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Ksenia Protasova

Publications and source records attributed to Ksenia Protasova.

16 recordsLinked to original sources
arXiv2020

Coarse structures on groups defined by conjugations

For a group $G$, we denote by $\stackrel{\leftrightarrow}{G}$ the coarse space on $G$ endowed with the coarse structure with the base $\{\{ (x,y)\in G\times G: y\in x^F \} : F \in [G]^{<ω} \}$, $x^F = \{z^{-1} xz : z\in F \}$. 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.

math.GM↗
arXiv2020

The dynamical approach to the conjugacy in groups

Given a discrete group $G$, we identify the Stone-$\check C$ech compactification $βG$ with the set of all ultrafilters on $G$ and put $G^\ast =βG\setminus G$. 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.

math.GR↗
arXiv2020

The normality of macrocubes and hyperballeans

For a bornology $\mathcal B$ on a cardinal $κ$, 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}.

math.GN↗
arXiv2020

Closeness and linkness in balleans

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.

math.GN↗
arXiv2019

A note on free vector balleans

A vector balleans is a vector space over $\mathbb{R}$ endowed with a coarse structure in such a way that the vector operations are coarse mappings. We prove that, for every ballean $(X, \mathcal{E})$, there exists the unique free vector ballean $\mathbb{V}(X, \mathcal{E})$ and describe the coarse structure of $\mathbb{V}(X, \mathcal{E})$. It is shown that normality of $\mathbb{V}(X, \mathcal{E})$ is equivalent to metrizability of $(X, \mathcal{E})$.

math.GN↗
arXiv2018

Lattices of coarse structures

We consider the lattice of coarse structures on a set $X$ and study metrizable, locally finite and cellular coarse structures on $X$ from the lattice point of view.

math.GN↗
arXiv2018

Free coarse groups

A coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. Let $(X, \mathcal{E})$ be a coarse space and let $\mathfrak{M}$ be a variety of groups different from the variety of singletons. We prove that there is a coarse group $F_{\mathfrak{M}} (X, \mathcal{E})\in \mathfrak{M}$ such that $(X, \mathcal{E}) $ is a subspace of $F_{\mathfrak{M}} (X, \mathcal{E})$, $X$ generates $F_{\mathfrak{M}} (X, \mathcal{E})$ and every coarse mapping $(X, \mathcal{E}) \longrightarrow (G, \mathcal{E}^{\prime}) $ where $G\in\mathfrak{M}$, $(G, \mathcal{E}^{\prime}) $ is a coarse group, can be extended to coarse homomorphism $F_{\mathfrak{M}} (X, \mathcal{E})\longrightarrow (G, \mathcal{E}^{\prime}) $. If $\mathfrak{M}$ is the variety of all groups, the groups $F_{\mathfrak{M}} (X, \mathcal{E})$ are asymptotic counterparts of Markov free topological groups over Tikhonov spaces.

math.GN↗
arXiv2018

Recent progress in subset combinatorics of groups

We systematize and analyze some results obtained in Subset Combinatorics of $G$ groups after publications the previous surveys [1-4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in $G$-spaces, new ideals in the Boolean algebra $\mathcal{P}_{G}$ of all subsets of a group $G$ and in the Stone-$\check{C}$ech compactification $βG$ of $G$ , the combinatorial derivation.

math.CO↗
arXiv2017

Metrically Ramsey ultrafilters

Given a metric space $(X,d)$, we say that a mapping $χ: [X]^{2}\longrightarrow\{0.1\}$ is an isometric coloring if $d(x,y)=d(z,t)$ implies $χ(\{x,y\})=χ(\{z,t\})$. A free ultrafilter $\mathcal{U}$ on an infinite metric space $(X,d)$ is called metrically Ramsey if, for every isometric coloring $χ$ of $[X]^{2}$, there is a member $U\in\mathcal{U}$ such that the set $[U]^{2}$ is $χ$-monochrome. We prove that each infinite ultrametric space $(X,d)$ has a countable subset $Y$ such that each free ultrafilter $\mathcal{U}$ on $X$ satisfying $Y\in\mathcal{U}$ is metrically Ramsey. On the other hand, it is an open question whether every metrically Ramsey ultrafilter on the natural numbers $\mathbb{N}$ with the metric $|x-y|$ is a Ramsey ultrafilter. We prove that every metrically Ramsey ultrafilter $\mathcal{U}$ on $\mathbb{N}$ has a member with no arithmetic progression of length 2, and if $\mathcal{U}$ has a thin member then there is a mapping $f:\mathbb{N}\longrightarrowω$ such that $f(\mathcal{U})$ is a Ramsey ultrafilter.

math.GN↗
arXiv2017

Ideals in $\mathcal{P} _G$ and $βG$

For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra $ \mathcal{P}_G$ of subsets of $G$ and closed subsets in the Stone-$\check{C}$ech compactifi-cation $βG$ as a right topological semigroup to introduce and characterize some new ideals in $βG$. We show that if a group $G$ is either countable or Abelian then there are no closed ideals in $βG$ maximal in $G^*$, $G^* = βG \setminus G$, but this statement does not hold for the group $S_κ$ of all permutations of an infinite cardinal $κ$. We characterize the minimal closed ideal in $βG$ containing all idempotents of $G^*$.

math.GN↗
arXiv2017

Ramsey-product subsets of a group

We say that a subset $S$ of an infinite group $G$ is a Ramsey-product subset if, for any infinite subsets $X$, $Y$ of $G$, there exist $x \in X$ and $y\in Y$ such that $x y \in S$ and $ y x \in S$ . We show that the family $φ$ of all Ramsey-product subsets of $G$ is a filter and $φ$ defines the subsemigroup $ \overline{G^*G^*}$ of the semigroup $G^*$ of all free ultrafilters on $G$.

math.GN↗
arXiv2017

On recurrence in G-spaces

We introduce and analyze the following general concept of recurrence. Let $G$ be a group and let $X$ be a G-space with the action $G\times X\longrightarrow X$, $(g,x)\longmapsto gx$. For a family $\mathfrak{F}$ of subset of $X$ and $A\in \mathfrak{F}$, we denote $Δ_{\mathfrak{F}}(A)=\{g\in G: gB\subseteq A$ for some $B\in \mathfrak{F}, \ B\subseteq A\}$, and say that a subset $R$ of $G$ is $\mathfrak{F}$-recurrent if $R\bigcap Δ_{\mathfrak{F}} (A)\neq\emptyset$ for each $A\in \mathfrak{F}$.

math.GN↗
arXiv2017

The descriptive look at the size of subsets of groups

We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-Čech compactification $βG$ of $G$, we prove, in particular, that the closure of the minimal ideal of $βG$ is of type $F_{σδ}$.

math.GN↗
arXiv2017

On hyperballeans of bounded geometry

A ballean (or coarse structure) is a set endowed with some family of subsets, the balls, is such a way that balleans with corresponding morphisms can be considered as asymptotic counterparts of uniform topological spaces. For a ballean $\mathcal{B}$ on a set $X$, the hyperballean $\mathcal{B}^{\flat}$ is a ballean naturally defined on the set $X^{\flat}$ of all bounded subsets of $X$. We describe all balleans with hyperballeans of bounded geometry and analyze the structure of these hyperballeans.

math.GN↗
arXiv2015

A note-question on partitions of semigroups

Given a semigroup $S$ and an $n$-partition $\mathcal{P}$ of $S$, $n\in \mathbb{N}$, do there exist $A\in \mathcal{P}$ and a subset $F$ of $S$ such that $S=F ^{-1} \{x \in S: x A \bigcap A\neq\emptyset\}$ and $|F |\leq n$? We give an affirmative answer provided that either $S$ is finite or $n=2$.

math.CO↗