Asymptotic property C of the countable direct sum of uniformly discrete $0$-hyperbolic spaces

TL;DR

The paper proves that the countable direct sum of 0-hyperbolic, D-discrete metric spaces has asymptotic property C, extending previous results to a broader class including free groups of finite rank.

Contribution

It introduces a new definition of the direct sum of pointed metric spaces and proves that such sums of 0-hyperbolic, D-discrete spaces possess asymptotic property C.

Findings

Countable direct sums of 0-hyperbolic, D-discrete spaces have asymptotic property C.

Includes the case of countable sums of free groups of finite rank.

Generalizes Yamauchi's result on sums of integers.

Abstract

We define the direct sum of a countable family of pointed metric spaces in a way resembling the direct sum of groups. Then we prove that if a family consists of $0$-hyperbolic (in the sense of Gromov) and $D$-discrete spaces, then its direct sum has asymptotic property C. The main example is a countable direct sum of free groups of (possibly varying) finite rank. This is a generalization of T. Yamauchi's result concernig the countable direct sum of the integers.

Equations

$$(x|y{)}_{x_0}=\frac{1}{2}\left(d(x,x_0)+d(y,x_0)-d(x,y)\right).$$

$$(x|z{)}_{x_0}\ge \min \{(x|y{)}_{x_0},(y|z{)}_{x_0}\}-\delta$$

$$\mathrm{mesh}(\mathcal{U}):=\sup \{\mathrm{diam}(U):U\in \mathcal{U}\}$$

$$\mathrm{dist}(A,B):=\inf \{d(a,b):a\in A,b\in B\}\ge R.$$

$$[\alpha ,\beta )=\{x\in X:\alpha \le d(x,x_0)<\beta \}.$$

$$x{\sim }_{\alpha ,r}y⟺(x|y)>\alpha -\frac{1}{2}r$$

$$d(x,y)<d(x,x_0)+d(y,x_0)-2\alpha +r<2(\alpha +d)-2\alpha +r=2d+r.$$

$$\begin{array}{rl}{\mathcal{V}}_0& =\bigcup _{n\in \mathbb{N}}[(2n-1)R,2nR)/{\sim }_{(2n-1)R,R},\\ {\mathcal{V}}_1& =\bigcup _{n\in {\mathbb{N}}_0}[2nR,(2n+1)R)/{\sim }_{2nR,R}.\end{array}$$

$$\begin{array}{rl}{\mathcal{C}}_l& =\bigcup _{n\in {\mathbb{N}}_0}[(2^m(n-1)+l)S,(2^{m}n+l)S-R_0)/{\sim }_{(2^m(n-1)+l)S,R_0},\\ {\mathcal{D}}_l& =\bigcup _{n\in {\mathbb{N}}_0}[(2^{m}n+l)S-R_0,(2^{m}n+l)S)/{\sim }_{(2^{m}n+l)S-R_0,2^{m}S-R_0}.\end{array}$$

$$\underset{i=1}{\overset{\infty }{⨁}}{X}_i=\left\{(x_i)\in \prod _{i=1}^{\infty }{X}_i:x_i=o_i\text{for all but finitely many}i\right\}.$$

$$d((x_i),(y_i))=\sum _{i=1}^{\infty }i{d}_i(x_i,y_i).$$

$${\mathcal{W}}_l^{(j)}=\left\{\prod _{i=1}^m{V}_i:V_i\in {\mathcal{V}}_{\psi (l{)}_i}^{(j+i-1)},i\in \{1,\dots ,m\}\right\},$$

$${\mathcal{U}}_0=\left\{\prod _{i=1}^k{C}_i\times W\times \prod _{i>k+m}\{y_i\}:(C_1,\dots ,C_k,W)\in \bigcup _{l=0}^{2^m-1}\left(\prod _{i=1}^k{\mathcal{C}}_l^{(i)}\times {\mathcal{W}}_l^{(k+1)}\right),(y_i)\in \underset{i=k+m+1}{\overset{\infty }{⨁}}{X}_i\right\}.$$

$$\begin{array}{rl}& {\mathcal{U}}_{2^k(s-1)+t}=\{\prod _{i=1}^{s-1}{V}_i\times D_s\times \prod _{i=s+1}^k{V}_i\times W\times \prod _{i>k+m}\{y_i\}:\\ & V_i\in {\mathcal{V}}_{\varphi (t{)}_i}^{(i)},i\in \{1,\dots ,s-1,s+1,\dots ,k\},(D_s,W)\in \bigcup _{l=0}^{2^m-1}\left({\mathcal{D}}_l^{(s)}\times {\mathcal{W}}_l^{(k+1)}\right),(y_i)\in \underset{i=k+m+1}{\overset{\infty }{⨁}}{X}_i\}.\end{array}$$

$$(x_i)\in \prod _{i=1}^{s-1}{V}_i\times D_s\times \prod _{i=s+1}^k{V}_i\times W\times \prod _{i>k+m}\{x_i\}\in {\mathcal{U}}_{2^k(s-1)+t}.$$

Full text

Keywords: asymptotic property C, Gromov hyperbolic spaces, direct sum.

2010 Mathematics Subject Classification: Primary 54F45; Secondary 20F67, 20F69.

1 Introduction

We recall some definitions and known facts about the notion of Gromov hyperbolicity. We refer to [6, Section 6.2] and [2, Section 3] for detailed exposition.

Let $(X,d)$ be a metric space with a fixed basepoint $x_{0}\in X$. For any $x,y\in X$ we denote by $(x|y)_{x_{0}}$ the Gromov product of $x$ and $y$ with respect to the basepoint $x_{0}$ defined by the following formula:

[TABLE]

Definition 1**.**

We say that a space $X$ is $\delta$-hyperbolic for $\delta\geq 0$ if and only if

[TABLE]

for any $x,y,z\in X$ and for any choice of a basepoint $x_{0}\in X$. We call $X$ (Gromov) hyperbolic if there exists $\delta\geq 0$ such that $X$ is $\delta$-hyperbolic.

It can be shown that if a space satisfies (2) for a fixed $x_{0}$, then it is $2\delta$-hyperbolic. Thus, in the case $\delta=0$ the condition defining [math]-hyperbolicity does not depend on the choice of a basepoint and we will omit it in the notation writing simply $(x|y)$ for the Gromov product.

Bonk and Schramm showed in [2, Theorem 4.1] that any $\delta$-hyperbolic metric space can be isometrically embedded into a complete $\delta$-hyperbolic geodesic metric space. For geodesic metric spaces there is a characterisation of hyperbolicity due to Rips using so called thin triangles. As we restrict our attention to the case $\delta=0$, a geodesic metric space is [math]-hyperbolic if and only if it is uniquely geodesic and for any $x,y,z\in X$ the geodesic $[x,y]$ is contained in the union $[x,z]\cup[z,y]$. Such spaces are called $\mathbb{R}$-trees.

Summarising, [math]-hyperbolic spaces are precisely subspaces of $\mathbb{R}$-trees. We prefer the term “[math]-hyperbolic” because our arguments will not depend on existence of geodesics, solely on the Gromov condition (2) for $\delta=0$.

Definition 2**.**

We call a metric space $(X,d)$ $D$-discrete if $d(x,y)\geq D$ for all $x\neq y$, $x,y\in X$. If there exists $D>0$ such that $X$ is $D$-discrete, we say that $X$ is uniformly discrete.

Example 3**.**

Each free group $\mathbb{F}_{n}$ on $n$ generators equipped with the usual word length metric becomes a $1$-discrete [math]-hyperbolic space. If we consider the Cayley graph of $\mathbb{F}_{n}$ with its edges realized as isometric copies of the unit interval $[0,1]$, we get a geodesic [math]-hyperbolic space (losing uniform discreteness).

We wil need some useful definitions.

Definition 4**.**

We say that a non-empty family $\mathcal{U}$ of metric spaces is uniformly bounded if the number

[TABLE]

is finite. Let $R>0$, we say that $\mathcal{U}$ is $R$-disjoint if for any different $A,B\in\mathcal{U}$ we have

[TABLE]

Let us recall the definition of the asymptotic property C, which was introduced by Dranishnikov [3]. It is one of several possible generalizations of the asymptotic dimension.

Definition 5**.**

We say that a metric space $X$ has the asymptotic property C if for any strictly increasing sequence $(a_{i})_{i\in\mathbb{N}}$ of natural numbers there is $n\in\mathbb{N}$ and a finite sequence $\{\mathcal{U}_{i}\}_{i=1}^{n}$ of uniformly bounded families such that $\bigcup_{i=1}^{n}\mathcal{U}_{i}$ covers $X$ and $\mathcal{U}_{i}$ is $a_{i}$-disjoint for $i=1,\dots,n$.

T. Radul showed in [4] that the asymptotic property C can be thought of as a transfinite extension of asymptotic dimension. T. Yamauchi in his paper [7] proved that the group $\bigoplus_{i=1}^{\infty}\mathbb{Z}$ with a proper invariant metric has asymptotic property C. Our goal is to extend this result to a more general context replacing each copy of $\mathbb{Z}$ with a [math]-hyperbolic space, all of them being $D$-discrete with common $D>0$. Of course, we have to define what direct sum means for pointed metric spaces (not necessarily groups) and choose a metric such that succeeding spaces are rescaled with multipliers tending to infinity.

2 Some auxiliary facts

We will restate some properties of [math]-hyperbolic spaces. They are essential for a proof that such spaces have asymptotic dimension at most $1$ (see [6, Proposition 9.8]).

From now, let $X$ be a [math]-hyperbolic space with a fixed basepoint $x_{0}\in X$. For real numbers $\alpha<\beta$ we define the interval $[\alpha,\beta)\subset X$ by the formula

[TABLE]

Note that we allow $\alpha,\beta$ to be negative, in particular the interval may be empty. It will not bother us and will be useful for keeping some notation concise. Fix $\alpha\in\mathbb{R}$ and $d,r>0$. Let us define an equivalence relation $\sim_{\alpha,r}$ in $[\alpha,\alpha+d)$ putting

[TABLE]

for any $x,y\in[\alpha,\alpha+d)$. We will briefly verify that this relation is reflexive and transitive. We have $(x|x)=d(x,x_{0})\geq\alpha>\alpha-\frac{1}{2}r$ for $x\in[\alpha,\alpha+d)$. Transitivity follows immediately from the definition of a [math]-hyperbolic space.

Lemma 6**.**

All equivalence classes of the relation $\sim_{\alpha,r}$ in $[\alpha,\alpha+d)$ are uniformly bounded by $2d+r$. Different equivalence classes are $r$-disjoint.

Proof.

Assume that $x\sim_{\alpha,r}y$. Then $(x|y)>\alpha-\frac{1}{2}r$, so $d(x,x_{0})+d(y,x_{0})-d(x,y)>2\alpha-r$. Thus

[TABLE]

Suppose now that $d(x,y)<r$. Then $(x|y)>\frac{1}{2}(2\alpha-r)=\alpha-\frac{1}{2}r$, which means that $x$ and $y$ belong to the same equivalence class. ∎

Observe that our result applies for all intervals $[\alpha,\alpha+d)$ and the bound $2d+r$ as well as the “disjointness” constant $r$ do not depend on $\alpha$. Moreover, they do not depend on a particular [math]-hyperbolic space $X$. Equivalently we can say that for any [math]-hyperbolic $X$ and $d>0$ the family $\{[\alpha,\alpha+d)\colon\alpha\in\mathbb{R}\}$ has asymptotic dimension [math] uniformly with type function $\tau(r)=2d+r$. We refer to [6, Example 9.14] for the terminology.

Suppose we are given a number $R>0$. Let us consider the following families:

[TABLE]

Both families $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$ are $R$-disjoint. If two members of a fixed $\mathcal{V}_{i}$ are contained in different intervals, then it is easy to see that their distance is not less than $R$. For different members lying in the same interval we apply Lemma 6. Using Lemma 6 we also conclude that both families are uniformly $3R$-bounded. Obviously $\bigcup(\mathcal{V}_{0}\cup\mathcal{V}_{1})=X$.

We will need some more families of subsets of $X$. Suppose that $0<R_{0}<S$ and $m\in\mathbb{N}$ are given. For any $l\in\{0,\dots,2^{m}-1\}$ define

[TABLE]

Similarly, $\mathcal{C}_{l}$ is $R_{0}$-disjoint and uniforlmy bounded for $l\in\{0,\dots,2^{m}-1\}$. Furthermore, $\bigcup_{l=0}^{2^{m}-1}\mathcal{D}_{l}$ is $(S-R_{0})$-disjoint and uniformly bounded. We have also that $\bigcup(\mathcal{C}_{l}\cup\mathcal{D}_{l})=X$ for any $l\in\{0,\dots,2^{m}-1\}$.

Note that the above construction works for any [math]-hyperbolic space $X$ and the boundedness and disjointness constants of the defined families remain the same.

3 The main result

Suppose that we have a sequence $(X_{i},d_{i})_{i\in\mathbb{N}}$ of $D$-discrete and [math]-hyperbolic spaces. Let $(o_{i})_{i\in\mathbb{N}}$ be a sequence of basepoints $o_{i}\in X_{i}$. Define the direct sum

[TABLE]

We equip $\bigoplus_{i=1}^{\infty}X_{i}$ with a metric

[TABLE]

Theorem 7**.**

The metric space $\bigoplus_{i=1}^{\infty}X_{i}$ has asymptotic property C.

Proof.

The general idea is to rewrite Yamauchi’s proof ([7]) replacing original families of subsets of $\mathbb{Z}$ with families (3), (4) of subsets of $X_{i}$ for $i\in\mathbb{N}$. The families will depend on $i$, which will be denoted by a superscript $(i)$. Nevertheless, the constants of disjointness and boundedness will suit for all $i\in\mathbb{N}$ uniformly. Without loss of generality we may assume that $D=1$, i.e. all the spaces $X_{i}$ are $1$-discrete.

Let $R_{0}<R_{1}<\dots$ be an arbitrary increasing sequence of natural numbers. Let $k=R_{0}+1$, $m=R_{k2^{k}}+1$ and $S=R_{0}+R_{k2^{k}}$. Take a bijection $\psi\colon\{0,1,\dots,2^{m}-1\}\to\{0,1\}^{m}$. For each $i\in\mathbb{N}$ define families $\mathcal{V}_{0}^{(i)},\mathcal{V}_{1}^{(i)}$ of subsets of $X_{i}$ as in (3) with $R=R_{k2^{k}}$. For each $i\in\mathbb{N}$ and $l\in\{0,1,\dots,2^{m}-1\}$ define similarly $\mathcal{C}_{l}^{(i)}$ and $\mathcal{D}_{l}^{(i)}$ according to (4). For any $j\in\mathbb{N}$ and $l\in\{0,1,\dots,2^{m}-1\}$ put also

[TABLE]

where $\psi(l)_{i}$ stands for the $i$-th coordinate of $\psi(l)\in\{0,1\}^{m}$. Note that $\mathcal{W}_{l}^{(j)}$ is a family of subsets of $X_{j}\times\dots\times X_{j+m-1}$. Since $\mathcal{V}_{0}^{(i)}\cup\mathcal{V}_{1}^{(i)}$ covers $X_{i}$, we conclude that $\bigcup_{l=0}^{2^{m}-1}\mathcal{W}_{l}^{(j)}$ covers $X_{j}\times\dots\times X_{j+m-1}$ for any $j\in\mathbb{N}$. Put

[TABLE]

We will show that $\mathcal{U}_{0}$ is $R_{0}$-disjoint. The elements of $\mathcal{U}_{0}$ depend on $l$, a choice of $C_{1},\dots,C_{k},W$, and a sequence $(y_{i})$. Suppose that $A,A^{\prime}$ are different members of $\mathcal{U}_{0}$. If the corresponding sequences $(y_{i}),(y^{\prime}_{i})$ are different, then $d(A,A^{\prime})\geq k+m+1>R_{0}$, which follows from the definition of the metric (5) and $1$-discreteness of all $X_{i}$. If $l\neq l^{\prime}$, then the corresponding $W$ and $W^{\prime}$ are disjoint (by construction of families $\mathcal{W}_{l}^{(k+1)}$) so $d(A,A^{\prime})\geq k+1>R_{0}$. In the remainig case we use the fact that each $\mathcal{C}_{l}^{(i)}$ is $R_{0}$-disjoint.

Let us fix a bijection $\phi\colon\{1,2,...,2^{k}\}\to\{0,1\}^{k}$. For $s\in\{1,2,\dots,k\}$ and $t\in\{1,2,\dots,2^{k}\}$ put

[TABLE]

We will show that for each $j\in\{1,2,\dots,k2^{k}\}$ the family $\mathcal{U}_{j}$ is $R_{k2^{k}}$-disjoint, in particular $R_{j}$-disjoint. The reasoning is similar to that for $\mathcal{U}_{0}$. Fix $s$ and $t$. Suppose that $A,A^{\prime}$ are different members of $\mathcal{U}_{2^{k}(s-1)+t}$. If the corresponding sequences $(y_{i}),(y^{\prime}_{i})$ are different, then $d(A,A^{\prime})\geq k+m+1>R_{k2^{k}}$. If $l\neq l^{\prime}$, then the corresponding $D_{s}$ and $D^{\prime}_{s}$ are $S-R_{0}=R_{k2^{k}}$-disjoint so $d(A,A^{\prime})\geq R_{k2^{k}}$. If $l$ is common for $A,A^{\prime}$ and $W\neq W^{\prime}$, we use $R_{k2^{k}}$-disjointness of $\mathcal{W}_{l}^{(k+1)}$. In the remainig case there exists $i\in\{1,\dots,s-1,s+1,\dots,k\}$ such that $V_{i}\neq V^{\prime}_{i}$ and we use $R_{k2^{k}}$-disjointness of $\mathcal{V}^{(i)}_{\phi(t)_{i}}$.

Let us show that $\bigcup_{j=0}^{k2^{k}}\mathcal{U}_{j}$ covers $\bigoplus_{i=1}^{\infty}X_{i}$. Let $(x_{i})\in\left(\bigoplus_{i=1}^{\infty}X_{i}\right)\setminus\bigcup\mathcal{U}_{0}$. Since $\bigcup_{l=0}^{2^{m}-1}\mathcal{W}_{l}^{(k+1)}$ covers $X_{k+1}\times\dots\times X_{k+m}$, there exist $l\in\{0,\dots,2^{m}-1\}$ and $W\in\mathcal{W}_{l}^{(k+1)}$ such that $(x_{i})_{i=k+1}^{k+m}\in W$. Since $(x_{i})\not\in\bigcup\mathcal{U}_{0}$, $(x_{i})_{i=1}^{k}$ does not belong to any set of the form $\prod_{i=1}^{k}C_{i}$ where $C_{i}\in\mathcal{C}_{l}^{(i)}$ for $i\in\{1,\dots,k\}$. Thus, there is $s\in\{1,\dots,k\}$ such that $x_{s}\not\in\bigcup\mathcal{C}_{l}^{(s)}$. Since $\bigcup(\mathcal{C}_{l}^{(s)}\cup\mathcal{D}_{l}^{(s)})=X_{s}$, there exists $D_{s}\in\mathcal{D}_{l}^{(s)}$ containing $x_{s}$. Since $\bigcup(\mathcal{V}_{0}^{(i)}\cup\mathcal{V}_{1}^{(i)})=X_{i}$ for each $i$, there exists $t\in\{1,\dots,2^{k}\}$ such that $(x_{i})_{i=1}^{k}\in\prod_{i=1}^{k}V_{i}$ for some $V_{i}\in\mathcal{V}_{\phi(t)_{i}}^{(i)}$, $i\in\{1,\dots,k\}$. Thus,

[TABLE]

Finally, the families $\mathcal{U}_{j}$, $j=\{0,\dots,k2^{k}\}$ are uniformly bounded. It follows from the fact that the construction of these families involves families $\mathcal{V}_{0}^{(i)}$, $\mathcal{V}_{1}^{(i)}$, $\mathcal{C}_{l}^{(i)}$, $\mathcal{D}_{l}^{(i)}$, $\mathcal{W}_{l}^{(i)}$, which are uniformly bounded with meshes independent of $i\in\mathbb{N}$. ∎

Remark 8**.**

In fact we have proved more, namely $\bigoplus_{i=1}^{\infty}X_{i}$ has transfinite asymptotic dimension not greater than $\omega$. For definition see [4, Section 3]. It follows from the fact that for fixed $R_{0}$ there exists a natural number $n=(R_{0}+1)2^{R_{0}+1}$ (depending only on $R_{0}$) such that for any sequence $R_{0}<R_{1}<\dots$ there exists a sequence $\{\mathcal{U}_{i}\}_{i=0}^{n}$ of uniformly bounded families such that $\bigcup_{i=0}^{n}\mathcal{U}_{i}$ covers $\bigoplus_{i=1}^{\infty}X_{i}$ and $\mathcal{U}_{i}$ is $R_{i}$-disjoint for $i=0,\dots,n$. Loosely speaking, the number of sufficiently disjoint and uniformly bounded families needed to cover the space is constant for fixed $R_{0}$.

We apply our result to the case of free groups. In the realm of dicrete spaces we call a metric proper if all balls with respect to this metric are finite.

Corollary 9**.**

Let $(G_{i})_{i\in\mathbb{N}}$ be a sequence of free groups of finite rank (which may depend on $i$) and $G=\bigoplus_{i=1}^{\infty}G_{i}$ their group-theoretical direct sum. Then $G$ equipped with any proper left-invariant metric has asymptotic property C.

Proof.

Let $d_{i}$ be the usual left-invariant word length metric in $G_{i}$ for $i\in\mathbb{N}$. Then each $(G_{i},d_{i})$ is [math]-hyperbolic and $1$-discrete. The metric $d$ defined by (5) in $G$ is proper and left-invariant. By Theorem 7 the space $(G,d)$ has asymptotic property C. The corollary follows from the fact that all proper left-invariant metrics on $G$ are coarsely equivalent and asymptotic property C is a coarse invariant. ∎

Roe proved in [5] that $\delta$-hyperbolic geodesic spaces with bounded growth have finite asymptotic dimension. This suggests posing the following problem:

Problem 10**.**

Can Theorem 7 be generalized to the case of $\delta$-hyperbolic spaces (with common $\delta$ and commonly bounded growth)?