On balanced coronas of groups

TL;DR

This paper investigates the structure of Higson's corona for infinite groups under certain coarse structures, showing it is a singleton in some cases and contains a complex space in others.

Contribution

It characterizes the Higson's corona of infinite groups with specific coarse structures, revealing when it is trivial or contains a large ultrafilter space.

Findings

Corona is singleton if <|G| or =|G| and  is singular.

Corona contains a space of -uniform ultrafilters if =|G| and  is regular.

Results depend on the cofinality of the cardinal .

Abstract

Let $G$ be an infinite group, $\kappa$ be an infinite cardinal, $\kappa\leq \mid G\mid$ and let $\mathcal{E}_{\kappa}$ denotes a coarse structure on $G$ with the base $\{\{ (x,y): y\in F x F\}: F\in [G]^{<\kappa}\}$. We prove that if either $\kappa< \mid G\mid$ or $\kappa= \mid G\mid$ and $\kappa$ is singular then the Higson's corona $\nu _{\kappa} (G)$ of the coarse space $(G, \mathcal{E}_{\kappa})$ is a singleton. If $\kappa= \mid G\mid$ and $\kappa$ is regular then $\nu _{\kappa} (G)$ contains a copy of the space $U_{\kappa}$ of $\kappa$-uniform ultrafilters on $\kappa$.

Equations

$$\{\{(x,y):y\in xF\}:F\in [G{]}^{<\kappa }\},$$

$$\{\{(x,y):y\in Fx\}:F\in [G{]}^{<\kappa }\},$$

$$\{\{(x,y):y\in FxF\}:F\in [G{]}^{<\kappa }\}$$

$$diamh(S_{\gamma +n}g{S}_{\gamma +n})<\frac{1}{n+1}$$

$$|h(zx)-h(x)|<\frac{\epsilon }{2},|h(zy)-h(y)|<\frac{\epsilon }{2}$$

$$|h(z{x}_{\alpha })-h(x_{\alpha })|<\frac{\epsilon }{2},|h(z{y}_{\alpha })-h(y_{\alpha })|<\frac{\epsilon }{2}.$$

$$U=\{p\in \beta G:X\in p,G\setminus G_{\alpha }\in pforeach\alpha <\kappa \}.$$

$$\{\{(x,y):y\in SxS\}:S\in [G{]}^{<\kappa }andSisasubgroupofG\}.$$

Full text

On balanced coronas of groups

Igor Protasov

Abstract. Let $G$ be an infinite group, $\kappa$ be an infinite cardinal, $\kappa\leq\mid G\mid$ and let $\mathcal{E}_{\kappa}$ denotes a coarse structure on $G$ with the base $\{\{(x,y):y\in FxF\}:F\in[G]^{<\kappa}\}$. We prove that if either $\kappa<\mid G\mid$ or $\kappa=\mid G\mid$ and $\kappa$ is singular then the Higson’s corona $\nu_{\kappa}(G)$ of the coarse space $(G,\mathcal{E}_{\kappa})$ is a singleton. If $\kappa=\mid G\mid$ and $\kappa$ is regular then $\nu_{\kappa}(G)$ contains a copy of the space $U_{\kappa}$ of $\kappa$-uniform ultrafilters on $\kappa$.

**MSC: ** 54E35, 20F69.

Keywords: coarse structure, slowly oscillating functions, balanced corona.

1. Introduction

Let $X$ be a set. A family $\mathcal{E}$ of subsets of $X\times X$ is called a *coarse structure * if

• •

each $E\in\mathcal{E}$ contains the diagonal $\bigtriangleup_{X}$, $\bigtriangleup_{X}=\{(x,x):x\in X\}$;

• •

if $E$, $E^{\prime}\in\mathcal{E}$ then $E\circ E^{\prime}\in\mathcal{E}$ and $E^{-1}\in\mathcal{E}$, where $E\circ E^{\prime}=\{(x,y):\exists z((x,z)\in E,\ \ (z,y)\in E^{\prime})\}$, $E^{-1}=\{(y,x):(x,y)\in E\}$;

• •

if $E\in\mathcal{E}$ and $\bigtriangleup_{X}\subseteq E^{\prime}\subseteq E$ then $E^{\prime}\in\mathcal{E}$;

• •

for any $x,y\in X$, there exists $E\in\mathcal{E}$ such that $(x,y)\in E$.

A subset $\mathcal{E}^{\prime}\subseteq\mathcal{E}$ is called a base for $\mathcal{E}$ if, for every $E\in\mathcal{E}$, there exists $E^{\prime}\in\mathcal{E}^{\prime}$ such that $E\subseteq E^{\prime}$. For $x\in X$, $A\subseteq X$ and $E\in\mathcal{E}$, we denote $E[x]=\{y\in X:(x,y)\in E\}$, $E[A]=\cup_{a\in A}\ \ E[a]$ and say that $E[x]$ and $E[A]$ are balls of radius $E$ around $x$ and $A$. We say that a subset $B$ of $X$ is bounded if there exist $E\in\mathcal{E}$ and $a\in X$ such that $B\subseteq E[a]$.

The pair $(X,\mathcal{E})$ is called a coarse space [14] or a *ballean * [10], [13].

Let $(X,\mathcal{E})$, $(X^{\prime},\mathcal{E}^{\prime})$ be balleans. A mapping $f:X\longrightarrow X^{\prime}$ is called macro uniform if, for every $E\in\mathcal{E}$, there exists $E^{\prime}\in\mathcal{E}^{\prime}$ such that $f(E[x])\subseteq E^{\prime}[f(x)]$ for each $x\in X$. If $f$ is a bijection such that $f$ and $f^{-1}$ are macro-uniform then $f$ is called an asymorphism.

Given a ballean $(X,\mathcal{E})$, we endow $X$ with the discrete topology and take the points of the Stone-$\check{C}$ech compactification $\beta X$ of $X$ to be the ultrafilters on $X$ with the points of $X$ identified with the principal ultrafilters. For every subset $A$ of $X$, we put $\bar{A}=\{q\in\beta X:A\in q\}$. The topology of $\beta X$ can be defined by stating that the family $\{\bar{A}:A\subseteq X\}$ is a base for the open sets.

If $Y$ is a compact Hausdorff space and $f:X\longrightarrow Y$ then $f^{\beta}$ denotes the extension of $f$ to $\beta X$.

We denote $X^{\sharp}=\{q\in\beta X:$ each member of $q$ is unbounded in $(X,\mathcal{E})\}$ and, given any $r,q\in X^{\sharp}$, write $r\|q$ if there exists $E\in\mathcal{E}$ such that $E[R]\in q$ for each $R\in r$. By [6, Lemma 4.1], $\|$ is an equivalence on $X^{\sharp}$. We denote by $\sim$ the minimal (by inclusion) closed (in $X^{\sharp}\times X^{\sharp}$) equivalence on $X$ such that $\|\subseteq\sim$. By [8, Proposition 1], $r\sim q$ if and only if $h^{\beta}(p)=h^{\beta}(q)$ for every slowly oscillating function $h:X\longrightarrow[0,1]$. We recall that a function $f:(X,\mathcal{E})\longrightarrow\mathbb{R}$ is slowly oscillating if, for every $\varepsilon>0$, there exists a bounded subset $B$ of $X$ such that $diam\ f(E[x])<\varepsilon$ for each $x\in X\setminus B$.

The quotient $X^{\sharp}/\sim$ is called the Higson’s corona [14] or corona [6], [8] of the ballean $(X,\mathcal{E})$.

Now let $G$ be an infinite group, $\kappa$ be an infinite cardinal, $\kappa\leq\mid G\mid$ and let $\mathcal{L}_{\kappa}$, $\mathcal{R}_{\kappa}$ and $\mathcal{E}_{\kappa}$ be coarse structures on $G$ with the bases

[TABLE]

[TABLE]

[TABLE]

and say that $(G,\mathcal{L}_{\kappa})$, $(G,\mathcal{R}_{\kappa})$ and $(G,\mathcal{E}_{\kappa})$ are left $\kappa$-ballean, right $\kappa$-ballean and balanced $\kappa$-ballean of $G$. We observe that the mapping $G\longrightarrow G$, $x\longmapsto x^{-1}$ is an asymorphism between $(G,\mathcal{L}_{\kappa})$ and $(G,\mathcal{R}_{\kappa})$.

We note that left or right $\omega$-balleans play important role in *Geometric Group Theory, * see [5, Chapter 4] and [3, Chapter 3]. Balanced $\omega$-balleans were introduce in [10, Chapter 12]. For right and balanced $\kappa$-balleans see [12], [1] and [11] respectively.

2. Results

We say that a function $h:(G,\mathcal{E}_{\kappa})\longrightarrow\mathbb{R}$ is convergent at infinity if there exists $r\in\mathbb{R}$ such that, for every $\varepsilon>0$, $|\{y\in G:|h(g)-r|>\varepsilon\}|<\kappa$. Equivalently, for every $\varepsilon>0$, there exists $B\in[G]^{<\kappa}$ such that $|h(x)-h(y)|<\varepsilon$ for all $x,y\in G\setminus B$. Thus, if $h$ is not a convergent at infinity then there exist $\varepsilon>0$ and the subsets $\{x_{\alpha}:\alpha<\kappa\}$, $\{y_{\alpha}:\alpha<\kappa\}$ of $G$ such that $|h(x_{\alpha})-h(y_{\alpha})|>\varepsilon$ for each $\alpha<\kappa$.

Theorem 1. If $\kappa<|G|$ then every slowly oscillating function $h:(G,\mathcal{E}_{\kappa})\longrightarrow\mathbb{R}$ is convergent at infinity, so the corona $\nu_{\kappa}(G)$ of $(G,\mathcal{E}_{\kappa})$ is a singleton.

Proof. We need the following auxiliary statement

$(\ast)$ for every subset $A$ of $G$ of cardinality $\kappa$, there exists a subgroup $S$ such that $A\subseteq S$, $|S|=\kappa$ and $h|_{SqS}$ is constant for each $g\in G\setminus S$.

For $\kappa=\omega$, $(\ast)$ follows from Theorem 3.1(i) in [4]. We suppose that $\kappa>\omega$, denote by $cf\ \kappa$ the cofinality of $\kappa$ and write $A$ as the union $\cup_{\alpha<cf\ \kappa}A_{\alpha}$, $|A_{\alpha}|<\alpha$ for each $\alpha<cf\ \kappa$.

We take the subgroup $S_{0}$ generated by $A_{0}$ and suppose that, for some $\beta<cf\ \kappa$, the subgroups $\{S_{\alpha}:\alpha<\beta\}$ have been chosen. If $\beta$ is a limit ordinal then we take the subgroup $S_{\beta}$ generated by $A_{\beta}\cup\{S_{\alpha}:\alpha<\beta\}$. If $\beta$ is a non-limit ordinal then $\beta=\gamma+n+1$, where $\gamma$ is a limit ordinal and $n\in\omega$. Since $h$ is slowly oscillating, there exists $K\in[G]^{<\kappa}$ such that

[TABLE]

for each $g\in G\setminus K$. We denote by $S_{\gamma+n+1}$ the subgroup generated by $S_{\gamma+n}\cup A_{\gamma+n+1}\bigcup K$.

After $cf\ \kappa$ steps we put $S=\cup_{\alpha<cf\ \kappa}S_{\alpha}$. By the construction, $diam\ h(SgS)<\frac{1}{n+1}$ for all $n\in\omega$ and $g\in G\setminus S$, so $(\ast)$ is proven.

We suppose that $h$ is not convergent at infinity and choose $\varepsilon>0$ and the subsets $\{x_{\alpha}:\alpha<\kappa$, $\{y_{\alpha}:\alpha<\kappa\}$ such that $|h(x_{\alpha})-h(y_{\alpha})|>\varepsilon$ for each $\alpha<\kappa$. Then we put $A=\{x_{\alpha},y_{\alpha}:\alpha<\kappa\}$ and choose the subgroup $S$ satisfying $(\ast)$.

We fix an arbitrary $z\in G\setminus S$. Since $h$ is slowly oscillating, there exists $B\in[G]^{<\kappa}$ such that

[TABLE]

for all $x,y\in G\setminus B$. Then we take $x_{\alpha},y_{\alpha}\in G\setminus B$ and get

[TABLE]

By the choice of $S,$ we have $x_{\alpha}\in S$, $y_{\alpha}\in S$ so $h(zx_{\alpha})=h(y_{\alpha}z)$ and $|h(x_{\alpha})-h(y_{\alpha})<\varepsilon$, a contradiction. $\Box$

For a cardinal $\kappa$, we denote $U_{\kappa}=\{p\in\beta\kappa:|P|=\kappa$ for each $P\in p\}$.

Theorem 2. If $\kappa=|G|$ and $\kappa$ is regular then $\nu_{\kappa}(G)$ contains a copy of $U_{\kappa}$.

Proof. If $\kappa=\omega$ then, by [13, Theorem 2.1.1], the ballean $(G,\mathcal{E}_{\omega})$ is metrizable and, by [9, Theorem 1], $\nu_{\omega}(G)$ contains a copy of $\beta\omega\setminus\omega$.

For $\kappa>\omega$, we write $G$ as the union of the increasing chain $\{G_{\alpha}:\alpha<\kappa\}$ of subgroups such that $|G_{\alpha}|<\kappa$ for each $\alpha<\kappa$. Since $\kappa$ is regular, every function $h:(G,\mathcal{E}_{\kappa})\longrightarrow\{0,1\}$ such that, for each $\alpha<\kappa$, $h|_{G_{\alpha+1}\setminus G_{\alpha}}=const$ is slowly oscillating. For every $\alpha<\kappa$, we pick $x_{\alpha}\in G_{\alpha+1}\setminus G_{\alpha}$ and put $X=\{x_{\alpha}:\alpha<\kappa\}.$ We denote

[TABLE]

If $r,q\in U$ and $r\neq q$ then there is a slowly oscillating function $h:(G,\mathcal{E}_{\kappa})\longrightarrow\{0,1\}$ such that $h^{\beta}(r)\neq h^{\beta}(q).$ Clearly, $U$ is homeomorphic to $U_{\kappa}$. $\Box$

For a ballean $(X,\mathcal{E})$, we consider an equivalence $\sim_{\{0,1\}}$ on $X^{\sharp}$ defined by the rule: $p\sim_{\{0,1\}}q$ if and only if $f^{\beta}(p)=f^{\beta}(q)$ for each slowly oscillating function $f:X\longrightarrow\{0,1\}$. The quotient $X^{\sharp}/\sim_{\{0,1\}}$ is called [6], [7] the binary corona of $(X,\mathcal{E})$. For a group $G$, the binary coronas of $(G,\mathcal{R}_{\omega})$ and $(G,\mathcal{E}_{\omega})$ are known as the spaces of ends and bi-ends of $G$ respectively [2], [15].

A ballean $(X,\mathcal{E})$ is called cellular if $\mathcal{E}$ has a base consisting of equivalence relations. If $(X,\mathcal{E})$ is cellular then, by [6, Lemma 4.3], $\sim\ =\ \sim_{\{0,1\}}$ so the binary corona and corona of $(X,\mathcal{E})$ coincide.

If $\kappa>\omega$ then the coarse structure $\mathcal{E}_{\kappa}$ of $G$ has the base

[TABLE]

It follows that the ballean $(G,\mathcal{E}_{\kappa})$ is cellular.

Theorem 3 [11]. *Let $G$ be a group of singular cardinality $\kappa$. For any finite partition $A_{1},\ldots,A_{n}$ of $G$, there exist $i\in\{1,\ldots,n\}$ and $S\in[G]^{<\kappa}$ such that * $G=SA_{i}S$.

Theorem 4. If $\kappa=|G|$ and $\kappa$ is singular then the corona $\nu_{\kappa}(G)$ is a singleton.

Proof. Since $(G,\mathcal{E}_{\kappa})$ is cellular, it suffices to show that $h^{\beta}(p)=h^{\beta}(q)$ for any $p,q\in G^{\sharp}$ and every slowly oscillating function $h:(G,\mathcal{E}_{\kappa})\longrightarrow\{0,1\}$. We assume the contrary and choose corresponding $p,q\in G^{\sharp}$ and $h$. Let $h^{\beta}(p)=1$, $h^{\beta}(q)=0$. Then $G$ can be partitioned $G=P\cup Q$ so that $h|_{P}=1$, $h|_{Q}=0$ and $|P|=|Q|=\kappa$. By Theorem 3, there exists $S\in[G]^{<\kappa}$ such that either $G=SPS$ or $G=SQS$ , which is impossible because $h$ is slowly oscillating. $\Box$

3. Comments

1. We suppose that $\kappa<|G|$, $A\subset G$, $|A|\geq\kappa$, $|G\setminus A|\geq\kappa$. We define a function $h:(G,\mathcal{E}_{\kappa})\longrightarrow\{0,1\}$ by $h|_{A}=1$, $h|_{G\setminus A}=0$. By Theorem 1, $h$ is not slowly oscillating. Hence, there exists $S\in[G]^{<\kappa}$ such that $SAS\cap(G\setminus A)\geq\kappa$. Taras Banakh noticed that, for $\kappa=\omega$, this statement follows also from Theorem 1.20 in [2].

2. With the same proof, Theorem 2 holds for the corona $\rho_{\kappa}(G)$ of the ballean $(G,\mathcal{R}_{\kappa})$ in place $\nu_{\kappa}(G)$. Let $X$ be a set of cardinality $|X|\geq\kappa$, $F(X)$ denotes the free group in the alphabet $X$. Example 3.3 from [4] and Theorem 4 from [1] show that $\rho_{\kappa}(F(X))$ is not a singleton. Hence, Theorem 1 and Theorem 4 do not hold for $G=F(X)$. On the other hand, by [4, Theorem 3.1], the corona $\rho_{\omega}(G)$ is a singleton provided that $|G|>\omega$ and each countable subset of $G$ is contained in some countable normal subgroup.

Question 1. How can one detect whether the corona $\rho_{\omega}(G)$ is a singleton?

Question 2. Are $\rho_{\omega}(G)$ and $\nu_{\omega}(G)$ homeomorphic for every countable group $G$?

By Theorem 2, for a countable group $G$, $|\rho_{\omega}(G)|=\mid\nu_{\omega}(G)\mid$ but the binary coronas of $(G,\mathcal{R}_{\omega})$ and $(G,\mathcal{E}_{\omega})$ could be of different cardinalities, see [2, Theorem 1.20].

3. In [4], [8] a function convergent at infinity is called constant at infinity. This change in terminology and reference [15] are suggested by Yves Cornulier. In the case of an uncountable abelian group $G$ of cardinality $\kappa$, the infinity of the space of $\kappa$-ends of $G$ is noticed in [15, Satz III].