},y_{2})\right\rvert\leq\theta(d_{X}(x_{1},x_{2})+d_{X}(\gamma(x_{1},y_{1},t),\gamma(x_{2},y_{2},s))).

The following reparametrization is used to construct ideal boundaries in Section 4

Definition 3.2.

Let $X$ be a metric space and let $\gamma\colon X\times X\times[0,1]\to X$ be a $(\lambda,k)$ -quasi-geodesic bicombing on $X$ . A reparametrised bicombing of $\gamma$ is a map

\displaystyle\textup{{rp}}\gamma\colon X\times X\times\mathbb{R}_{\geq 0}\to X

defined by

\displaystyle\textup{{rp}}\gamma(x,y,t):=\begin{cases}\gamma\left(x,y,{t}/{d_{X}(x,y)}\right)&\text{if }t\leq d_{X}(x,y)\\ y&\text{if }t>d_{X}(x,y)\end{cases}.

4. Ideal boundary

In this section, we review the construction of the ideal boundary of a coarsely convex space.

Let $(X,d_{X})$ be a metric space equipped with a $(\lambda,k,E,C,\theta)$ -coarsely convex bicombing $\gamma\colon X\times X\times[0,1]\to X$ . We choose a base point $e\in X$ .

4.1. Gromov product and sequential boundary

Definition 4.1.

Set $k_{1}=\lambda+k$ , $D=2(1+E)k_{1}+C$ , and $D_{1}=2D+2$ . We define a product $(\cdot\mid\cdot)_{e}\colon X\times X\to\mathbb{R}_{\geq 0}$ by

\displaystyle(x\mid y)_{e}:=\min\{d_{X}(e,x),\ d_{X}(e,y),\ \sup\{t\in\mathbb{R}_{\geq 0}:d_{X}(\textup{{rp}}\gamma(e,x,t),\textup{{rp}}\gamma(e,y,t))\leq D_{1}\}\},

We abbreviate $(x\mid y)_{e}$ by $(x\mid y)$ .

Lemma 4.2 ([11, Lemma 4.8]).

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

\displaystyle(x\mid z)\geq D_{2}^{-1}\min\{(x\mid y),(y\mid z)\}.

We use a sequential model of the ideal boundary introduced in [12]. The idea is based on [14] and [13, Chapitre 7].

Definition 4.3.

Let

\displaystyle S_{\infty}(X)=\{(x_{i}):(x_{i}\mid x_{j})\to\infty\text{ as }i,j\to\infty\}

and define a relation $\sim$ on $S_{\infty}(X)$ as follows. For every $(x_{i}),(y_{i})\in S_{\infty}(X)$ , we have $(x_{i})\sim(y_{i})$ if

\displaystyle(x_{i}\mid y_{i})\to\infty\text{ as }i\to\infty

Lemma 4.4.

The relation $\sim$ is an equivalence relation on $S_{\infty}(X)$ .

Proof..

It is clear that the relation $\sim$ is reflexive and symmetric. Lemma 4.2 implies that it is transitive. ∎

Definition 4.5.

Let

\displaystyle\partial X=S_{\infty}(X)/\sim\quad\text{ and }\quad\overline{X}=X\cup\partial X.

For $x\in X$ and a sequence $(x_{i})$ in $X$ , we write $(x_{i})\in x$ if $x_{i}=x$ for every $i\in\mathbb{N}$ . We extend the Gromov product $(\cdot\mid\cdot)\colon X\times X\to\mathbb{R}_{\geq 0}$ to a symmetric function $(\cdot\mid\cdot)\colon\overline{X}\times\overline{X}\to\mathbb{R}_{\geq 0}\cup\{\infty\}$ by letting

(1)

\displaystyle(x\mid y)

\displaystyle=\sup\{\liminf_{i\to\infty}(x_{i}\mid y_{i}):(x_{i})\in x,(y_{i})\in y\}

for $x,y\in\overline{X}$ .

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

	$\displaystyle V_{n}$	$\displaystyle=\{(x,y)\in\overline{X}\times\overline{X}:(x\mid y)>n\}\cup\{(x,y)\in X\times X:d_{X}(x,y)<1/n\},$
	$\displaystyle V_{n}[x]$	$\displaystyle=\{y\in\overline{X}:(x,y)\in V_{n}\},\quad(x\in\overline{X}).$

By [12, Lemma 2.6.], $\{V_{n}:n\in\mathbb{N}\}$ is a base of a metrizable uniformity on $\overline{X}$ . See also [11, Section 4.3].

Definition 4.6.

Let $\overline{X}$ be equipped with the topology $\mathcal{T}_{(\cdot\mid\cdot)}$ generated by the family $\{V_{n}[x]:x\in\overline{X},n\in\mathbb{N}\}$ . We call the subspace $\partial X$ of $\overline{X}$ the Gromov boundary of $X$ with respect to $(\cdot\mid\cdot)$ . We also call $\partial X$ the ideal boundary of $X$ .

Lemma 4.7 ([12, Lemma 2.8, Theorem 2.9]).

We have the following:

(i)

The relative topology $\mathcal{T}_{(\cdot\mid\cdot)}\!\!\upharpoonright_{X}$ on $X$ with respect to $\overline{X}$ coincides with the topology $\mathcal{T}_{d}$ induced by the metric $d$ .
(ii)

$X$ is a dense open subset in $\overline{X}$ .
(iii)

If $(X,d)$ is complete, then so is $(\overline{X},d)$ .
(iv)

If $(X,d)$ is proper, then $(\overline{X},d)$ is compact.

4.2. Combing at infinity

Lemma 4.8.

Suppose that $X$ is proper. Then there exists a map

\displaystyle\textup{{rp}}\bar{\gamma}\colon X\times\overline{X}\times\mathbb{R}_{\geq 0}\to X

satisfying the following:

(i)

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

For $(e,x)\in X\times\partial X$ , there exists a sequence $(x_{n})$ in $X$ such that the sequence of maps $(\textup{{rp}}\gamma(e,x_{n},-)\!\!\upharpoonright_{\mathbb{N}}:\mathbb{N}\to X)_{n}$ converges to $\textup{{rp}}\bar{\gamma}(e,x,-)\!\!\upharpoonright_{\mathbb{N}}$ pointwise.
(iii)

For $(e,x)\in X\times\partial X$ , we have

$\displaystyle(\textup{{rp}}\bar{\gamma}(e,x,t)\mid x)_{e}\to\infty\quad\text{as}\quad t\to\infty$
(iv)

For $(e,x)\in X\times\partial X$ , the map $\textup{{rp}}\bar{\gamma}(e,x,-)\colon\mathbb{R}_{\geq 0}\to X$ is a $(\lambda,k_{1})$ -quasi geodesic, where $k_{1}:=\lambda+k$ .

Proof..

For $(x,y)\in X\times X$ , we define $\textup{{rp}}\gamma(x,y,-)$ by the equality in (i). We use [11, Proposition 4.17] to construct maps $\textup{{rp}}\bar{\gamma}(e,x,-)$ for $(e,x)\in X\times\partial X$ satisfying (ii) and (iii). Finally, (iv) follows from [11, Lemma 4.1]. ∎

Definition 4.9.

We call the map $\textup{{rp}}\bar{\gamma}$ given in Lemma 4.8 an extended bicombing on $X\times\overline{X}$ corresponding to $\gamma$ . For $(e,x)\in X\times\partial X$ , we abbreviate $\textup{{rp}}\bar{\gamma}(e,x,-)$ by $\gamma_{e}^{x}(-)$ .

Lemma 4.10 ([11, Proposition 4.2]).

The extended bicombing $\textup{{rp}}\bar{\gamma}$ on $X\times\overline{X}$ satisfies the following:

(1)

Let $(e_{1},x_{1}),(e_{2},x_{2})\in X\times\partial X$ and let $a,b\in\mathbb{R}_{\geq 0}$ . Set $x_{1}^{\prime}:=\textup{{rp}}\bar{\gamma}(e_{1},x_{1},a)$ and $x_{2}^{\prime}:=\textup{{rp}}\bar{\gamma}(e_{2},x_{2},b)$ . Then, for $c\in[0,1]$ , we have

\displaystyle d_{X}(\textup{{rp}}\bar{\gamma}(e_{1},x_{1},ca),\textup{{rp}}\bar{\gamma}(e_{2},x_{2},cb))\leq(1-c)Ed_{X}(e_{1},e_{2})+cEd_{X}(x_{1}^{\prime},x_{2}^{\prime})+D

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

(2)

We define a non-decreasing function $\tilde{\theta}\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ by $\tilde{\theta}(t):=\theta(t+1)+1$ . Let $(e_{1},x_{1}),(e_{2},x_{2})\in X\times\partial X$ Then for $t,s\in\mathbb{R}_{\geq 0}$ , we have

\displaystyle\left\lvert t-s\right\rvert\leq\tilde{\theta}(d_{X}(e_{1},e_{2})+d_{X}(\textup{{rp}}\bar{\gamma}(e_{1},x_{1},t),\textup{{rp}}\bar{\gamma}(e_{2},x_{2},s))).

Definition 4.11.

For $x,y\in\partial X$ , we define

\displaystyle(\gamma_{e}^{x}\mid\gamma_{e}^{y})_{e}:=\sup\{t\in\mathbb{R}_{\geq 0}:d_{X}(\gamma_{e}^{x}(t),\gamma_{e}^{y}(t))\leq D_{1}\}.

For $x\in\partial X$ and for $p\in X$ , we define

\displaystyle(\gamma_{e}^{x}\mid p)_{e}=(p\mid\gamma_{e}^{x})_{e}:=\min\{d_{X}(e,p),\sup\{t\in\mathbb{R}_{\geq 0}:d_{X}(\gamma(e,p,t),\gamma_{e}^{x}(t))\leq D_{1}\}\}.

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 $x,y\in\partial X$ , we have

$\displaystyle(\gamma_{e}^{x}\mid\gamma_{e}^{y})\leq(x\mid y)\leq\Omega(\gamma_{e}^{x}\mid\gamma_{e}^{y}).$

(2)

For triplet $x,y,z\in\partial X$ , we have

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

(3)

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

$\displaystyle(x\mid z)\geq\Omega^{-1}\min\{(x\mid y),(y\mid z)\}.$
(4)

Let $x,y\in\partial X$ . For all $t\in\mathbb{R}_{\geq 0}$ with $t\leq(\gamma_{e}^{x}\mid\gamma_{e}^{y})$ , we have

$\displaystyle d_{X}(\gamma_{e}^{x}(t),\gamma_{e}^{y}(t))\leq\Omega.$
(5)

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

$\displaystyle d_{X}(\textup{{rp}}\bar{\gamma}(e,x,t),\textup{{rp}}\bar{\gamma}(e,y,t))\leq\Omega.$
(6)

Let $e\in X$ and $x\in\partial X$ . For $v\in X$ and $t\in[0,1]$ , we have

$\displaystyle(x\mid\gamma(e,v,t))_{e}\geq\Omega^{-1}\min\{(x\mid v)_{e},td_{X}(e,v)\}.$

Proof..

(1) to (5) are [8, Lemma 2.9.]. We give a proof of (6). By the definition, we have $(x\mid\gamma(e,v,t))_{e}\geq td_{X}(e,v)$ . We suppose that $(x\mid v)\leq td_{X}(e,v)$ . Set $s=(x\mid v)$ . By [11, lemma 4.7.], we have $d_{X}(\gamma_{e}^{x}(s),\textup{{rp}}\gamma(e,v,s))\leq D_{1}+k_{1}$ . Set $s^{\prime}\coloneqq s/(E(D_{1}+k_{1}))$ . Then,

\displaystyle d_{X}(\gamma_{e}^{x}(s^{\prime}),\textup{{rp}}\gamma(e,v,s^{\prime}))\leq\frac{E}{E(D_{1}+k_{1})}d_{X}(\gamma_{e}^{x}(s),\textup{{rp}}\gamma(e,v,s))+D\leq D+1\leq D_{1}

Thus $(x\mid\gamma(e,v,t))\geq(E(D_{1}+k_{1}))^{-1}(x\mid v)$ .

∎

4.3. Visual maps

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. We say that a map $f\colon X\to Y$ is a large scale Lipschitz map, if there exists $L>1$ such that

\displaystyle d_{Y}(f(x),f(y))\leq Ld_{X}(x,y)+L.

Let $\gamma_{x}$ and $\gamma_{y}$ be coarsely convex bicombings on $X$ and $Y$ , respectively. Let $a\in X$ and $b\in Y$ be base points of $X$ and $Y$ , respectively.

Definition 4.13.

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

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 sequence $(x_{n})_{n}$ representing $x$ . Then we define $\partial f(x)$ to be the equivalence class of the sequence $(f(x_{n}))_{n}$ . By [8, Proposition 3.5], $\partial f(x)$ does not depend on the choice of $(x_{n})_{n}$ . We say that $\partial f$ is induced by $f$ .

Proposition 4.14 ([8, Corollary 3.6]).

Let $f$ be a visual map. Then the induced map $\partial f\colon\partial X\to\partial Y$ is continuous.

5. Geodesic coarsely convex spaces

5.1. Geodesic combing at infinity

Let $(X,d_{X})$ be a metric space. A geodesic bicombing on $X$ is a map $\gamma:X\times X\times[0,1]\to X$ such that for $x,y\in X$ , we have $\gamma(x,y,0)=x$ , $\gamma(x,y,1)=y$ , and

\displaystyle d_{X}(\gamma(x,y,t),\gamma(x,y,s))=\left\lvert t-s\right\rvert d_{X}(x,y)\quad(t,s\in[0,1])

Definition 5.1.

Let $E\geq 1$ and $C\geq 0$ be constants. An $(E,C)$ -geodesic coarsely convex bicombing on a metric space $(X,d_{X})$ is a geodesic bicombing $\gamma\colon X\times X\times[0,1]\to X$ satisfying the following:

Let $x_{1},x_{2},y_{1},y_{2}\in X$ and let $a,b\in[0,1]$ . Set $y_{1}^{\prime}:=\gamma(x_{1},y_{1},a)$ and $y_{2}^{\prime}:=\gamma(x_{2},y_{2},b)$ . Then, for $c\in[0,1]$ , we have

\displaystyle d_{X}(\gamma(x_{1},y_{1},ca),\gamma(x_{2},y_{2},cb))\leq(1-c)Ed_{X}(x_{1},x_{2})+cEd_{X}(y_{1}^{\prime},y_{2}^{\prime})+C.

Lemma 5.2.

An $(E,C)$ -geodesic coarsely convex bicombing $\gamma$ on a metric space $(X,d_{X})$ is a $(1,0,E,C,\mathrm{id}_{\mathbb{R}_{\geq 0}})$ -coarsely convex bicombing on $(X,d_{X})$ .

Proof..

It follows from the definition of the $(E,C)$ -geodesic coarsely convex bicombing, that $\gamma$ satisfies (i) of Definition 3.1.

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

\displaystyle\left\lvert td_{X}(x_{1},y_{1})-sd_{X}(x_{2},y_{2})\right\rvert\leq d_{X}(x_{1},x_{2})+d_{X}(\gamma(x_{1},y_{1},t),\gamma(x_{2},y_{2},s)).

by the triangle inequality. Therefore $\gamma$ satisfies (ii) of Definition 3.1 with $\theta=\textrm{id}_{\mathbb{R}_{\geq 0}}$ . ∎

Example 5.3.

Let $(X,d_{X})$ be a geodesic $\delta$ -hyperbolic space. For $x,y\in X$ , we choose a geodesic $\gamma_{x,y}\colon[0,d_{X}(x,y)]\to X$ such that $\gamma_{x,y}(0)=x$ and $\gamma_{x,y}(d_{X}(x,y))=y$ . We define $\gamma\colon X\times X\times[0,1]\to X$ by

\displaystyle\gamma(x,y,t):=\gamma_{x,y}(t/d_{X}(x,y)).

Then $\gamma$ is a $(1,8\delta)$ -geodesic coarsely convex bicombing. See [13, Chapitre 2. 25.- Proposition] for details.

Let $X$ be a space equipped with an $(E,C)$ -geodesic coarsely convex bicombing $\gamma\colon X\times X\times[0,1]\to X$ .

By Lemma 5.2 and Lemma 4.8, we have an extended bicombing on $X\times\overline{X}$ . Since $\gamma$ is a geodesic bicombing, we can upgrade it.

Lemma 5.4.

Suppose that $X$ is proper and $\gamma$ is an $(E,C)$ -geodesic coarsely convex bicombing on $X$ . Then there exists a map

\displaystyle\textup{{rp}}\bar{\gamma}\colon X\times\overline{X}\times\mathbb{R}_{\geq 0}\to X

satisfying the following:

(1)

For $x,y\in X$ , we have $\textup{{rp}}\bar{\gamma}(x,y,-)=\textup{{rp}}\gamma(x,y,-)$ .
(2)

For each $(e,x)\in X\times\partial X$ , there exists a sequence $(x_{n})$ in $X$ such that the sequence of maps $(\textup{{rp}}\gamma(e,x_{n},-):\mathbb{R}_{\geq 0}\to X)_{n}$ converges to $\textup{{rp}}\bar{\gamma}(e,x,-)$ uniformly on compact sets.
(3)

For each $(e,x)\in X\times\partial X$ , we have

$\displaystyle(\textup{{rp}}\bar{\gamma}(e,x,t)\mid x)_{e}\to\infty\quad\text{as}\quad t\to\infty$
(4)

For each $(e,x)\in X\times\partial X$ , the map $\textup{{rp}}\bar{\gamma}(e,x,-)\colon\mathbb{R}_{\geq 0}\to X$ is a geodesic.

Proof..

We can use the same construction of geodesics as the one described in [13, 5-25 Théorèm]. Here we only give a sketch of the proof.

Let $(e,x)\in X\times\partial X$ . We choose $(x_{n})\in S_{\infty}(X)$ such that $(x_{n})\in x$ . By Arzelà–Ascoli Theorem and standard diagonal arguments, We can take a subsequence $x_{n_{k}}$ such that the sequence of maps $(\textup{{rp}}\gamma(e,x_{n_{k}},-):\mathbb{R}_{\geq 0}\to X)_{k}$ converges uniformly on compact sets. This defines a map $\textup{{rp}}\bar{\gamma}(e,x,-)\colon\mathbb{R}_{\geq 0}\to X$ . From the construction, it is easy to see that this map is a geodesic, and $(\textup{{rp}}\bar{\gamma}(e,x,t)\mid x)_{e}\to\infty$ as $t\to\infty$ . ∎

Definition 5.5.

We call the map $\textup{{rp}}\bar{\gamma}$ given in Lemma 5.4 an extended geodesic bicombing on $X\times\overline{X}$ corresponding to $\gamma$ . For $(e,x)\in X\times\partial X$ , we abbreviate $\textup{{rp}}\bar{\gamma}(e,x,-)$ by $\gamma_{e}^{x}$ .

Lemma 5.6.

The extended geodesic bicombing $\textup{{rp}}\bar{\gamma}$ on $X\times\overline{X}$ satisfies the following: Let $(e_{1},x_{1}),(e_{2},x_{2})\in X\times\partial X$ and let $a,b\in\mathbb{R}_{\geq 0}$ . Set $x_{1}^{\prime}:=\textup{{rp}}\bar{\gamma}(e_{1},x_{1},a)$ and $x_{2}^{\prime}:=\textup{{rp}}\bar{\gamma}(e_{2},x_{2},b)$ . Then, for $c\in[0,1]$ , we have

\displaystyle d_{X}(\textup{{rp}}\bar{\gamma}(e_{1},x_{1},ca),\textup{{rp}}\bar{\gamma}(e_{2},x_{2},cb))\leq(1-c)Ed_{X}(e_{1},e_{2})+cEd_{X}(x_{1}^{\prime},x_{2}^{\prime})+C.

Proof..

The statement follows immediately from (ii) of Lemma 5.4. ∎

Remark 5.7.

All statements in Lemma 4.12 hold for $\gamma_{e}^{x}$ .

Lemma 5.8.

For $e,x,y\in X$ , we have

\displaystyle(x\mid y)_{e}\geq\frac{\min\{d_{X}(e,x),d_{X}(e,y)\}}{2Ed_{X}(x,y)}.

Proof..

Set $u:=\min\{d_{X}(e,x),d_{X}(e,y)\}$ . We have

\displaystyle d_{X}(\textup{{rp}}\gamma(e,x,u),\textup{{rp}}\gamma(e,y,u))\leq d_{X}(x,y)+\left\lvert d_{X}(e,x)-d_{X}(e,y)\right\rvert\leq 2d_{X}(x,y).

Set $c:=1/(2Ed_{X}(x,y))$ , we have

\displaystyle d_{X}(\textup{{rp}}\gamma(e,x,cu),\textup{{rp}}\gamma(e,y,cu))\leq cEd_{X}(\textup{{rp}}\gamma(e,x,u),\textup{{rp}}\gamma(e,y,u))+C\leq 1+C.

Therefore

\displaystyle(x\mid y)_{e}\geq cu=\frac{\min\{d_{X}(e,x),d_{X}(e,y)\}}{2Ed_{X}(x,y)}.

∎

5.2. $\gamma$ -convex subspaces

Definition 5.9.

Let $X$ be a metric space with $(E,C)$ -geodesic coarsely convex bicombing $\gamma\colon X\times X\times[0,1]\to X$ . We say that a subspace $Y\subset X$ is $\gamma$ -convex if for all $x,y\in Y$ and all $t\in[0,1]$ , we have $\gamma(x,y,t)\in Y$ .

Proposition 5.10.

Let $X$ be a metric space with $(E,C)$ -geodesic coarsely convex bicombing and $Y\subset X$ be a $\gamma$ -convex subspace. Then the restriction $\gamma_{Y}$ of $\gamma$ on $\colon Y\times Y\times[0,1]$ is a $(E,C)$ -geodesic coarsely convex bicombing on $Y$ . Moreover, the inclusion $\iota\colon Y\hookrightarrow X$ extends to a topological embedding $\bar{\iota}\colon\overline{Y}\hookrightarrow\overline{X}$ .

Corollary 5.11.

Let $X$ be a metric space with $(E,C)$ -geodesic coarsely convex bicombing and $Y\subset X$ be a $\gamma$ -convex subspace. Then the ideal boundary of $Y$ with respect to $\gamma_{Y}$ is homeomorphic to $\overline{\iota(Y)}\setminus\iota(Y)$ , where $\overline{\iota(Y)}$ is the closure of $\iota(Y)$ in $\overline{X}$ .

6. Continuous at infinity

We study maps between geodesic coarsely convex spaces which are not necessarily continuous, although, induce continuous maps between ideal boundaries. The main result of this section will be used to prove Proposition 9.2, which is a key proposition for results in Section 9

Definition 6.1.

Let $X$ be a space with a geodesic coarsely convex bicombing. Let $A\subset X$ be a subspace and $\overline{A}$ be the closure of $A$ in $\overline{X}$ . Let $U$ be a topological space. We say that a map $F\colon\overline{A}\to U$ is continuous at infinity if $F$ is continuous at all $x\in\overline{A}\setminus A$ .

Let $(X,d_{X})$ be a space equipped with an $(E,C)$ -geodesic coarsely convex bicombing $\gamma\colon X\times X\times[0,1]\to X$ .

Lemma 6.2.

Let $A\subset X$ be a subspace and $\overline{A}$ be the closure of $A$ in $\overline{X}$ . Let $f\colon\overline{A}\to\overline{X}$ be a map such that:

(1)

$\sup\{d_{X}(x,f(x)):x\in A\}<\infty$ , and
(2)

for $x\in\partial X\cap\overline{A}$ , $f(x)=x$ .

Then $f$ is continuous at infinity.

Proof..

Let $x\in\partial X\cap\overline{A}$ . We show that $f$ is continuous at $x$ . Set $K:=\sup\{d_{X}(x,f(x)):x\in A\}$ . Let $(x_{n})$ be a sequence in $\overline{A}$ converging to $x$ . We can assume without loss of generality that $x_{n}\in A$ for all $n$ . Set $t_{n}:=\min\{d_{X}(e,f(x_{n})),d_{X}(e,x_{n})\})$ . Since $\left\lvert d_{X}(e,f(x_{n}))-d_{X}(e,x_{n})\right\rvert\leq K$ , we have

\displaystyle d_{X}(\textup{{rp}}\gamma(e,f(x_{n}),t_{n}),\textup{{rp}}\gamma(e,x_{n},t_{n}))\leq K+d_{X}(x_{n},f(x_{n}))\leq 2K.

Set

	$\displaystyle p_{n}$	$\displaystyle:=\textup{{rp}}\gamma(e,f(x_{n}),(1/(2KE)t_{n})),$
	$\displaystyle q_{n}$	$\displaystyle:=\textup{{rp}}\gamma(e,x_{n},(1/(2KE)t_{n})).$

We have

\displaystyle d_{X}(p_{n},q_{n})

\displaystyle\leq\frac{1}{2KE}Ed_{X}(\textup{{rp}}\gamma(e,f(x_{n}),t_{n}),\textup{{rp}}\gamma(e,x_{n},t_{n}))+C\leq C+1

So we have $(x_{n}\mid f(x_{n}))_{e}\geq 1/(2KE)t_{n}$ . Then,

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

It follows that $(f(x_{n}))$ converges to $x$ . ∎

6.1. Image of the exponential map

We fix a base point $e\in X$ . We define a map $\exp\colon\partial X\times\mathbb{R}_{\geq 0}\to X$ by $\exp(x,t):=\textup{{rp}}\bar{\gamma}(e,x,t)$ . Set

	$\displaystyle\exp(\mathcal{O}\partial X)$	$\displaystyle:=\{\textup{{rp}}\bar{\gamma}(e,x,t):x\in\partial X_{k},t\in\mathbb{R}_{\geq 0}\}$
	$\displaystyle\overline{\exp(\mathcal{O}\partial X)}$	$\displaystyle:=\exp(\mathcal{O}\partial X)\cup\partial X_{k}.$

We remark that $\exp(\mathcal{O}\partial X)$ is not necessarily coarsely dense in $X$ . In [11, Section 5.5], a coarsely equivalence map $\tilde{\varphi}\colon X\to\exp(\mathcal{O}\partial X_{k})$ is contracted. We review the construction.

Let $X^{(0)}$ be a subset of $X$ such that $X^{(0)}$ is 2-dense in $X$ and 1-discrete, that is, for all $v,w\in X^{(0)}$ , if $v\neq w$ then $d_{X}(v,w)\geq 1$ , and, for all $v\in X$ , there exists $v^{\prime}\in X^{(0)}$ with $d_{X}(v,v^{\prime})\leq 2$ . Set $D_{6}:=2D_{1}E+D$ and $Y:=B_{D_{6}}(\exp(\mathcal{O}\partial X))$ .

Let $\chi\colon\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ is a proper map given in [11, Section 5.5]. By the construction of $\chi$ , for $v\in X^{(0)}$ , we have $\textup{{rp}}\gamma(e,v,\chi(d_{X}(e,v)))\in Y$ , so we define a map $\varphi\colon X^{(0)}\to Y$ by

\displaystyle\varphi(v):=\textup{{rp}}\gamma(e,v,\chi(d_{X}(e,v))).

We define a map $\overline{\varphi}\colon\overline{X^{(0)}}\to\overline{Y}$ by

\displaystyle\overline{\varphi}(v):=\begin{cases}\varphi(v)&\text{ if }v\in X^{(0)}\\ v&\text{ if }v\in\partial X\end{cases}.

Here we remark that $\overline{X^{(0)}}\setminus X^{(0)}=\partial X$ since $X^{(0)}$ is 2-dense in $X$ .

Lemma 6.3.

The map $\overline{\varphi}\colon\overline{X^{(0)}}\to\overline{Y}$ is continuous at infinity.

Proof..

Let $x\in\partial X$ . We show that $\overline{\varphi}$ is continuous at $x$ . Let $(x_{n})$ be a sequence in $\overline{X^{(0)}}$ converging to $x$ . We can assume without loss of generality that $x_{n}\in X^{(0)}$ for all $n$ . We remark $(x\mid x_{n})_{e}\to\infty$ . By (6) of Lemma 4.12, we have

\displaystyle(x\mid\varphi(x_{n}))_{e}\geq\Omega\min\{(x\mid x_{n}),\chi(d_{X}(e,x_{n}))\}\to\infty

since $\chi$ is proper. It follows that $(\varphi(x_{n}))$ converges to $x$ . ∎

Let $\iota\colon X\to X^{(0)}$ be a restriction such that for $x\in X$ , we have $d_{X}(x,\iota(x))\leq 2$ . Let $j\colon Y\to\exp(\mathcal{O}\partial X)$ be a retraction such that, for $x\in Y$ , we have $d_{X}(x,j(x))\leq D_{6}$ . Here we do not require that $i$ and $j$ are continuous. We define a map $\overline{\iota}\colon\overline{X}\to\overline{X^{(0)}}$ and $\overline{j}\colon\overline{Y}\to\overline{\exp(\mathcal{O}\partial X)}$ by

\displaystyle\overline{\iota}(v):=\begin{cases}\iota(v)&\text{ if }v\in X^{(0)}\\ v&\text{ if }v\in\partial X\end{cases}\quad\text{and}\quad\overline{j}(v):=\begin{cases}j(v)&\text{ if }v\in Y\\ v&\text{ if }v\in\partial X\end{cases}.

By Lemma 6.2, the maps $\overline{\iota}$ and $\overline{j}$ are continuous at infinity. We define a map $\Phi\colon\overline{X}\to\overline{\exp(\mathcal{O}\partial X)}$ by

\displaystyle\Phi(x):=\begin{cases}\overline{j}\circ\overline{\varphi}\circ\overline{\iota}(x)&\text{if }x\in X\\ x&\text{if }x\in\partial X\end{cases}.

Lemma 6.4.

The map $\Phi\colon\overline{X}\to\overline{\exp(\mathcal{O}\partial X)}$ is continuous at infinity.

6.2. Angle map

We define a map $\arg\colon\overline{\exp(\mathcal{O}\partial X)}\to\partial X$ as follows: For $x\in\exp(\mathcal{O}\partial X_{k})$ , we choose $(p_{x},t_{x})\in\partial X_{k}\times\mathbb{R}_{\geq 0}$ such that $x=\textup{{rp}}\bar{\gamma}(e,p_{x},t_{x})$ . Then we define

\displaystyle\arg(x)=p_{x}.

Lemma 6.5.

The map $\arg\colon\exp(\mathcal{O}\partial X_{k})\to\partial X_{k}$ is continuous at infinity.

We define a map $\Psi\colon\overline{X}\to\partial X$

\displaystyle\Psi(x):=\begin{cases}\arg\circ\Phi(x)&\text{if }x\in X\\ x&\text{if }x\in\partial X\end{cases}

By Lemmata 6.4 and 6.5, we have the following.

Proposition 6.6.

The map $\Psi\colon\overline{X}\to\partial X$ is continuous at infinity.

7. Trees of spaces

7.1. Trees of spaces

Definition 7.1.

Let $(Z,d_{Z})$ be a geodesic metric space and $T$ be a bipartite tree. Set $V(T):=K\sqcup L$ . Let $\xi\colon Z\to T$ be a continuous map. We suppose the following:

(1)

For each $l\in L$ , the inverse image $\xi^{-1}(l)$ consists of a single point.
(2)

For each $k\in K$ , the inverse image $X_{k}:=\xi^{-1}(\mathsf{star}(k))$ is a geodesic space.

Then we say that $Z$ is a tree of spaces associated with the map $\xi\colon Z\to T$ and the bipartite structure $V(T)=K\sqcup L$ .

Definition 7.2.

Let $(Z,d_{Z})$ be a tree of spaces associated with the map $\xi\colon Z\to T$ and the bipartite structure $V(T)=K\sqcup L$ .

Let $E^{\prime}\geq 1$ and $C^{\prime}\geq 0$ be constants. We suppose that for each $k\in K$ , the inverse image $X_{k}:=\xi^{-1}(\mathsf{star}(k))$ admits an $(E^{\prime},C^{\prime})$ -geodesic coarsely convex bicombing

\displaystyle\gamma_{k}\colon X_{k}\times X_{k}\times[0,1]\to X_{k}.

Then we say that $Z$ is a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex spaces.

Definition 7.3.

We say that a metric space $Z$ is a tree of geodesic coarsely convex spaces if $Z$ is $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex space for some $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ .

Theorem 7.4 ([10]).

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex space. For $k\in K$ , set $X_{k}:=\xi^{-1}(\mathsf{star}(k))$ .

Then the space $Z$ admits an $(E,C)$ -geodesic coarsely convex bicombing

\displaystyle\gamma\colon Z\times Z\times[0,1]\to Z

where $E=E^{\prime}+2$ and $C=6C^{\prime}$ . The restriction of $\gamma$ to $X_{k}\times X_{k}\times[0,1]$ is equal to $\gamma_{k}$ .

Definition 7.5.

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex space. We call the bicombing $\gamma$ on $Z$ constructed in Theorem 7.4 the geodesic coarsely convex bicombing associated with the family $\{\gamma_{k}\}$ .

Remark 7.6.

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex space, and let $\gamma$ be the geodesic coarsely convex bicombing on $Z$ associated with the family $\{\gamma_{k}\}$ . For $k\in K$ , set $X_{k}:=\xi^{-1}(\mathsf{star}(k))$ .

By the construction, it is clear that for each $k\in K$ , the subspace $X_{k}$ is $\gamma$ -convex in $Z$ . So by Corollary 5.11, the boundary of $X_{k}$ in $\overline{Z}$ is homeomorphic to $\partial X_{k}$ .

7.2. Free products of metric spaces

In [9], we introduce free products of metric spaces, which can be regarded as trees of metric spaces.

Definition 7.7.

Let $(X,d_{X})$ be a metric space and let $X_{0}$ be a set with the discrete topology. Let $i_{X}:X_{0}\to X$ be a proper map. We choose a base point $e_{X}\in i_{X}(X_{0})$ . We call $(X_{0},i_{X},e_{X})$ a net of $X$ . For $x_{0}\in X_{0}$ , we denote by $\overline{x_{0}}$ the image $i_{X}(x_{0})$ . We say that $(X,d_{X},X_{0},i_{X},e_{X})$ is a metric space with a net.

Remark 7.8.

We often abbreviate specifying a net $(X_{0},i_{X},e_{X})$ and say that $(X,d_{X})$ is a metric space with a net.

Example 7.9.

Let $(X,d_{X},e_{X})$ be a metric space with a base point $e_{X}$ . Let $G$ be a group acting on $X$ properly by isometries. We define a map $\mathsf{orb}\colon G\to X$ by $\mathsf{orb}(g)\coloneqq g\cdot e_{X}$ . Then $(G,o(e_{X}),e_{X})$ is a net of $X$ , called G-net.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces with nets. In [9, Section 1], we constructed the free product $X*Y$ of $X$ and $Y$ . In [10], we constructed a bipartite tree $T$ and continuous map $\xi\colon X*Y\to T$ such that $X*Y$ is a tree of spaces associated with $\xi$ .

Example 7.10.

Let $G$ and $H$ be groups acting on metric spaces $X$ and $Y$ . We consider a $G$ -net of $X$ , and an $H$ -net of $Y$ , respectively. Then the free product $G*H$ acts on $X*Y$ properly. If the action of $G$ and $H$ are cocompact, then so is the action of $G*H$ .

8. Augmented space

8.1. $n$ -th augmented spaces

Definition 8.1.

Let $(Z,d_{Z})$ be a tree of spaces associated with the map $\xi\colon Z\to T$ and the bipartite structure $V(T)=K\sqcup L$ .

For $k\in K$ , set $X_{k}\colon=\xi^{-1}(\mathsf{star}(k))$ . We choose an 1-discrete, 2-dense subset $X_{k}^{(0)}\subset X_{k}$ . So the pair $(X_{k},X_{k}^{(0)})$ is a space with a lattice. We choose a bijection $k\colon\mathbb{N}\to K$ .

Let $n\in\mathbb{N}\sqcup\{0\}$ . The $n$ -th augmented space $A_{n}(Z)$ of $Z$ is a tree of spaces associated with the map $A_{n}(\xi)\colon A_{n}(Z)\to T$ satisfying the following:

(1)

For $i\in\mathbb{N}$ with $i\leq n$ , the space $A_{n}(\xi)^{-1}(\mathsf{star}(k(i)))$ is isometric to $X_{k(i)}$ .
(2)

For $i\in\mathbb{N}$ with $i>n$ , the space $A_{n}(\xi)^{-1}(\mathsf{star}(k(i)))$ is isometric to the metric horoball $\mathcal{H}(X_{k(i)},X_{k(i)}^{(0)})$ of the space with a lattice $(X_{k(i)},X_{k(i)}^{(0)})$ .

We call $0$ -th augmented space $A_{0}(Z)$ the augmented space, and we denote by $A(Z)$ .

Remark 8.2.

In the above definition, the coarse equivalence class of the $n$ -th augmented space does not depend on the choice of nets $X_{k}^{(0)}$ . Therefore, when we consider $n$ -th augmented spaces, we implicitly choose nets, and we say that $A_{n}(Z)$ is the $n$ -th augmented space of $Z$ , without mentioning nets. We abbreviate $\mathcal{H}(X_{k},X_{k}^{(0)})$ by $\mathcal{H}(X_{k})$

The augmented space $A(Z)$ does not depend on the choice of the bijection $k\colon\mathbb{N}\to K$ .

We can explicitly construct a $n$ -th augmented space $A_{n}(Z)$ from a tree of spaces $Z$ by choosing nets and gluing combinatorial horoballs along nets by the way described in Definition 2.5.

8.2. Bicombings on $n$ -th augmented spaces

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex spaces. Let $\gamma$ be the geodesic coarsely convex bicombing associated with the family $\{\gamma_{k}\}$ . We choose a pair of constants $(E^{\prime},C^{\prime})$ such that $\gamma$ is $(E^{\prime},C^{\prime})$ geodesic coarsely convex.

For $k\in K$ , set $X_{k}\colon=\xi^{-1}(\mathsf{star}(k))$ . For $k\in K$ , we define an $(E^{\prime},C^{\prime}+8\delta)$ -geodesic coarsely convex bicombing $A_{n}(\gamma_{k})$ on $A_{n}(\xi)^{-1}(\mathsf{star}(k))=A_{n}(X_{k})$ by the following:

(1)

if $i\leq n$ then $A_{n}(\gamma_{k(i)})=\gamma_{k(i)}$ ,
(2)

Suppose $i>n$ . Note that $A_{n}(X_{k(i)})$ is equal to $\mathcal{H}(X_{k(i)})$ . For $x,y\in A_{n}(X_{k(i)})$ , we define a map

$\displaystyle A_{n}(\gamma_{k(i)})(x,y,-)\colon[0,1]\to A_{n}(X_{k(i)})$

to be the normal geodesic segment from $x$ to $y$ given in Proposition 2.6.

We choose a bijection $k\colon\mathbb{N}\to K$ . Let $A_{n}(Z)$ be the $n$ -th augmented space of $Z$ . Then $A_{n}(Z)$ is a $(A_{n}(\xi),K,L,\{A_{n}(\gamma_{k})\}_{k\in K})$ -tree of geodesic coarsely convex space. Let $A_{n}(\gamma)$ be an geodesic coarsely convex bicombing on $A_{n}(Z)$ associated with $\{A_{n}(\gamma_{k})\}$ .

Definition 8.3.

We call the bicombing $A_{n}(Z)$ constructed in the above an augmented bicombing associated with the family $\{\gamma_{k}\}_{k\in K}$ .

Remark 8.4.

For $i\leq n$ , $X_{k(i)}$ is an $A_{n}(\gamma)$ -convex subspace, so by Corollary 5.11, $\partial X_{k(i)}$ is a closed subspace of $\partial A_{n}(Z)$ .

For $i>n$ , $\mathcal{H}(X_{k(i)})$ is an $A_{n}(\gamma)$ -convex subspace, the boundary $\partial\mathcal{H}(X_{k(i)})=\overline{\mathcal{H}(X_{{k(i)}})}\setminus\mathcal{H}(X_{k(i)})$ consists of a single point. We call this the center of the horoball $\mathcal{H}(X_{k(i)})$ , and denote by $c_{k(i)}$ .

9. (co)Homology if Ideal boundary of a tree of spaces

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex spaces. We fix a bijection $k\colon\mathbb{N}\to K$ . Let $A(Z)$ be the augmented space of $Z$ , and let $A_{n}(\gamma)$ be the augmented bicombing on $A_{n}(Z)$ associated with the family $\{\gamma_{k}\}_{k\in K}$ .

Remark 9.1.

It is clear that the inclusion $\iota_{n}\colon A_{n}(Z)\hookrightarrow A_{n-1}(Z)$ is a visual map in the sense of Definition 4.13. By Proposition 4.14, it induces a continuous map

\displaystyle\partial\iota_{n}\colon\partial A_{n}(Z)\to\partial A_{n-1}(Z).

The family $\{\partial A_{n}(Z),\partial\iota_{n}\}$ forms a projective system, and the projective limit is $\partial Z$ , that is,

\displaystyle\partial Z=\varprojlim_{n}\partial A_{n}(Z).

The following is a key proposition for computing (co)homologies of $\partial Z$ .

Proposition 9.2.

For $n\in\mathbb{N}$ , there exists a continuous retraction

\displaystyle s_{k(n)}\colon\partial A_{n}(Z)\to\partial X_{k(n)}.

We will prove Proposition 9.2 in Section 10

9.1. Cohomology

Let $M^{*}=(M^{n})_{n\in\mathbb{N}}$ be the $K$ -theory or the Alexander-Spanier cohomology, and let $\tilde{M}^{*}$ be the reduced one. For a compact Hausdorff space $Y$ , and $e\in Y$ , we have $\tilde{M}^{*}(Y)\cong M^{*}(Y,\{e\})$ .

We consider the long exact sequence for the triple $(\partial A_{n}(Z),\partial X_{k(n)},\{e_{n}\})$ .

	$\displaystyle\dots$	$\displaystyle\to M^{p}(\partial A_{n}(Z),\partial X_{k(n)})\to M^{p}(\partial A_{n}(Z),\{e_{n}\})\to M^{p}(\partial X_{k(n)},\{e_{n}\})$
		$\displaystyle\to M^{p+1}(\partial A_{n}(Z),\partial X_{k(n)})\to\cdots$

By Proposition 9.2, the above exact sequence splits, so we have

(2)

\displaystyle 0\to M^{p}(\partial A_{n}(Z),\partial X_{k(n)})\to\tilde{M}^{p}(\partial A_{n}(Z))\to\tilde{M}^{p}(\partial X_{k(n)})\to 0

By the strong excision axiom, we have

	$\displaystyle M^{p}(\partial A_{n}(Z),\partial X_{k(n)})$	$\displaystyle\cong M^{p}((\partial A_{n}(Z)\setminus\partial X_{k(n)})^{+},\{c_{k(n)}\})$
		$\displaystyle\cong M^{p}(\partial A_{n-1}(Z),\{c_{k(n)}\})$
		$\displaystyle=\tilde{M}^{p}(\partial A_{n-1}(Z)).$

Here $(\partial A_{n}(Z)\setminus\partial X_{k(n)})^{+}$ denotes the one-point compactification

\displaystyle(\partial A_{n}(Z)\setminus\partial X_{k(n)})\sqcup\{c_{k(n)}\}.

It follows that $\tilde{M}^{p}(\partial A_{n}(Z))\cong\tilde{M}^{p}(\partial A_{n-1}(Z))\oplus\tilde{M}^{p}(\partial X_{k(n)})$ . By the continuity of $M^{*}$ , we have $\tilde{M}^{p}(\partial Z)\cong\varinjlim\tilde{M}^{p}(\partial A_{n}(Z))$ . Here we summarize the computation.

Theorem 9.3.

Let $Z$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex spaces. Suppose that $Z$ is proper. Set $X_{k}:=\xi^{-1}(\mathsf{star}(k))$ . Let $A(Z)$ be the augmented space of $Z$ . Then,

\displaystyle\tilde{M}^{p}(\partial Z)\cong\tilde{M}^{p}(\partial A(Z))\oplus\bigoplus_{k\in K}\tilde{M}^{p}(\partial X_{k})

Here we remark that the topological type of $\partial A(Z)$ is described in Theorem 11.4.

9.2. Homology

Let $M_{*}=(M_{n})_{n\in\mathbb{N}}$ be the $K$ -homology or the Steenrod homology, and let $\tilde{M}_{*}$ be the reduced one. For a compact Hausdorff space $X$ , and $e\in X$ , we have $\tilde{M}_{*}(X)\cong M_{*}(X,\{e\})$ .

Theorem 9.4 ([19, Theorem 4.]).

Let $Y_{1}\leftarrow Y_{2}\leftarrow Y_{3}\leftarrow\cdots$ be an inverse system of compact metric spaces. Then there exists an exact sequence

\displaystyle 0\to\varprojlim{\!}^{1}M_{p+1}(Y_{i})\to M_{p}(\varprojlim Y_{i})\to\varprojlim M_{p}(Y_{i})\to 0.

By the same argument as the one in Section 9.1, we obtain

\displaystyle\tilde{M}_{p}(\partial A_{n}(Z))\cong\tilde{M}_{p}(\partial A_{n-1}(Z))\oplus\tilde{M}_{p}(\partial X_{k(n)}).

By Theorem 9.4, we have

\displaystyle 0\to\varprojlim{\!}^{1}M_{p+1}(\partial A_{n}(Z))\to M_{p}(\partial Z)\to\varprojlim M_{p}(\partial A_{n}(Z))\to 0.

By Proposition 9.2, the maps $M_{p+1}(\partial A_{n+1}(Z))\to M_{p+1}(\partial A_{n}(Z))$ are surjective, so the Mittag-Leffler condition holds. Thus $\varprojlim{\!}^{1}M_{p+1}(\partial A_{n}(Z))$ vanishes.

Theorem 9.5.

Let $Z$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex spaces. Set $X_{k}:=\xi^{-1}(\mathsf{star}(k))$ . Suppose that $Z$ is proper. Let $A(Z)$ be the augmented space of $Z$ . Then,

\displaystyle\tilde{M}_{p}(\partial Z)\cong\tilde{M}_{p}(\partial A(Z))\times\prod_{k\in K}\tilde{M}_{p}(\partial X_{k}).

9.3. $K$ -theory of the Roe algebra

Let $V$ be a proper metric space. We denote by $C^{*}(V)$ the Roe algebra of $V$ . We also denote by $KX_{*}(V)$ the coarse $K$ -homology of $V$ . For detail, see [16], [21] and [17].

Suppose that $V$ is a proper coarsely convex space. By [11, Theorem 1.3], the following coarse assembly map is an isomorphism.

\displaystyle\mu_{*}\colon KX_{*}(V)\to K_{*}(C^{*}(V)).

By [11, Theorem 6.7], the following transgression map is an isomorphism.

\displaystyle{T_{\partial V}}\colon KX_{*}(V)\to\tilde{K}_{*-1}(\partial V).

Thus we have an isomorphism

(3)

\displaystyle{T_{\partial V}}\circ\mu_{*}^{-1}\colon K_{*}(C^{*}(V))\to\tilde{K}_{*-1}(\partial V).

Theorem 9.6.

\displaystyle K_{p}(C^{*}(Z))

\displaystyle\cong\tilde{K}_{p-1}(\partial A(Z))\times\prod_{k\in K}\tilde{K}_{p-1}(\partial X_{k})

Proof..

Combining Theorem 9.5 for $K$ -homology with an isomorphism

\displaystyle K_{*}(C^{*}(Z))\cong\tilde{K}_{*-1}(\partial Z)

obtained by applying 3 for geodesic coarsely convex space $Z$ , we obtain the desired result. ∎

Corollary 9.7.

Let $Z$ be the tree of geodesic coarsely convex spaces described in Theorem 9.6. We have

\displaystyle K_{p}(C^{*}(Z))

\displaystyle\cong\tilde{K}_{p-1}(\partial A(Z))\times\prod_{k\in K}K_{p}(C^{*}(X_{k}))

Proof..

Combining Theorem 9.6 with isomorphisms

\displaystyle K_{*}(C^{*}(X))\cong\tilde{K}_{*-1}(\partial X)\quad\text{and}\quad K_{*}(C^{*}(Y))\cong\tilde{K}_{*-1}(\partial Y),

we obtain the desired result. ∎

9.4. Case of free products

Proof of Theorems 1.1, 1.2 and 1.3.

Let $X$ and $Y$ be proper geodesic coarsely convex spaces with nets. Suppose that the net of $X$ and that of $Y$ are coarsely dense in $X$ and $Y$ , respectively. Let $X*Y$ be the free product of $X$ and $Y$ . Then $X*Y$ is a tree of space. Since the net of $X$ and $Y$ are coarsely dense in $X$ and $Y$ , respectively, by Theorem 11.4, the ideal boundary $\partial A(X*Y)$ of the augmented space has no isolated point. So, $\partial A(X*Y)$ is homeomorphic to the Cantor space $\mathcal{C}$ . Then applying Theorems 9.3, 9.5, 9.6 and 9.7 for $X*Y$ , we obtain the desired result. ∎

10. Construction of retractions

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex space. We suppose that $Z$ is proper. Let $\gamma$ be the geodesic coarsely convex bicombing on $Z$ associated with the family $\{\gamma_{k}\}_{k\in K}$ . By Remark 7.6, we can regard $\partial X_{k}$ as a closed subspace of $\partial Z$ . The purpose of this section is to construct a retraction $s_{k}\colon\partial Z\to\partial X_{k}$ .

Let $k\in K$ . For $x\in\partial Z$ , set

\displaystyle t(k;x):=\sup\{t\in\mathbb{R}:\textup{{rp}}\bar{\gamma}(e,x,t)\in X_{k}\}.

We define a map $\pi_{k}\colon\partial Z\to\overline{X_{k}}$ by

\displaystyle\pi_{k}(x):=\textup{{rp}}\bar{\gamma}(e,x,t(k;x)).

Remark 10.1.

For $x\in\partial Z\setminus\partial X_{k}$ , we have $t(k;x)=d(e,\textup{{rp}}\bar{\gamma}(e,x,t(k;x)))$ . For $x\in\partial X_{k}$ , we have $t(k;x)=\infty$ and $\pi_{k}(x)=\textup{{rp}}\bar{\gamma}(e,x,\infty)=x$ .

Lemma 10.2.

We fix $k\in K$ . Let $e\in X_{k}$ be a base point of $X_{k}$ , and let $v,w\in X_{k}$ such that $v\neq w$ . Then for $x\in\pi_{k}^{-1}(v)$ and $y\in\pi_{k}^{-1}(w)$ , we have

\displaystyle(x\mid y)_{e}<\Omega(\max\{d_{Z}(e,v),d_{Z}(e,w)\}+\Omega).

Here $\Omega$ is a constant given in Lemma 4.12.

Proof..

We define $t\in\mathbb{R}$ by

\displaystyle t=\max\{t(k;x),t(k;y)\}+\Omega=\max\{d_{Z}(e,v),d_{Z}(e,w)\}+\Omega

The geodesic from $\gamma_{e}^{x}(t)$ to $\gamma_{e}^{y}(t)$ is obtained by concatenating $\gamma(\gamma_{e}^{x}(t),v,-)$ , $\gamma(v,w,-)$ and $\gamma(w,\gamma_{e}^{y}(t),-)$ . So we have

	$\displaystyle d_{Z}(\gamma_{e}^{x}(t),\gamma_{e}^{y}(t))$	$\displaystyle=d_{Z}(\gamma_{e}^{x}(t),v)+d_{Z}(v,w)+d_{Z}(w,\gamma_{e}^{y}(t))$
		$\displaystyle\geq t-t(k;x)+d_{Z}(v,w)+t-t(k;y)$
		$\displaystyle\geq\Omega.$

Thus by (4) of Lemma 4.12, we have $(\gamma_{e}^{x}\mid\gamma_{e}^{y})_{e}<t$ . Then by (2) of Lemma 4.12, we have $(x\mid y)_{e}\leq\Omega(\gamma_{e}^{x}\mid\gamma_{e}^{y})_{e}\leq\Omega t$ . ∎

Lemma 10.3.

We fix $k\in K$ . Let $e\in X_{k}$ be a base point of $X_{k}$ , and let $v\in X_{k}$ . Then for $x\in\partial X_{k}$ and $y\in\pi_{k}^{-1}(v)$ , we have

\displaystyle(x\mid y)_{e}<\Omega(d_{Z}(e,v)+\Omega).

Here $\Omega$ is a constant given in Lemma 4.12.

Proof..

We can prove the lemma by a similar argument as the proof of Lemma 10.2. ∎

Lemma 10.4.

The map $\pi_{k}\colon\partial Z\to\overline{X_{k}}$ is continuous.

Proof..

Let $x\in\partial Z$ . We will show that $\pi_{k}$ is continuous at $x$ . First, we suppose that $x\in\partial Z\setminus\partial X_{k}$ . Set $v=\pi_{k}(x)$ . Then by Lemma 10.2, $\pi_{k}^{-1}(v)$ is an open set. This implies that $\pi_{k}$ is continuous at $x$ .

Now we suppose that $x\in\partial X_{k}$ . We remark that $\pi_{k}(x)=x$ . For $r\in\mathbb{R}_{\geq 0}$ , set $V_{r}[x]:=\{y\in\overline{X}:(x\mid y)>r\}$ . The family $\{V_{n}[x]:n\in\mathbb{N}\}$ forms a system of the fundamental neighborhood of $x$ . We will show that for each $n\in\mathbb{N}$ , there exists $T\in\mathbb{R}_{\geq 0}$ such that

(4)

\displaystyle\pi_{k}(V_{T}[x])\subset V_{n-1}[\pi_{k}(x)]=V_{n-1}[x].

We fix $n\in\mathbb{N}$ . Set $T=EC\Omega(n+\Omega)$ . Let $y\in V_{T}[x]$ . Thus $(x\mid y)_{e}>T$ . Set $v=\pi_{k}(y)$ . By Lemma 10.3, we have

\displaystyle t(k;y)=d_{Z}(e,v)\geq\Omega^{-1}(x\mid y)_{e}-\Omega>ECn.

By (4) of Lemma 4.12, we have $d_{Z}(\gamma_{e}^{x}(T),\gamma_{e}^{y}(T))\leq\Omega$ . Set $T^{\prime}\coloneqq ECn$ . Then we have

\displaystyle d_{Z}(\gamma_{e}^{y}(T^{\prime}),\textup{{rp}}\gamma(e,v,T^{\prime}))

\displaystyle\leq Ed_{Z}(v,v)+C=C

Here we used that $\gamma_{e}^{y}(t(k;y))=v=\textup{{rp}}\gamma(e,v,t(k;y)))$ . We have

	$\displaystyle d_{Z}(\gamma_{e}^{x}(n),\textup{{rp}}\gamma(e,v,n))$	$\displaystyle\leq\frac{1}{EC}Ed_{Z}(\gamma_{e}^{y}(T^{\prime}),\textup{{rp}}\gamma(e,v,T^{\prime}))+C$
		$\displaystyle\leq C+1.$

It follows that $(x\mid v)\geq n$ . Thus $v=\pi_{k}(y)\in V_{n-1}[x]$ . Therefore 4 holds. This implies that $\pi_{k}$ is continuous at $x$ . ∎

Let $\Psi_{k}\colon\overline{X_{k}}\to\partial X_{k}$ be the map constructed in Section 6.2.

Definition 10.5.

We define a retraction $s_{k}\colon\partial Z\to\partial X_{k}$ by

\displaystyle s_{k}=\Psi_{k}\circ\pi_{k}.

Proposition 10.6.

The map $s_{k}\colon\partial Z\to\partial X_{k}$ is continuous.

Proof..

Let $x\in\partial Z$ . We will show that $\pi_{k}$ is continuous at $x$ .

Let $x_{n}$ be a sequence in $\partial Z$ converging to $x$ . Since $\pi_{k}$ is continuous, the sequence $\pi_{k}(x_{n})$ converges to $\pi_{k}(x)\in\partial X_{k}$ . Then by Proposition 6.6, the sequence $(\Psi_{k}(\pi_{k}(x_{n})))$ converges to $\Psi_{k}(\pi_{k}(x))$ . ∎

Proof of Proposition 9.2.

Applying Proposition 10.6 to the tree of spaces $A_{n}(Z)$ associated with the map $A_{n}(\xi)\colon A_{n}(Z)\to T$ , we obtain the desired result. ∎

11. Ideal boundaries of the augmented spaces

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex space. We suppose that $Z$ is proper.

Let $A(Z)$ be the augmented space of $Z$ . For simplicity, we denote by $\gamma$ an augmented bicombing on $A(Z)$ associated with the family $\{\gamma_{k}\}_{k\in K}$ .

For $x\in\overline{A(Z)}$ , set $W_{n}[x]=\{y\in\partial A(Z):(\gamma_{e}^{x}\mid\gamma_{e}^{y})_{e}\geq n\}$ . Recall that $\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})^{(0)}$ denotes the vertex set of the combinatorial horoball $\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})$ . We consider the family

\displaystyle\mathcal{T}:=\bigsqcup_{k\in K}\{W_{n}[x]:x\in\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})^{(0)},n\in\mathbb{N}\}.

Since $Z$ is proper, for any bounded subset $B\subset Z$ , the cardinality of the set $B\cap X_{k}^{(0)}$ is finite. It follows that $\mathcal{T}$ is countable.

Lemma 11.1.

The family $\mathcal{T}$ is a countable family of open basis of $\partial A(Z)$ .

Proof..

We have already seen that $\mathcal{T}$ is countable. Let $x\in\partial A(Z)$ , and $n\in\mathbb{N}$ . We will show that there exists $U\in\mathcal{T}$ such that

\displaystyle x\in U\subset V_{n}[x]\cap\partial A(Z).

We choose $t\in\mathbb{R}_{\geq 0}$ such that:

	$\displaystyle t$	$\displaystyle\geq 4E{\Omega^{3}}n+2,$
	$\displaystyle\textup{{rp}}\bar{\gamma}(e,x,t)$	$\displaystyle=\gamma_{e}^{x}(t)\in\mathcal{H}(X_{k},X_{k}^{(0)})\text{ for some }k\in K.$

There exists $x^{\prime}\in\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})^{(0)}$ such that $d_{A(Z)}(\gamma_{e}^{x}(t),x^{\prime})\leq 2$ since $\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})^{(0)}$ is 2-dense in $\mathcal{H}(X_{k},X_{k}^{(0)})$ . By Lemma 5.8, we have

\displaystyle(\gamma_{e}^{x}(t)\mid x^{\prime})

\displaystyle\geq\frac{\min\{d_{A(Z)}(e,\gamma_{e}^{x}(t)),\,d_{A(Z)}(e,x^{\prime})\}}{2Ed_{A(Z)}(\gamma_{e}^{x}(t),x^{\prime})}\geq\frac{t-2}{4E}.

\displaystyle(\gamma_{e}^{x}\mid x^{\prime})

\displaystyle\geq\Omega^{-1}\min\{(\gamma_{e}^{x}\mid\gamma_{e}^{x}(t)),(\gamma_{e}^{x}(t)\mid x^{\prime})\}\geq\frac{t-2}{4E\Omega}\geq\Omega^{2}n.

Then for all $y\in W_{t}[x^{\prime}]$ ,

\displaystyle(x\mid y)\geq\Omega^{-2}\min\{(x\mid\gamma_{e}^{x}),(\gamma_{e}^{x}\mid x^{\prime}),(x^{\prime}\mid y)\}\geq n.

Here we used that $(x^{\prime}\mid y)\geq(\gamma_{e}^{x^{\prime}}\mid\gamma_{e}^{y})\geq t\geq n$ . Therefore $x\in W_{t}[x^{\prime}]\subset V_{n}[x]\cap\partial A(Z)$ . This implies that $\mathcal{T}$ is an open basis of $\partial A(Z)$ .∎

Lemma 11.2.

Each $W\in\mathcal{T}$ is a clopen set.

Proof..

Let $k\in K$ . Let $x\in\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})^{(0)}$ and $n\in\mathbb{N}$ . We will show that $W_{n}[x]$ is closed. Let $y\in\partial A(Z)\setminus W_{n}[x]$ . We will show that there exists $T>0$ such that

\displaystyle W_{2T}[y]\subset\partial A(Z)\setminus W_{n}[x].

For $p\in\partial A(Z)$ , set

\displaystyle s(k;p):=\inf\{t\in\mathbb{R}:\textup{{rp}}\bar{\gamma}(e,p,t)\in X_{k}^{(0)}\},\quad t(k;p):=\sup\{t\in\mathbb{R}:\textup{{rp}}\bar{\gamma}(e,p,t)\in X_{k}\}

Here we use the convention that $\inf\emptyset=\infty$ and $\sup\emptyset=-\infty$ . First, we suppose $y=c_{k}\in\overline{\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})^{(0)}}$ . Then $s(k;y)<\infty$ and $t(k;y)=\infty$ . We remark that the geodesic $\textup{{rp}}\bar{\gamma}(\gamma_{e}^{y}(s(k;y)),y,-)$ consists of vertical line in $\mathcal{H}(X_{k},X_{k}^{(0)})$ . Let $h$ be the depth of $x$ in $\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})^{(0)}$ . Set $T:=h+s(k;y)+n+D_{1}$ . Let $y^{\prime}\in W_{2T}[y]$ . We have

\displaystyle\textup{{rp}}\bar{\gamma}(e,y,t)=\textup{{rp}}\bar{\gamma}(e,y^{\prime},t)\qquad(\forall t\in[0,T]).

See fig. 1. If $y^{\prime}\in W_{n}[x]$ , then

\displaystyle d_{Z}(\gamma_{e}^{x}(n),\textup{{rp}}\bar{\gamma}(e,y,n))=d_{Z}(\gamma_{e}^{x}(n),\textup{{rp}}\bar{\gamma}(e,y^{\prime},n))\leq D_{1}

Thus $(\gamma_{x}^{x}\mid\gamma_{e}^{y})\geq n$ . This contradicts $y\notin W_{n}[x]$ . Therefore $y^{\prime}\notin W_{n}[x]$ , so we have $W_{2T}[y]\cap W_{n}[x]=\emptyset$ .

Next, we suppose $y\notin\overline{\mathcal{H}_{\mathsf{comb}}(X_{k}^{(0)})^{(0)}}$ . If $s(k;y)=\infty$ , then set $T=2n+D_{1}$ . it is easy to see that $W_{2T}[y]\cap W_{n}[x]=\emptyset$ . If $s(k;y)<\infty$ , then set $T=t(k;y)+2n+D_{1}$ . For $y^{\prime}\in W_{2T}[y]$ , we have

\displaystyle\textup{{rp}}\bar{\gamma}(e,y,t)=\textup{{rp}}\bar{\gamma}(e,y^{\prime},t)\qquad(\forall t\in[0,T]).

See fig. 2. Then by a similar argument as before, $y^{\prime}\notin W_{n}[x]$ . Therefore $W_{2T}[y]\cap W_{n}[x]=\emptyset$ .

Figure 1. geodesics

\gamma_{e}^{y}

and

\gamma_{e}^{y^{\prime}}

where

y=c_{k}

Figure 2. geodesics

\gamma_{e}^{y}

and

\gamma_{e}^{y^{\prime}}

where

y\neq c_{k}

∎

Lemma 11.3.

Let $k\in K$ . The center $c_{k}$ of the horoball $\mathcal{H}(X_{k})$ is an isolated point if and only if $X_{k}\cap\xi^{-1}(L^{\circ})$ is bounded, where $L^{\circ}$ denotes a set of vertices $l\in L$ which adjacent to at least two different vertices in $K$ .

Proof..

For $p\in\partial A(Z)$ , set

\displaystyle t(k;p):=\sup\{t\in\mathbb{R}:\textup{{rp}}\bar{\gamma}(e,p,t)\in X_{k}\}.

First, we suppose that $c_{k}$ is an accumulation point. Thus there exists a sequence $(x_{n})$ in $\partial A(Z)\setminus\{c_{k}\}$ converging to $c_{k}$ . Then we see that $t(k;x_{n})\to\infty$ . We remark that $\textup{{rp}}\bar{\gamma}(e,x_{n},t(k;x_{n}))\in X_{k}\cap\xi^{-1}(L^{\circ})$ . Therefore $X_{k}\cap\xi^{-1}(L^{\circ})$ is unbounded.

Next, we suppose that $X_{k}\cap\xi^{-1}(L^{\circ})$ is unbounded. Then there exists an unbounded sequence $(v_{n})$ in $X_{k}\cap\xi^{-1}(L^{\circ})$ . By the definition of $L^{\circ}$ , for each $v_{n}$ , there exists $k_{n}\in K\setminus\{k\}$ which adjacent to $\xi(v_{n})$ . Then we see that $c_{k_{n}}$ converges to $c_{k}$ . See fig. 3

Figure 3.

c_{k_{n}}

converges to

c_{k}

∎

Theorem 11.4.

Let $Z$ be a tree of geodesic coarsely convex spaces, and let $A(Z)$ be the augmented space of $Z$ .

(1)

Let $\partial A(Z)_{\mathsf{iso}}$ denote the set of isolated points in $\partial A(Z)$ .

$\displaystyle\partial A(Z)_{\mathsf{iso}}=\{c_{k}:k\in K,\ X_{k}\cap\xi^{-1}(L^{\circ})\text{ is bounded }\}$
(2)

If $\partial A(Z)$ has no isolated point, then $\partial A(Z)$ is the Cantor space.

Proof..

By Lemma 11.3,

\displaystyle\partial A(Z)_{\mathsf{iso}}\cap\{c_{k}:k\in K\}=\{c_{k}:k\in K,\ X_{k}\cap\xi^{-1}(L)\text{ is bounded }\}.

Let $x\in\partial A(Z)_{\mathsf{iso}}\setminus\{c_{k}:k\in K\}$ . Then there exists a sequence $\{t_{n}\}\subset\mathbb{R}_{\geq 0}$ and $\{k_{n}\}\subset K$ such that $\textup{{rp}}\bar{\gamma}(e,x,t_{n})\in X_{k_{n}}$ for all $n\in\mathbb{N}$ , and $k_{n}\neq k_{m}$ for all $n\neq m$ . Then a sequence $\{c_{k}\}$ converges to $x$ . Thus (1) holds.

By Lemmata 11.1 and 11.2, $\partial A(Z)$ has countable bases consisting of clopen sets. Thus, if $\partial A(Z)$ has no isolated points, $\partial A(Z)$ is homeomorphic to the Cantor space by [4]. ∎

Remark 11.5.

It is known by [1, Lemma 4.6(2)] that the augmented space $A(Z)$ is a Gromov hyperbolic space. The ideal boundary of $A(Z)$ constructed by the geodesic bicombing coincides with the Gromov boundary of $A(Z)$ as a Gromov hyperbolic space.

12. Topological dimension of ideal boundaries

In this section, we study the topological dimensions of ideal boundaries of trees of spaces. Then we compare the results with a well-known formula for the cohomological dimension of free products of groups.

12.1. Topological dimension

Theorem 12.1.

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex space. Suppose that $Z$ is proper. For $k\in K$ , set $X_{k}:=\xi^{-1}(\mathsf{star}(k))$ . Then,

\displaystyle\mathop{\mathrm{dim}}\nolimits(\partial Z)=\sup\{\mathop{\mathrm{dim}}\nolimits(\partial X_{k}):k\in K\}.

Theorem 1.4 is obtained by applying Theorem 12.1 to free products of metric spaces.

Let $(Z,d_{Z})$ be a $(\xi,K,L,\{\gamma_{k}\}_{k\in K})$ -tree of geodesic coarsely convex space. We fix a bijection $k\colon\mathbb{N}\to K$ . Let $A(Z)$ be the augmented space of $Z$ .

For simplicity, we denote by $\gamma$ an augmented bicombing on $A_{n}(Z)$ associated with the family $\{\gamma_{k}\}_{k\in K}$ .

Lemma 12.2.

Let $A_{n}(Z)$ be the $n$ -th augmented space as above.

\displaystyle\mathop{\mathrm{dim}}\nolimits\partial A_{n}(Z)\leq\max\{\mathop{\mathrm{dim}}\nolimits\partial X_{k(i)}:1\leq i\leq n\}.

Proof..

We suppose that $N:=\max\{\mathop{\mathrm{dim}}\nolimits\partial X_{k(i)}:1\leq i\leq n\}<\infty$ . Let $\mathcal{U}=\{U_{\alpha}\}_{\alpha\in\Lambda}$ be a finite open cover of $\partial A_{n}(Z)$ . We can suppose without loss of generality that $\mathcal{U}$ consists of fundamental neighbourhoods, that is, for $\alpha\in\Lambda$ , there exists $(x_{\alpha},n_{\alpha})\in A_{n}(Z)\cup\mathbb{N}$ such that

\displaystyle U_{\alpha}=V_{n_{\alpha}}[x_{\alpha}]\cap\partial A_{n}(Z)=\{y\in\partial A_{n}(Z):(x\mid y)>n_{\alpha}\}.

Furthermore, since $A_{n}(Z)$ is normal, and each $\partial X_{k(i)}$ is closed, we can assume that for each $\alpha\in\Lambda$ , there exists at most one $i\in\{1,\dots,n\}$ such that $U_{\alpha}\cap\partial X_{k(i)}\neq\emptyset$ . Set

	$\displaystyle\Lambda_{i}$	$\displaystyle:=\{\alpha:U_{\alpha}\cap\partial X_{k(i)}\neq\emptyset\}$
	$\displaystyle W_{i}$	$\displaystyle:=\bigcup_{\alpha\in\Lambda_{i}}U_{\alpha}.$

We can assume that for $i\neq i^{\prime}$ and for $\alpha\in\Lambda_{i}$ and $\alpha^{\prime}\in\Lambda_{i^{\prime}}$ , we have

(5)

\displaystyle\overline{U_{\alpha}}\cap\overline{U_{\alpha^{\prime}}}=\emptyset.

By taking a common refinement of the cover $\{U_{\alpha}:\alpha\in\Lambda_{i}\}$ and $\{V_{n}[x]:(x,n)\in X_{k(n)}\times\mathbb{N}\}$ , we can assume that for $\alpha\in\Lambda_{i}$ , we have $(x_{\alpha},n_{\alpha})\in X_{k(n)}\times\mathbb{N}$ .

Since $\mathop{\mathrm{dim}}\nolimits\partial X_{k(i)}\leq N$ , the cover $\{U_{\alpha}:\alpha\in\Lambda_{i}\}$ has a refinement $\mathcal{V}_{i}$ of order $N+1$ .

Now set

\displaystyle F_{i}:=\bigcup_{\alpha\in\Lambda_{i}}U_{\alpha}=\bigcup_{V\in\mathcal{V}_{i}}V,

\displaystyle Z^{\prime}:=A_{n}(Z)\setminus\left(\bigcup_{1\leq i\leq n}F_{i}\right)

Since $F_{i}$ ’s are open, $Z^{\prime}$ is closed, so $Z^{\prime}$ is a compact metrizable space.

Claim 12.3.

For each $i=1,\dots,n$ , $F_{i}$ is closed. Thus $Z^{\prime}$ is open.

We will show the claim by a similar argument as in the proof of Lemma 11.2.

Let $x\in\partial A_{n}(Z)\setminus F_{i}$ . We will show that there exists $T_{x}>0$ such that $V_{T_{x}}[x]\cap F_{i}=\emptyset$ . Since $\Lambda_{i}$ is a finite set, it is enough to show that for each $\alpha\in\Lambda_{i}$ , there exists $T_{x,\alpha}>0$ such that $V_{2T_{x,\alpha}}[x]\cap U_{\alpha}=\emptyset$ . We remark that $U_{\alpha}=V_{n_{\alpha}}[x_{\alpha}]$ with $(x_{\alpha},n_{\alpha})\in X_{k}\times\mathbb{N}$ .

For $k=k(i)\in K$ with $i\geq n$ , let $c_{k}=c_{k(i)}$ be the center of the horoball $\mathcal{H}(X_{k})$ . First, we suppose that $x=c_{k}$ for some $k\in K$ . Set

\displaystyle s(k;x):=\inf\{t\in\mathbb{R}:\textup{{rp}}\bar{\gamma}(e,x,t)\in X_{k}^{(0)}\}.

The geodesic $\textup{{rp}}\bar{\gamma}(\gamma_{e}^{x}(s(k;x)),x,-)$ consists of the vertical line. Set $T_{x,\alpha}:=s(k;x)+4D_{1}+n_{\alpha}$ . Then for $y\in V_{2T_{x,\alpha}}[x]$ , we have

\displaystyle\textup{{rp}}\bar{\gamma}(e,x,t)=\textup{{rp}}\bar{\gamma}(e,y,t)\quad(\forall t\in[0,T_{x,\alpha}])

If $y\in U_{\alpha}=V_{n_{\alpha}}[x_{\alpha}]$ , then

\displaystyle d_{Z}(\textup{{rp}}\bar{\gamma}(e,x_{\alpha},n_{\alpha}),\textup{{rp}}\bar{\gamma}(e,x,n_{\alpha}))=d_{Z}(\textup{{rp}}\bar{\gamma}(e,x_{\alpha},n_{\alpha}),\textup{{rp}}\bar{\gamma}(e,y,n_{\alpha}))\leq D_{1}

Thus $(x\mid x_{\alpha})_{e}\geq n_{\alpha}$ . This contradicts $x\notin U_{\alpha}$ . Therefore $y\notin U_{\alpha}$ , so we have

\displaystyle V_{2T_{x,\alpha}}[x]\cap U_{\alpha}=\emptyset.

Next, we suppose $x\neq c_{k}$ for all $k\in K$ . Then there exists $k\in K$ such that

\displaystyle s(k;x):=\inf\{t\in\mathbb{R}:\textup{{rp}}\bar{\gamma}(e,x,t)\in X_{k}\}

is finite, and $d_{Z}(\textup{{rp}}\bar{\gamma}(e,x,s(k;x)),x_{\alpha})>2n_{\alpha}$ . Set $T_{x,\alpha}=2(s(k;x)+n_{\alpha}+D_{1})$ . Then for all $y\in V_{T_{x,\alpha}}[x]$ , we have

\displaystyle\textup{{rp}}\bar{\gamma}(e,x,t)=\textup{{rp}}\bar{\gamma}(e,y,t)\quad(\forall t\in[0,n_{x,\alpha}])

By the same argument as before, $y\notin U_{\alpha}$ , so we have $V_{2T_{x,\alpha}}[x]\cap U_{\alpha}=\emptyset$ . This completes a proof of the claim.

Now by a similar argument as in Section 11, we see that $Z^{\prime}$ has countable bases consisting of clopen sets. Thus $Z^{\prime}$ is zero dimensional. So the open cover

\displaystyle\{U_{\alpha}\cap Z^{\prime}:\alpha\notin\Lambda_{i},\,1\leq i\leq n\}

has a refinement $\mathcal{V}_{\infty}$ of order $1$ . It follow that $\mathcal{U}$ has a refinement $\mathcal{V}\cup\bigcup_{i}\mathcal{V}_{i}$ of order $N+1$ . Thus $\mathop{\mathrm{dim}}\nolimits\partial A_{n}(Z)\leq N$ . ∎

Proof of Theorem 12.1.

Since each $\partial X_{k}$ is a closed subset, we have

\displaystyle\mathop{\mathrm{dim}}\nolimits(\partial Z)\geq\sup\{\mathop{\mathrm{dim}}\nolimits(\partial X_{k}):k\in K\}.

By a formula for the topological dimensions of projective limits in [18], we have

\displaystyle\mathop{\mathrm{dim}}\nolimits(\partial Z)\leq\sup_{n}\mathop{\mathrm{dim}}\nolimits\partial A_{n}(Z).

By Lemma 12.2, we have

\displaystyle\sup_{n}\mathop{\mathrm{dim}}\nolimits\partial A_{n}(Z)\leq\sup_{n}\mathop{\mathrm{dim}}\nolimits\partial X_{k(n)}.

This completes the proof. ∎

12.2. Comparison with the formula for cohomological dimensions of groups

We remark that Theorem 1.4 is an analogue of a well-known formula for the cohomological dimensions of free products. For a group $G$ , we denote by $\operatorname{cd}G$ the cohomological dimension of $G$ . By [5, VIII (2.4) Proposition, (a) and (c)] we have the following.

Theorem 12.4 ([5]).

Let $G$ and $H$ be groups. Let $G*H$ be the free product of $G$ and $H$ . Then we have

\displaystyle\operatorname{cd}(G*H)=\max\{\operatorname{cd}G,\operatorname{cd}H\}.

The following theorem relates the cohomological dimensions of group $G$ acting coarsely convex space $X$ and the topological dimension of the ideal boundary $\partial X$ .

Theorem 12.5 ([11, Corollary 8.10]).

Let $G$ be a group acting geometrically on a proper coarsely convex space $X$ . If $G$ admits a finite model for the classifying space $BG$ , then

\displaystyle\operatorname{cd}G=\mathop{\mathrm{dim}}\nolimits\partial X+1.

Let $G$ and $H$ be groups acting on geodesic coarsely convex spaces $X$ and $Y$ respectively. Suppose $G$ and $H$ admit finite models for the classifying spaces $BG$ and $BH$ respectively. In this case, $\mathop{\mathrm{dim}}\nolimits\partial(X*Y)$ can be computed by Theorem 12.4 as follows.

By Theorem 12.5,

\displaystyle\operatorname{cd}(G)=\mathop{\mathrm{dim}}\nolimits\partial X+1,\quad\operatorname{cd}(H)=\mathop{\mathrm{dim}}\nolimits\partial Y+1.

Since the free product $G*H$ is hyperbolic relative to $\{G,H\}$ , by [11, Theorem A.1.], $G*H$ admits a finite model for the classifying spaces $B(G*H)$ . By Example 7.10, $G*H$ acts on $X*Y$ geometrically, so using Theorem 12.5 again,

\displaystyle\operatorname{cd}(G*H)=\mathop{\mathrm{dim}}\nolimits(\partial(X*Y))+1.

Therefore by Theorem 12.4, we have

\displaystyle\mathop{\mathrm{dim}}\nolimits(\partial(X*Y))=\max\{\mathop{\mathrm{dim}}\nolimits\partial X,\mathop{\mathrm{dim}}\nolimits\partial Y\}.

References

[1] Hadi Bigdely and Eduardo Martínez-Pedroza, Relative dehn functions, hyperbolically embedded subgroups and combination theorems, Glasgow Mathematical Journal (2023), 1–23.
[2] Bruce Blackadar, $K$ -theory for operator algebras, second ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031 (99g:46104)
[3] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
[4] L. E. J. Brouwer, On the structure of perfect sets of points, Proc. Koninklijke Akademie van Wetenschappen 12 (1910), no. 4, 785–794.
[5] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994, Corrected reprint of the 1982 original. MR 1324339 (96a:20072)
[6] Jérémie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, and Damian Osajda, Helly groups, 2020, arXiv:2002.06895.
[7] Dominic Descombes and Urs Lang, Convex geodesic bicombings and hyperbolicity, Geom. Dedicata 177 (2015), 367–384. MR 3370039
[8] Yuuhei Ezawa and Tomohiro Fukaya, Visual maps between coarsely convex spaces, Kobe J. Math. 40 (2021), no. 1-2, 7–45.
[9] Tomohiro Fukaya and Takumi Matsuka, Free products of coarsely convex spaces and the coarse Baum-Connes conjecture, arXiv:2303.13701, to appear in Kyoto J. Math (2023).
[10] Tomohiro Fukaya, Takumi Matsuka, and Eduardo Martínez-Pedroza, Geodesic coarsely convex group pair, in preparation.
[11] 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
[12] Tomohiro Fukaya, Shin-ichi Oguni, and Takamitsu Yamauchi, Coarse compactifications and controlled products, J. Topol. Anal. 14 (2022), no. 4, 875–900. MR 4523562
[13] É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648
[14] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829 (89e:20070)
[15] Daniel Groves and Jason Fox Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008), 317–429. MR 2448064 (2009h:57030)
[16] 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)
[17] by same author, Analytic $K$ -homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000, Oxford Science Publications. MR 1817560 (2002c:58036)
[18] Yûkiti Katuta, On the covering dimension of inverse limits, Proc. Amer. Math. Soc. 84 (1982), no. 4, 588–592. MR 643755
[19] John Milnor, On the Steenrod homology theory, Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 79–96. MR 1388297 (98d:55005)
[20] Damian Osajda and Piotr Przytycki, Boundaries of systolic groups, Geom. Topol. 13 (2009), no. 5, 2807–2880. MR 2546621
[21] Guoliang Yu, Coarse Baum-Connes conjecture, $K$ -Theory 9 (1995), no. 3, 199–221. MR 1344138 (96k:58214)

Boundary of free products of metric spaces

Abstract.

Key words and phrases:

1. Introduction

Theorem 1.1.

Theorem 1.2.

Corollary 1.3.

Theorem 1.4.

1.1. Outline

2. Combinatorial and metric horoballs

Definition 2.1.

Remark 2.2.

Definition 2.3.

Definition 2.4.

Definition 2.5.

Proposition 2.6 ([15], Lemma 3.10).

Proof..

Claim 2.7.

Claim 2.8.

Proposition 2.9 ([15]).

Proof..

3. coarsely convex bicombing

Definition 3.1.

Definition 3.2.

4. Ideal boundary

4.1. Gromov product and sequential boundary

Definition 4.1.

Lemma 4.2 ([11, Lemma 4.8]).

Definition 4.3.

Lemma 4.4.

Proof..

Definition 4.5.

Definition 4.6.

Lemma 4.7 ([12, Lemma 2.8, Theorem 2.9]).

4.2. Combing at infinity

Lemma 4.8.

Proof..

Definition 4.9.

Lemma 4.10 ([11, Proposition 4.2]).

Definition 4.11.

Lemma 4.12.

Proof..

4.3. Visual maps

Definition 4.13.

Proposition 4.14 ([8, Corollary 3.6]).

5. Geodesic coarsely convex spaces

5.1. Geodesic combing at infinity

Definition 5.1.

Lemma 5.2.

Proof..

Example 5.3.

Lemma 5.4.

Proof..

Definition 5.5.

Lemma 5.6.

Proof..

Remark 5.7.

Lemma 5.8.

Proof..

5.2. γ\gamma-convex subspaces

Definition 5.9.

Proposition 5.10.

Corollary 5.11.

6. Continuous at infinity

Definition 6.1.

Lemma 6.2.

Proof..

6.1. Image of the exponential map

Lemma 6.3.

Proof..

Lemma 6.4.

6.2. Angle map

Lemma 6.5.

Proposition 6.6.

7. Trees of spaces

7.1. Trees of spaces

Definition 7.1.

Definition 7.2.

Definition 7.3.

Theorem 7.4 ([10]).

5.2. $\gamma$ -convex subspaces

8.1. $n$ -th augmented spaces

8.2. Bicombings on $n$ -th augmented spaces

9.3. $K$ -theory of the Roe algebra