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Sławomir Solecki

Publications and source records attributed to Sławomir Solecki.

At least 19 recordsLinked to original sources
arXiv2026

Idealistic equivalence relations and the Lusin derivative

We give a proof, that does not rely on analytic determinacy, of the existence of an idealistic analytic equivalence relation with Borel classes that is not classwise Borel isomorphic to an orbit equivalence relation. This result answers a question of Becker, and Calderoni and Motto Ros. Our proof proceeds through an analysis of Borel complexity of certain sets of countable rooted trees defined via the notion of the Lusin derivative.

math.LO↗
arXiv2026

Homeomorphisms of continua through projective Fraïssé limits

We study homeomorphisms and the homeomorphism groups of compact metric spaces using the automorphism groups of projective Fraïssé limits. In our applications, we investigate the Polish group ${\rm Homeo}(P)$ of all homeomorphisms of the pseudoarc $P$ using the automorphism group ${\rm Aut}(\mathbb{P})$ of the pre-pseudoarc $\mathbb{P}$. Strengthening results from the literature, we show that the diagonal conjugacy action of ${\rm Homeo}(P)$ on ${\rm Homeo}(P)^{\mathbb{N}}$ has a dense orbit. In our second application, we show that there exists a homeomorphism of $P$ that is not conjugate in ${\rm Homeo}(P)$ to an element of ${\rm Aut}(\mathbb{P})$.

math.LO↗
arXiv2025

Simplicial complexes, stellar moves, and projective amalgamation

We explore connections between stellar moves on simplicial complexes (these are fundamental operations of combinatorial topology) and projective Fra{ï}ss{é} limits (this is a model theoretic construction with topological applications). We identify a class of simplicial maps that arise from the stellar moves of welding and subdividing. We call these maps weld-division maps. The core of the paper is the proof that the category of weld-division maps fulfills the projective amalgamation property. This gives an example of an amalgamation class that substantially differs from known classes. The weld-division amalgamation class naturally gives rise to a projective Fra{ï}ss{é} class. We compute the canonical limit of this projective Fra{ï}ss{é} class and its canonical quotient space. This computation gives a combinatorial description of the geometric realization of a simplicial complex and an example of a combinatorially defined projective Fra{ï}ss{é} class whose canonical quotient space has topological dimension strictly bigger than $1$. The method of proof of the amalgamation theorem is new. It is not geometric or topological, but rather it consists of combinatorial calculations performed on finite sequences of finite sets and functions among such sequences. Set theoretic nature of the entries of the sequences is crucial to the arguments.

math.CO↗
arXiv2024

Groups without unitary representations, submeasures, and the escape property

We give new examples of topological groups that do not have non-trivial continuous unitary representations, the so-called exotic groups. We prove that all groups of the form $L^0(ϕ, G)$, where $ϕ$ is a pathological submeasure and $G$ is a topological group, are exotic. This result extends, with a different proof, a theorem of Herer and Christensen on exoticness of $L^0(ϕ,\mathbb{R})$ for $ϕ$ pathological. It follows that every topological group embeds into an exotic one. In our arguments, we introduce the escape property, a geometric condition on a topological group, inspired by the solution to Hilbert's fifth problem and satisfied by all locally compact groups, all non-archimedean groups, and all Banach--Lie groups. Our key result involving the escape property asserts triviality of all continuous homomorphisms from $L^0(ϕ, G)$ to $L^0(μ, H)$, where $ϕ$ is pathological, $μ$ is a measure, $G$ is a topological group, and $H$ is a topological group with the escape property.

math.RT↗
arXiv2024

$κ$-Borel sets, $κ$-Baire spaces, and filtrations between topologies

Filtrations are certain transfinite sequences of topologies increasing in strength and interpolating between two given topologies $σ$ and $τ$, with $τ$ being stronger than $σ$. We prove general results on stabilization at $τ$ of filtrations interpolating between $σ$ and $τ$. These topological results involve an interplay between $κ$-Borel sets with respect to the topology $σ$ and $κ$-Baireness of the topology $τ$.

math.GN↗
arXiv2022

Polish modules over subrings of $\mathbb Q$

We give a method of producing a Polish module over an arbitrary subring of $\mathbb Q$ from an ideal of subsets of $\mathbb N$ and a sequence in $\mathbb N$. The method allows us to construct two Polish $\mathbb Q$-vector spaces, $U$ and $V$, such that -- both $U$ and $V$ embed into $\mathbb R$ but -- $U$ does not embed into $V$ and $V$ does not embed into $U$, where by an embedding we understand a continuous $\mathbb Q$-linear injection. This construction answers a question of Frisch and Shinko. In fact, our method produces a large number of incomparable with respect to embeddings Polish $\mathbb Q$-vector spaces.

math.LO↗
arXiv2022

Generic measure preserving transformations and the closed groups they generate

We show that, for a generic measure preserving transformation $T$, the closed group generated by $T$ is not isomorphic to the topological group $L^0(λ, {\mathbb T})$ of all Lebesgue measurable functions from $[0,1]$ to $\mathbb T$ (taken with pointwise multiplication and the topology of convergence in measure). This result answers a question of Glasner and Weiss. The main step in the proof consists of showing that Koopman representations of ergodic boolean actions of $L^0(λ, {\mathbb T})$ possess a non-trivial spectral property not shared by all unitary representations of $L^0(λ, {\mathbb T})$. The main tool underlying our arguments is a theorem on the form of unitary representations of $L^0(λ, {\mathbb T})$ from our earlier work.

math.DS↗
arXiv2022

Finite Ramsey Theory through Category Theory

We present a new, category theoretic point of view on finite Ramsey theory. Our aims are as follows: -- to define the category theoretic notions needed for the development of finite Ramsey Theory, -- to state, in terms of these notions, the general fundamental Ramsey results (of which various concrete Ramsey results are special cases), and -- to give self-contained proofs within the category theoretic framework of these general results. We also provide some concrete illustrations of the general method.

math.CO↗
arXiv2021

The generic combinatorial simplex

We employ projective Fraïssé theory to define the "generic combinatorial $n$-simplex" as the pro-finite, simplicial complex that is canonically associated with a family of simply defined selection maps between finite triangulations of the simplex. The generic combinatorial $n$-simplex is a combinatorial object that can be used to define the geometric realization of a simplicial complex without any reference to the Euclidean space. It also reflects dynamical properties of its homeomorphism group down to finite combinatorics. As part of our study of the generic combinatorial simplex, we define and prove results on domination closure for Fraïssé classes, and we develop further the theories of stellar moves and cellular maps. We prove that the domination closure of selection maps contains the class of face-preserving simplicial maps that are cellular on each face of the $n$-simplex and is contained in the class of simplicial, face-preserving near-homeomorphisms. Under the PL-Poincaré conjecture, this gives a characterization of the domination closure of selections.

math.LO↗
arXiv2021

Tukey order among $F_σ$ ideals

We investigate the Tukey order in the class of $F_σ$ ideals of subsets of $ω$. We show that no nontrivial $F_σ$ ideal is Tukey below a $G_δ$ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets.

math.LO↗
arXiv2020

Concentration of measure, classification of submeasures, and dynamics of $L_{0}$

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set of the product. Our proof combines the Herbst argument with an analogue of Shearer's lemma for differential entropy. We give a quantitative "geometric" classification of diffuse submeasures into elliptic, parabolic, and hyperbolic. We prove that any non-elliptic submeasure (for example, any measure, or any pathological submeasure) has a property that we call covering concentration. Our results have strong consequences for the dynamics of the corresponding topological $L_{0}$-groups.

math.PR↗
arXiv2020

Random continuum and Brownian motion

We describe a probabilistic model involving iterated Brownian motion for constructing a random chainable continuum. We show that this random continuum is indecomposable.

math.PR↗
arXiv2020

Dual Ramsey theorem for trees

The classical Ramsey theorem was generalized in two major ways: to the dual Ramsey theorem, by Graham and Rothschild, and to Ramsey theorems for trees, initially by Deuber and Leeb. Bringing these two lines of thought together, we prove the dual Ramsey theorem for trees. Galois connections between partial orders are used in formulating this theorem, while the abstract approach to Ramsey theory, we developed earlier, is used in its proof.

math.CO↗
arXiv2019

Coherent extension of partial automorphisms, free amalgamation, and automorphism groups

We give strengthened versions of the Herwig-Lascar and Hodkinson-Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraisse classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraisse limit of any Fraisse class that is the class of all $\mathcal{F}$-free structures (in the Herwig--Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language, all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

math.LO↗
arXiv2018

Monoid actions and ultrafilter methods in Ramsey theory

First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg--Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers' Ramsey theorem.

math.CO↗
arXiv2015

Tukey reduction among analytic directed orders

This is a survey of recent work on the structure of Tukey reductions among analytic $σ$-ideals of compact subsets of compact metric spaces and analytic P-ideals of sets of natural numbers. An attempt is made to organize the results into a unified whole. This organization makes it possible to identify natural unresolved problems. Some new joint results with Stevo Todorcevic are announced.

math.LO↗