Coarse structures on groups defined by $T$-sequences

TL;DR

This paper investigates the coarse geometric properties of groups equipped with topologies induced by $T$-sequences, showing infinite asymptotic dimension and conditions for having one end, using Hamming space embeddings.

Contribution

It introduces a new coarse structure on groups based on $T$-sequences and analyzes their asymptotic dimension and end structure, revealing infinite asymptotic dimension and conditions for one end.

Findings

Asymptotic dimension of the coarse group is infinite for all non-trivial $T$-sequences.

The coarse group has exactly one end if the $T$-sequence generates the group.

Embeddings of Hamming space are used to establish these properties.

Abstract

A sequence $(a_{n}) $ in an Abelian group is called a $T$-sequence if there exists a Hausdorff group topology on $G$ in which $(a_{n}) $ converges to $0$. For a $T$-sequence $(a_{n}) $, $\tau_{(a_{n}) } $ denotes the strongest group topology on $G$ in which $(a_{n}) $ converges to $0$. The ideal $\mathcal{I}_{(a_{n})} $ of all precompact subsets of $(G, \tau_{(a_{n}) } )$ defines a coarse structure on $G$ with base of entourages $\{(x, y): x-y \in P \}$, $P\in\mathcal{I}_{(a_{n})}. $ We prove that $asdim \ \ (G, \mathcal{I}_{(a_{n}) }) =\infty $ for every non-trivial $T$-sequence $(a_{n})$ on $G$, and the coarse group $(G, \mathcal{I}_{(a_{n}) })$ has 1 end provided that $(a_{n}) $ generates $G$. The keypart play asymorphic copies of the Hamming space in $(G, \mathcal{I}_{(a_{n})})$.

Equations

$$E\circ E^{\prime }=\{(x,y):(\exists z\in X)((x,z)\in E,(z,y)\in E^{\prime })\}\text{mbox}and{E}^{-1}=\{(y,x):(x,y)\in E\};$$

$$E[x]=\{y\in X:(x,y)\in E\},\text{mbox}andE[A]={\cup }_{a\in A}E[a],$$

$$FS(b_n{)}_{n\in \omega }=\left\{\sum _{i\in F}{b}_i:F\in [\omega {]}^{<\omega }\right\}$$

$$\sum _{i\in H}{b}_i=\sum _{i\in F}{b}_i⟹F=H.$$

$$b_{n+1}\notin \left\{\sum _{i\in F}{b}_i-\sum _{i\in H}{b}_i:H,F\subseteq \{0,\dots ,n\}\right\}.$$

$$\text{mbox}ifb=\sum _{i\in F}{b}_i,a\in A_n\text{mbox}andb+a=\sum _{i\in H}{b}_i\text{mbox}then|F△H|\le n.$$

$$\text{mbox}if{i}_0<i_1<\dots <i_n<\omega \text{mbox}and{t}_{i_0},\dots ,t_{i_n}\in \{1,-1\}\text{mbox}then{t}_{i_0}{b}_{i_0}+\dots +t_{i_n}{b}_{i_n}\notin A_n.$$

$$\sum _{s=0}^k{t}_{i_s}{b}_{i_s}+t_{n+1}{b}_{n+1}\notin A_{k+1}\text{mbox}for{i}_0<\dots <i_k\le n\text{mbox}and{t}_i\in \{1,-1\}\text{mbox}fori\in \{i_0,\dots ,i_k,n+1\}.$$

$$f{|}_{G\setminus A_m}=const.$$

$$y=b_1+\dots +b_k\text{mbox}andz=c_1+\dots +c_k,$$

$$a_{m_1}+y\notin A_{m+1}\text{mbox}and{a}_{m_1}+z\notin A_{m+1},$$

$$f(a_{m_1}+b_2+\dots +b_k)=f(y)\text{mbox}andf(a_{m_1}+c_2+\dots +c_k)=f(z).$$

Full text

Coarse structures on groups defined by $T$-sequences

D. Dikranjan, I. Protasov

**Abstract. ** A sequence $(a_{n})$ in an Abelian group is called a $T$-sequence if there exists a Hausdorff group topology on $G$ in which $(a_{n})$ converges to [math]. For a $T$-sequence $(a_{n})$, $\tau_{(a_{n})}$ denotes the strongest group topology on $G$ in which $(a_{n})$ converges to [math]. The ideal $\mathcal{I}_{(a_{n})}$ of all precompact subsets of $(G,\tau_{(a_{n})})$ defines a coarse structure on $G$ with base of entourages $\{(x,y):x-y\in P\}$, $P\in\mathcal{I}_{(a_{n})}.$ We prove that $asdim\ \ (G,\mathcal{I}_{(a_{n})})=\infty$ for every non-trivial $T$-sequence $(a_{n})$ on $G$, and the coarse group $(G,\mathcal{I}_{(a_{n})})$ has 1 end provided that $(a_{n})$ generates $G$. The keypart play asymorphic copies of the Hamming space in $(G,\mathcal{I}_{(a_{n})})$.

**MSC: ** 22B05, 54E99.

Keywords: Coarse structure, group ideal, asymptotic dimension, end, Hamming space.

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 $\bigtriangleup_{X}$, $\bigtriangleup_{X}=\{(x,x):x\in X\}$;

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if $E$, $E^{\prime}\in\mathcal{E}$ then $E\circ E^{\prime}\in\mathcal{E}$ and $E^{-1}\in\mathcal{E}$, where

[TABLE]

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

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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 write

[TABLE]

and we say that $E[x]$ and $E[A]$ are balls of radius $E$ around $x$ and $A$.

The pair $(X,\mathcal{E})$ is called a coarse space [9]. Alternatively, a coarse space can be defined in terms of balls [5], [7]. In this case, it is called a ballean. As pointed out in [1], balleans and coarse structures as two faces of the same coin.

A subset $A$ of a coarse space $(X,\mathcal{E})$ is called

• •

bounded if $A\subseteq E[x]$ for some $x\in X$ and $E\in\mathcal{E}$;

• •

large if $E[A]=X$ for some $E\in\mathcal{E}$.

Given two coarse spaces $(X,\mathcal{E})$ , $(X^{\prime},\mathcal{E}^{\prime})$ a mapping $f:X\longrightarrow X^{\prime}$ is called macro-uniform if, for any $E\in\mathcal{E}$, there exists $E^{\prime}\in\mathcal{E}^{\prime}$ such that, for each $x\in X$, $f(E[x])\subseteq E^{\prime}[f(x)]$. If $f$ is a bijection such that $f$ and $f^{\prime}$ are macro-uniform then $f$ is called an asymorphism. The spaces $(X,\mathcal{E})$ , $(X^{\prime},\mathcal{E}^{\prime})$ are coarsely equivalent if $X,X^{\prime}$ have large asymorphic subspaces.

Now let $G$ be a group. A family $\mathcal{I}$ of subsets of $G$ is called a group ideal [6] if $\mathcal{I}$ contains all finite subset and, for any $A,B\in\mathcal{I}$ and $C\subseteq A$, $\ AB^{-1}\in\mathcal{I}$ and $C\in\mathcal{I}$. Every group ideal $\mathcal{I}$ defines a coarse structure on $G$ with the base of entourages $\{(x,y):x\in Fy\}$, $\ F\in\mathcal{I}$. The group $G$ endowed with this coarse structure is called a coarse group, it is denoted by $(G,\mathcal{I})$.

We recall that a sequence $(a_{n})_{n\in\omega}$ in an Abelian group $G$ is a $T$-sequence if there exists a Hausdorff group topology on $G$ in which $(a_{n})_{n\in\omega}$ converges to [math]. For a $T$-sequences $(a_{n})_{n\in\omega}$ on $G$, we denote by $\tau_{(a_{n})}$ the strongest group topology on $G$ in which $(a_{n})_{n\in\omega}$ converges to [math]. We put $A=\{0,a_{n},-a_{n}:n\in\omega\}$ and denote by $A_{n}$ the sum on $n$ copies of $A$.

By [8, Theorem 2.3.11], $(G,\tau_{(a_{n})})$ is complete. Hence, a subset $S$ of $G$ is totally bounded in $(G,\tau_{(a_{n})})$ if and only if $S$ is precompact.

The ideal $\mathcal{I}_{(a_{n})}$ of all precompact subsets of $(G,\tau_{(a_{n})})$ defines a coarse structure on $G$ with the base of entourages $\{(x,y):x-y\in P\}$, $P\in\mathcal{I}_{(a_{n})}$. We denote the obtained coarse group by $(G,\mathcal{I}_{(a_{n})})$.

Theorem 1. For any $T$-sequences $(a_{n})_{n\in\omega}$ on $G$, the family $\{F+A_{n}:F\in[G]^{<\omega}\}$, $n\in\omega\}$ is a base for the ideal $\mathcal{I}_{(a_{n})}$. If $G$ is generated by the set $\{a_{n}:n\in\omega\}$ then $\{A_{n}:n\in\omega\}$ is a base for $\mathcal{I}_{(a_{n})}$

Proof. Apply Lemma 2.3.2 from [8]. $\Box$

Given an arbitrary subset $S$ of $G$, the Cayley graph Cay $(G,S)$ is a graph with the set of vertices $G$ and the set of edges $\{(x,y):x-y\in S\cup(-S)\}$.

Theorem 2. If a $T$-sequences $(a_{n})_{n\in\omega}$ generates $G$ then the coarse group $(G,\mathcal{I}_{(a_{n})})$ is asymorphic to Cay $(G,\{a_{n}:n\in\omega\})$.

Proof. Apply and Theorem 1 and Theorem 5.1.1 from [7]. $\Box$

**Example. ** Let $G$ be the direct sum of groups $\{<a_{n}>:n\in\omega\}$ of order 2. Clearly, $(<a_{n}>)_{n\in\omega}$ is a $T$-sequence on $G$. By Theorem 1, the canonical bijection between $(G,\mathcal{I}_{(a_{n})})$ and the Hamming space $\mathbb{H}$ of all finite subsets of $\omega$ is an asymorphism.

A $T$-sequences $(a_{n})_{n\in\omega}$ is called trivial if $a_{n}=0$ for all but finitely many $n\in\omega$.

Theorem 3. For any non-trivial $T$-sequences $(a_{n})_{n\in\omega}$ on $G$, the coarse group $(G,\mathcal{I}_{(a_{n})})$ contains a subspace asymorphic to the Hamming space $\mathbb{H}$ so $asdim(G,\mathcal{I}_{(a_{n})})=\infty$.

Proof. Without loss of generality, we suppose that $\{a_{n}:n\in\omega\}$ generates $G$ and $a_{n}\neq 0$ for each $n\in\omega$.

Given an arbitrary $T$-sequence $(b_{n})_{n\in\omega}$ in $G$, we denote

[TABLE]

and say that $(b_{n})_{n\in\omega}$ is FS-strict if, for any $H,F\in[\omega]^{<\omega}$,

[TABLE]

We note that $(b_{n})_{n\in\omega}$ is FS-strict if, for each $n\in\omega$,

[TABLE]

We assume that $(b_{n})_{n\in\omega}$ is FS-strict and

[TABLE]

Then the canonical bijection $f:\mathbb{H}\longrightarrow FS(b_{n}),\ \ f(H)=\sum_{i\in H}b_{i}$ is an asymorphism.

To construct the desired sequence $(b_{n})_{n\in\omega}$ we rewrite $(2)$ in the following equivalent form

[TABLE]

We put $b_{0}=a_{0}$ and assume that $b_{0},\ldots,b_{n}$ have been chosen. We show how to choose $b_{n+1}$ to satisfy $(1)$ and

[TABLE]

We assume that there exists a subsequence $(c_{m})_{m\in\omega}$ of $(a_{n})_{n\in\omega}$ such that $t_{i_{0}}b_{i_{0}}+\ldots+t_{i_{k}}b_{i_{k}}+tc_{m}\in A_{k+1}$ for $t\in\{1,-1\}$ and for each $m\in\omega$. Every infinite subset of $A_{k+1}$ has a limit point in $A_{k}$. Hence, $t_{i_{0}}b_{i_{0}}+\ldots+t_{i_{k}}b_{i_{k}}\in A_{k+1}$ contradicting the choice of $b_{0},\ldots,b_{n}$. Thus, $b_{n+1}$ can be taken from $\{a_{m_{0}},a_{m_{0}+1},\ldots\}$ for some $m_{0}\in\omega$. $\Box$

Let $(X,\mathcal{E})$ be a coarse space. A function $f:X\longrightarrow\{0,1\}$ is called slowly oscillating if, for every $E\in\mathcal{E}$ , there exists a bounded subset $B$ of $X$ such that $f|_{E[x]}=const$ for each $x\in X\setminus B$. We endow $X$ with the discrete topology, identify the Stone-$\check{C}$ech compactification $\beta X$ of $X$ with the set of ultrafilters on $X$ and denote $X^{\sharp}=\{p\in\beta X:$ each $P\in p$ is unbounded $\}$. We define an equivalence $\sim$ on $X^{\sharp}$ by the rule: $p\sim q$ if and only if $f^{\beta}(p)=f^{\beta}(q)$ for every slowly oscillating function $f:X\longrightarrow\{0,1\}$. The quotient $X^{\sharp}/\sim$ is called a space of *ends * or binary corona of $(X,\mathcal{E})$, see [3], [4].

Theorem 4. If a non-trivial $T$-sequences $(a_{n})_{n\in\omega}$ generates $G$ then the space of ends $(G,(a_{n})_{n\in\omega})$ is a singleton.

Proof. First we show that for every slowly oscillating function $f:G\longrightarrow\{0,1\}$ there exists an $m$ such that

[TABLE]

Indeed, by the definition of slow oscillation and Theorem 1, there exists $m\in\omega$ such that $f|_{x+A}=const$ for each $x\in G\setminus A_{m}$. We show nw that (1) holds true for this $m$.

We take arbitrary $y,z\in G\setminus A_{m}$. Since $A$ generates $G$ and contains 0, there exists an index $k$ such that $y,z\in A_{k}$, i.e.,

[TABLE]

for appropriate $\ \ b_{1},\ldots,b_{k}$, $\ \ c_{1},\ldots,c_{k}\in A$. By a property of $T$-sequences established at the end of the proof of Theorem 3, there exists a member $a_{m_{1}}$ of $(a_{n})$ such that

[TABLE]

since $(a_{n})$ is a $T$-sequence. Then $a_{m_{1}}+b_{2}+\ldots+b_{k}\notin A_{m}$, $\ \ a_{m_{1}}+c_{2}+\ldots+c_{k}\notin A_{m}$. Therefore,

[TABLE]

Repeating this trick $k$ times, we can replace $b_{2},\ldots,b_{k}$ and $c_{2},\ldots,c_{k}$, by appropriate members $a_{m_{2}},\ldots,a_{m_{k}}$ of $(a_{n})$, as before. Hence, we can replace $b_{2}+\ldots+b_{k}$ and $c_{2}+\ldots+c_{k}$, by $a_{m_{2}}+\ldots+a_{m_{k}}$ in (6). This obviously gives $f(y)=f(z)$ and proves (1).

Finally, the prove the assertion of the theorem, pick $p,q\in X^{\sharp}$. In order to check that $p\sim q$ fix an arbitrary slowly oscillating function $f:X\longrightarrow\{0,1\}$. We have prove that $f^{\beta}(p)=f^{\beta}(q)$. Pick an $m$ with (1). Since $G\setminus A_{m}\in p\cap q$, for every $P\in p$ and for every $Q\in q$ we have $P_{1}:=P\setminus A_{m}\in p$ and $Q_{1}:=Q\setminus A_{m}\in q$ and $f|_{P_{1}}=f|_{Q_{1}}$ is constant in view of (5). This proves that $f^{\beta}(p)=f^{\beta}(q)$. $\Box$

Remark 1. An $FS$-strict sequence generating $G$ is called a base for $G$. By [2, Theorem 3.3], a countable group $G$ has a base if and only if $G$ has no elements of odd order. If $(a_{n})_{n\in\omega}$ is a base of $G$ then we have the natural bijection $f:\mathbb{H}\longrightarrow G$, $\ \ f(F)=\sum_{i\in F}a_{i}$. If in addition $(a_{n})_{n\in\omega}$ is a $T$-sequence, we cannot state that $f$ is an asymorphism between $\mathbb{H}$ and $(G,\mathcal{I}_{(a_{n})})$. For example, we may take the base $((-1)^{n}2^{n})_{n\in\omega}$ of $\mathbb{Z}$.

We denote by $\mathbb{H}(n)$ the direct sum of $\omega$ copies of the cyclic group of order $n$ and endow $\mathbb{H}(n)$ with the Hamming metric $h$, where for $a,b\in\mathbb{H}(n)$ $h(a,b)$ is the number of distinct coordinates of $a$ and $b$. If $n,m$ have the same set of prime divisors that $\mathbb{H}(n)$ and $\mathbb{H}(m)$ are asymorphic.

Question 1. When precisely $\mathbb{H}(n)$ and $\mathbb{H}(m)$ are asymorphic? Coarsely equivalent?

Remark 2. Given a group ideal $\mathcal{I}$ on $G$, the coarse group $(G,\mathcal{I})$ has bounded geometry if exists $F\in\mathcal{I}$ such that, for each $H\in\mathcal{I}$, we have $H\subseteq FK$ for some $K\in[G]^{<\omega}$. For each non-trivial $T$-sequence $(a_{n})_{n\in\omega}$ on $G$, the coarse group $(G,\mathcal{I}_{(a_{n}))}$ is not of bounded geometry.

Remark 3. We say that a subset $S$ of a topological group $(G,\tau)$ is $R$-bounded (see [10]) if, for any neighbourhood $U$ of the identity of $G$, there exist $K\in[G]^{<\omega}$ and $n\in\omega$ such that $S\subseteq(UK)^{n}$. The family $\mathfrak{R}_{\tau}$ of all $R$-bounded subsets of $(G,\tau)$ is a group ideal. For any $T$-sequence on $G$, we have $\mathfrak{R}_{\tau(a_{n})}=\mathcal{I}_{a_{n}}$ .

Question 2. Let $G$ be a non-discrete countable metrizable Abelian group and let $\mathcal{I}_{seq}$ denotes the smallest group ideal on $G$ containing all converging sequences. Is $asdim\ (G,\mathcal{I}_{seq})=\infty$ ?