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Aleksandra Kwiatkowska

Publications and source records attributed to Aleksandra Kwiatkowska.

At least 19 recordsLinked to original sources
arXiv2026

Group-extensive embeddings into Fraïssé structures and stationary weak independence relations

Let $M$ be a Fraïssé structure (a countably infinite ultrahomogeneous structure). We call an embedding $f : A \to M$ group-extensive if each automorphism of its image extends to an automorphism of $M$, where the extension map respects composition. We say that $M$ has group-extensible $ω$-age if each substructure admits a group-extensive embedding into $M$. We investigate the relationship between the following two properties: the presence of a stationary weak independence relation (SWIR) on $M$, and group-extensibility of the $ω$-age of $M$. We show that linearly ordered Fraïssé structures with a SWIR have group-extensible $ω$-age, but also we give examples of Fraïssé structures where only one of the two properties holds. Finally, we consider whether a wide range of examples of Fraïssé structures have group-extensible $ω$-age or a finite SWIR expansion, including all countably infinite ultrahomogeneous oriented graphs (with one exception).

math.LO↗
arXiv2026

Universal homogeneous two-sorted ultrametric spaces

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining isometries and linear order embeddings. We show that the class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé, and that the limit is the countable rational Urysohn ultrametric space $\mathbb{U}$. The space $\mathbb{U}$ is dc-universal for all countable ultrametric spaces, and its Cauchy completion $\overline{\mathbb{U}}$ is dc-universal for all separable ultrametric spaces, which is in contrast with the situation of classical ultrametric spaces and isometric embeddings, where no such universal space can exist. We study further properties of $\mathbb{U}$, of its variants, and of its automorphism group, which is richer than its group of isometries. In particular, we provide two types of tree representations of the two-sorted ultrametric spaces, discuss connections to valued fields, and characterize the automorphism group of $\mathbb{U}$ as the semidirect product of a group of order preserving bijections and a group of isometries. Furthermore, we show universality of $\operatorname{Aut}(\mathbb{U})$ and identify its universal minimal flow.

math.LO↗
arXiv2026

Generic dc-automorphisms of two-sorted ultrametric spaces

We continue to study ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances, together with the dc-embeddings, which we introduced in our earlier paper "Universal homogeneous two-sorted ultrametric spaces". The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé whose limit we denote by $\mathbb{U}$. The main result of the article is that $\operatorname{Aut}(\mathbb{U})$ has a comeager conjugacy class. For that we show the cofinal amalgamation property of partial automorphisms and characterize amalgamation bases. In fact we develop a general strategy for showing cofinal amalgamation property for a broad class of categories. Furthermore, we show that there is no generic pair of automorphisms, we provide a detailed description of single orbits under dc-automorphisms, and we prove that any finite partial dc-automorphism, even in the presence of other orbits, can be extended to one that is closed or monotone.

math.LO↗
arXiv2026

Inter-qubit correlation dynamics driven by mutual interactions

A particularly useful tool for characterizing multi-qubit systems is the correlation tensor, providing an experimentally friendly and theoretically concise representation of quantum states. In this work, we analyze the evolution of the correlation tensor elements of quantum systems composed of $\frac12$-spins, generated by mutual interactions and the influence of the external field. We focus on two-body interactions in the form of anisotropic Heisenberg as well as antisymmetric exchange interaction models. The evolution of the system is visualized in the form of a trajectory in a suitable correlation space, which, depending on the system's frequencies, exhibits periodic or nonperiodic behavior. In the case of two $\frac12$-spins we study the stationary correlations for several classes of Hamiltonians, which allows a full characterization of the families of density matrices invariant under the evolution generated by the Hamiltonians. We discuss some common properties shared by the 2- and 3-qubit systems and show how a strong external field can play a stabilizing factor with respect to certain correlation characteristics.

quant-ph↗
arXiv2025

Embeddings into the generic poset

Let M be the generic poset, defined as the Fraïssé limit of the class of finite posets. We show that every countably infinite poset A can be embedded with coinfinite image into M so that each automorphism of the image of A extends uniquely to an automorphism of M.

math.LO↗
arXiv2024

The projective Fra\"ıssé limit of the family of all connected finite graphs with confluent epimorphisms

We investigate the projective Fra\"ıssé family of finite connected graphs with confluent epimorphisms and the continuum obtained as the topological realization of its projective Fra\"ıssé limit. This continuum was unknown before. We prove that it is indecomposable, but not hereditarily indecomposable, one-dimensional, Kelley, pointwise self-homeomorphic, but not homogeneous. It is hereditarily unicoherent and each point is the top of the Cantor fan. Moreover, the universal solenoid, the universal pseudo-solenoid, and the pseudo-arc may be embedded in it.

math.GN↗
arXiv2024

Projective Fraïssé limits of trees with confluent epimorphisms

We continue the study of projective Fraïssé limits developed by Irwin-Solecki and Panagiotopoulos-Solecki by investigating families of epimorphisms between finite trees and finite rooted trees. Ideas of monotone, confluent, and light mappings from continuum theory as well as several properties of continua are modified so as to apply them to topological graphs. As the topological realizations of the projective Fraïssé limits we obtain the dendrite $D_3$, the Mohler-Nikiel universal dendroid, as well as new, interesting continua for which we do not yet have topological characterizations.

math.GN↗
arXiv2024

Projective Fraïssé limits and generalized Ważewski dendrites

We continue the study of projective Fraïssé limits of trees initiated by Charatonik and Roe and we construct many generalized Ważewski dendrites as the topological realization of a projective Fraïssé limit of families of finite trees with (weakly) coherent epimorphisms. Moreover we use the categorical approach to Fraïssé limits developed by Kubiś to construct all generalized Ważewski dendrites as topological realizations of Fraïssé limits of suitable categories of finite structures. As an application we recover a homogeneity result for countable dense sets of endpoints in generalized Ważewski dendrites.

math.LO↗
arXiv2023

Large orbits, projective Fraisse limits, and the pseudo-arc

We show that the conjugacy action of the automorphism group ${\rm Aut}(\mathbb{P})$, of the projective Fraïssé limit $\mathbb{P}$, whose natural quotient is the pseudo-arc, on the set of involutions of $\mathbb{P}$, has a comeager orbit.

math.LO↗
arXiv2020

Topologically 2-generated groups

We prove that for a number of ultrahomogeneous structures $M$, including those with the free amalgamation property, the powers of the automorphism group ${\rm{Aut}}(M)^n$, $n=1,2,\ldots$, and the group $L_0({\rm{ Aut}}(M))$ of measurable functions with values in ${\rm{Aut}}(M)$, have a cyclically dense conjugacy class, in particular, are topologically 2-generated. This provides a number of new examples of groups with this property. Moreover, we will show that each of these groups is cyclically generated by a pair generating the free group.

math.LO↗
arXiv2019

Ordered structures and large conjugacy classes

This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager $2$-dimensional diagonal conjugacy class. We also provide general conditions implying that there is no comeager conjugacy class, comeager $2$-dimensional diagonal conjugacy class or non-meager $2$-dimensional topological similarity class in the automorphism group of an ordered Fraisse limit. We provide a number of applications of these results.

math.LO↗
arXiv2018

Automorphism groups of countable structures and groups of measurable functions

Let $G$ be a topological group and let $μ$ be the Lebesgue measure on the interval $[0,1]$. We let $L_0(G)$ to be the topological group of all $μ$-equivalence classes of $μ$-measurable functions defined on [0,1] with values in $G$, taken with the pointwise multiplication and the topology of convergence in measure. We show that for a Polish group $G$, if $L_0(G)$ has ample generics, then $G$ has ample generics, thus the converse to a result of Kaïchouh and Le Maître. We further study topological similarity classes and conjugacy classes for many groups ${\rm{Aut}}(M)$ and $L_0({\rm{Aut}}(M))$, where $M$ is a countable structure. We make a connection between the structure of groups generated by tuples, the Hrushovski property, and the structure of their topological similarity classes. In particular, we prove the trichotomy that for every tuple $ \bar{f}$ of ${\rm{Aut}}(M)$, where $M$ is a countable structure such that algebraic closures of finite sets are finite, either the countable group $\langle \bar{f} \rangle$ is precompact, or it is discrete, or the similarity class of $\bar{f}$ is meager, in particular the conjugacy class of $\bar{f}$ is meager. We prove an analogous trichotomy for groups $L_0({\rm{Aut}}(M))$.

math.LO↗
arXiv2018

Universal minimal flows of generalized Ważewski dendrites

We study universal minimal flows of the homeomorphism groups of generalized Ważewski dendrites $W_P$, $P\subset\{3,4,\ldots,ω\}$. If $P$ is finite, we prove that the universal minimal flow of of the homeomorphism group $H(W_P)$ is metrizable and we compute it explicitly. This answers a question of B. Duchesne. If $P$ is infinite, we show that the universal minimal flow of $H(W_P)$ is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a non-metrizable universal minimal flow with a comeager orbit.

math.LO↗