arXiv subjects
Publications and source records attributed to Igor Protasov.
Given a $T$-sequence on a countable abelian group $G$, we prove that there exists $2^{2^{|G|}}$ Hausdorff group topologies in which this sequence converges to $0$. This answers a question posed in Intern. J. Math. Math. Sci. {\bf 24}(3) (2000), 145-148.
A coarse space $X$, endowed with a linear order compatible with the coarse structure of $X$, is called linearly ordered. We prove that every linearly ordered coarse space $X$ is locally convex and the asymptotic dimension of $X$ is either $0$ or $1$. If $X$ is metrizable then the family of all right bounded subsets of $X$ has a selector.
We consider a connected graph $\Gamma$ as a coarse space and prove that $\Gamma$ admits a 2-selector if and only if $\Gamma$ is either bounded or coarsely equivalent to $\mathbb{N}$ or $\mathbb{Z}$. We apply this result to geodesic metric spaces admitting linear orders compatible with coarse structures.
For a group $G$, $\mathcal{F}_G$ denotes the set of all non-empty finite subsets of $G$. We extend the finitary coarse structure of $G$ from $G\times G$ to $\mathcal{F}_G\times \mathcal{F}_G$ and say that a macro-uniform mapping $f: \mathcal{F}_G \rightarrow \mathcal{F}_G$ (resp. $f: [G]^2 \rightarrow G$) is a finitary selector (resp. 2-selector) of $G$ if $f(A)\in A$ for each $A\in \mathcal{F}_G$ (resp. $ A \in [G]^2 $). We prove that a group $G$ admits a finitary selector iff $G$ admits a 2-selector and iff $G$ is a finite extension of an infinite cyclic subgroup or $G$ is countable and locally finite. We use this result to characterize groups admitting linear orders compatible with finitary coarse structures.
Given a coarse space $(X, \mathcal{E})$, we consider linear orders on $X$ compatible with the coarse structure $\mathcal E$ and explore interplays between these orders and macro-uniform selectors of $(X, \mathcal{E})$.
Given a coarse space $(X, \mathcal{E})$ with the bornology $\mathcal B$ of bounded subsets, we extend the coarse structure $\mathcal E$ from $X\times X$ to the natural coarse structure on $(\mathcal B \backslash \lbrace \emptyset\rbrace)\times (\mathcal B \backslash \lbrace \emptyset\rbrace)$ and say that a macro-uniform mapping $f: (\mathcal B \backslash \lbrace \emptyset\rbrace)\rightarrow X$ (resp. $f: [ X]^2 \rightarrow X$) is a selector (resp. 2-selector) of $(X, \mathcal{E})$ if $f(A)\in A$ for each $A\in \mathcal B\setminus \lbrace\emptyset\rbrace$ (resp. $A \in [X]^2 )$. We prove that a discrete coarse space $(X, \mathcal{E})$ admits a selector if and only if $(X, \mathcal{E})$ admits a 2-selector if and only if there exists a linear order $\leq$ on $X$ such that the family of intervals $\lbrace [a, b]: a,b\in X, \ a\leq b \}$ is a base for the bornology $\mathcal B$.
For a group $G$, we denote by $\stackrel{\leftrightarrow}{G}$ the coarse space on $G$ endowed with the coarse structure with the base $\{\{ (x,y)\in G\times G: y\in x^F \} : F \in [G]^{<\omega} \}$, $x^F = \{z^{-1} xz : z\in F \}$. Our goal is to explore interplays between algebraic properties of $G$ and asymptotic properties of $\stackrel{\leftrightarrow}{G}$. In particular, we show that $asdim \ \stackrel{\leftrightarrow}{G} = 0$ if and only if $G / Z_G$ is locally finite, $Z_G$ is the center of $G$. For an infinite group $G$, the coarse space of subgroups of $G$ is discrete if and only if $G$ is a Dedekind group.
Given a discrete group $G$, we identify the Stone-$\check C$ech compactification $\beta G$ with the set of all ultrafilters on $G$ and put $G^\ast =\beta G\setminus G$. The action $G$ on $G$ by the conjugations $(g,x)\mapsto g^{-1}xg$ induces the action of $G$ on $G^\ast$ by $(g, p)\mapsto p^g $, $p^g = \{ g^{-1} Pg: P\in p\}$. We study interplays between the algebraic properties of $G$ and the dynamical properties of $(G, G^\ast)$. In particular, we show that $p^G$ is finite for each $p\in G^\ast$ if and only if the commutant of $G$ is finite.
For a bornology $\mathcal B$ on a cardinal $\kappa$, we prove that the $\mathcal B$-macrocube is normal if and only if $\mathcal B$ has a linearly ordered base. As a corollary, we get that the hyperballean of bounded subsets of an ultradiscrete ballean is not normal. These answer Question 1 from \cite{b2} and Question 14.4 from \cite{b1}.
In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on $\omega$, about the interplay between properties of a coarse space and its Higson corona, about some special ultrafilters ($T$-points and cellular $T$-points) related to finitary coarse structures on $\omega$, about partitions of coarse spaces into thin pieces, and also about coarse groups having some extremal properties.
For a coarse space $(X, \mathcal{E})$, $X^\sharp$ denotes the set of all unbounded ultrafilters on $X$ endowed with the parallelity relation: $p||q$ if there exists $E \in \mathcal{E} $ such that $ E[P]\in q $ for each $P\in p$. If $(X, \mathcal{E})$ is finitary then there exists a group $G $ of permutations of $X$ such that the coarse structure $\mathcal{E}$ has the base $\{\{ (x,gx): x\in X$, $g\in F\}: F\in [G]^{<\omega}, \ id \in F \}.$ We survey and analyze interplays between $(X, \mathcal{E})$, $X^\sharp$ and the dynamical system $(G, X^\sharp)$.
For any compact Hausdorff space $K$ we construct a canonical finitary coarse structure $\mathcal E_{X,K}$ on the set $X$ of isolated points of $K$. This construction has two properties: $\bullet$ If a finitary coarse space $(X,\mathcal E)$ is metrizable, then its coarse structure $\mathcal E$ coincides with the coarse structure $\mathcal E_{X,\bar X}$ generated by the Higson compactification $\bar X$ of $X$; $\bullet$ A compact Hausdorff space $K$ coincides with the Higson compactification of the coarse space $(X,\mathcal E_{X,K})$ if the set $X$ is dense in $K$ and the space $K$ is Frechet-Urysohn. This implies that a compact Hausdorff space $K$ is homeomorphic to the Higson corona of some finitary coarse space if one of the following conditions holds: (i) $K$ is perfectly normal; (ii) $K$ has weight $w(K)\le\omega_1$ and character $\chi(K)<\mathfrak p$. Under CH every (zero-dimensional) compact Hausdorff space of weight $\le\omega_1$ is homeomorphic to the Higson (resp. binary) corona of some cellular finitary coarse space.
Let $G$ be a non-discrete countable metrizable abelian topological group endowed with the coarse structure $ \mathcal{C} $ generated by compact subsets of $G$. We prove that $asdim (G, \mathcal{C} ) = \infty$. For an infinite cyclic subgroup $G$ of the circle, this answers a question of Dikranjan and Zava [3].
A set $X$ endowed with a coarse structure is called ballean or coarse space. For a ballean $(X, \mathcal{E})$, we say that two subsets $A$, $B$ of $X$ are close (linked) if there exists an entourage $E\in \mathcal{E}$ such that $A\subseteq E [B]$, $B\subseteq E[A]$ (either $A, B$ are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness.
A bornology $\mathcal{B}$ on a set $X$ is called minmax if the smallest and the largest coarse structures on $X$ compatible with $\mathcal{B}$ coincide. We prove that $\mathcal{B}$ is minmax if and only if the family $\mathcal B^\sharp=\{p\in\beta X:\{X\setminus B:B\in\mathcal B\}\subset p\}$ consists of ultrafilters which are pairwise non-isomorphic via $\mathcal B$-preserving bijections of $X$. Also we construct a minmax bornology $\mathcal B$ on $\omega$ such that the set $\mathcal B^\sharp$ is infinite. We deduce this result from the existence of a closed infinite subset in $\beta\omega$ that consists of pairwise non-isomorphic ultrafilters.
Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F: X \longrightarrow B_{X}$ be a set-valued mapping such that $x \in F(x)$, $sup_{x\in X} | F(x)|< \kappa$, $sup_{x\in X} | F^{-1}(x)|< \kappa$ for all $x\in X$ and some infinite cardinal $\kappa$. Then there exists a family $\mathcal{F}$ of bijective selectors of $F$ such that $|\mathcal{F}|<\kappa$ and $F(x) = \{ f(x): f\in\mathcal{F}\}$ for each $x\in X$. We apply this result to $G$-space representations of balleans.
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 $\kappa$.
We endow a topological group $(G, \tau)$ with a coarse structure defined by the smallest group ideal $S_{\tau} $ on $G$ containing all converging sequences with their limits and denote the obtained coarse group by $(G, S_{\tau})$. If $G$ is discrete then $(G, S_{\tau})$ is a finitary coarse group studding in Geometric Group Theory. The main result: if a topological abelian group $(G, \tau)$ contains a non-trivial converging sequence then $asdim \ (G, S_{\tau})= \infty $.