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Aristotelis Panagiotopoulos

Publications and source records attributed to Aristotelis Panagiotopoulos.

At least 19 recordsLinked to original sources
arXiv2026

The class and dynamics of $α$-balanced Polish groups

For each ordinal $α<ω_1$, we introduce the class of $α$-balanced Polish groups. These classes form a hierarchy that completely stratifies the space between the class of Polish groups admitting a two-side-invariant metric (TSI) and the class of Polish groups admitting a complete left-invariant metric (CLI). We establish various closure properties, provide connections to model theory, and we develop a boundedness principle for CLI groups by showing that $α$-balancedness is an initial segment of a regular coanalytic rank. In the spirit of Hjorth's turbulence theory we also introduce "generic $α$-unbalancedness": a new dynamical condition for Polish $G$-spaces which serves as an obstruction to classification by actions of $α$-balanced Polish groups. We use this to provide, for each $α<ω_1$, an action of an $α$-balanced Polish group whose orbit equivalence relation is strongly generically ergodic against actions of any $β$-balanced Polish group with $β<α$.

math.LO↗
arXiv2024

The definable content of homological invariants I: $\mathrm{Ext}$ & $\mathrm{lim}^1$

This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting definable $\mathrm{Ext}(B,F)$ for pairs of countable abelian groups $B,F$ and definable $\mathrm{lim}^{1}(\boldsymbol{A})$ for towers $\boldsymbol{A}$ of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable $\textrm{Ext}(-,\mathbb{Z})$ is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups $Λ$ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups $Λ$ with isomorphic classical invariants $\textrm{Ext}(Λ,\mathbb{Z}) $. To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of the $p$-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem $\mathcal{R}(\mathrm{Aut}(Λ)\curvearrowright\mathrm{Ext}(Λ,\mathbb{Z}))$ of classifying all group extensions of $Λ$ by $\mathbb{Z}$ up to base-free isomorphism, when $Λ=\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $ d\geq 1$.

math.LO↗
arXiv2024

Quasi-invariant measures concentrating on countable structures

Countable $\mathcal{L}$-structures $\mathcal{N}$ whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman-Freer-Patel to be precisely those $\mathcal{N}$ which have no algebraicity. Here we characterize those countable $\mathcal{L}$-structure $\mathcal{N}$ whose isomorphism class supports a quasi-invariant probability measure. These turn out to be precisely those $\mathcal{N}$ which are not "highly algebraic" -- we say that $\mathcal{N}$ is highly algebraic if outside of every finite $F$ there is some $b$ and a tuple $\bar{a}$ disjoint from $b$ so that $b$ has a finite orbit under the pointwise stabilizer of $\bar{a}$ in $\mathrm{Aut}(\mathcal{N})$. As a bi-product of our proof we show that whenever the isomorphism class of $\mathcal{N}$ admits a quasi-invariant measure, then it admits one with continuous Radon--Nikodym cocycles.

math.LO↗
arXiv2024

The definable content of homological invariants II: Čech cohomology and homotopy classification

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors $\check{\mathrm{H}}^n$ on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ taking values in the category $\mathsf{GPC}$ of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from $\mathsf{GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of $d$-spheres or $d$-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors $\check{\mathrm{H}}^n_{\mathrm{def}}$ to show that a seminal problem in the development of algebraic topology, namely Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3\backslashΣ$ to the $2$-sphere, is essentially hyperfinite but not smooth. In the course of this work, we record Borel definable versions of a number of classical results bearing on both the combinatorial and homotopical formulations of Čech cohomology; in aggregate, this work may be regarded as laying foundations for the descriptive set theoretic study of the homotopy relation on the space of maps from a locally compact Polish space to a polyhedron, a relation which embodies a substantial variety of classification problems arising throughout mathematics.

math.LO↗
arXiv2024

Definable (co)homology, pro-torus rigidity, and (co)homological classification

We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact metrizable spaces up to homotopy. In particular, we show that pro-tori are completely classified up to homeomorphism by their definable homology. This is in contrast with the fact that, for example, there exist uncountably many pairwise non-homeomorphic solenoids with the same Steenrod homology groups. We similarly develop a definable cohomology theory which strengthens Čech cohomology and we show that it completely classifies complements of pro-tori up to homeomorphism. We also apply definable cohomology theory to the study of the space $\left[ X,S^{2}\right] $ of homotopy classes of continuous functions from a solenoid complement $X$ to the $2$-sphere, which was initiated by Borsuk and Eilenberg in 1936. It was proved by Eilenberg and Steenrod in 1940 that the space $\left[ X,S^{2}\right] $ is uncountable. We will strengthen this result, by showing that each orbit of the canonical action $\mathrm{Homeo}% \left( X\right) \curvearrowright \left[ X,S^{2}\right] $ is countable, and hence that such an action has uncountably many orbits. This can be seen as a rigidity result, and will be deduced from a rigidity result for definable automorphisms of the Čech cohomology of $X$. We will also show that these results still hold if one replaces solenoids with pro-tori. We conclude by applying the machinery developed herein to bound the Borel complexity of several well-studied classification problems in mathematics, such as that of automorphisms of continuous-trace $C^{*}$-algebras up to unitary equivalence, or that of Hermitian line bundles, up to isomorphism, over a locally compact second countable space.

math.AT↗
arXiv2024

Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension

The algebraic dimension of a Polish permutation group $Q\leq \mathrm{Sym}(\mathbb{N})$ is the smallest $n\inω$, so that for all $A\subseteq \mathbb{N}$ of size $n+1$, the orbit of every $a\in A$ under the pointwise stabilizer of $A\setminus\{a\}$ is finite. We study the Bernoulli shift $P\curvearrowright \mathbb{R}^{\mathbb{N}}$ for various Polish permutation groups $P$ and we provide criteria under which the $P$-shift is generically ergodic relative to the injective part of the $Q$-shift, when $Q$ has algebraic dimension $\leq n$. We use this to show that the sequence of pairwise $*$-reduction-incomparable equivalence relations defined in [KP21] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal $\aleph_1^{+}$ which strongly resembles the equivalence relation of pinned cardinal $\aleph_1^{+}$ from [Zap11], but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions. Our proofs rely on the study of symmetric models whose symmetries come from the group $Q$. We show that when $Q$ is "locally finite" -- e.g. when $Q=\mathrm{Aut}(\mathcal{M})$, where $\mathcal{M}$ is a locally finite countable structure with no algebraicity -- the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.

math.LO↗
arXiv2023

Incompleteness Theorems for Observables in General Relativity

The quest for complete observables in general relativity has been a longstanding open problem. We employ methods from descriptive set theory to show that no complete observable on rich enough collections of spacetimes is Borel definable. In fact, we show that it is consistent with the Zermelo-Fraenkel and Dependent Choice axioms that no complete observable for rich collections of spacetimes exists whatsoever. In a nutshell, this implies that the Problem of Observables is to 'analysis' what the Delian Problem was to 'straightedge and compass'. Our results remain true even after restricting the space of solutions to vacuum solutions. In other words, the issue can be traced to the presence of local degrees of freedom. We discuss the next steps in a research program that aims to further uncover this novel connection between theoretical physics and descriptive set theory.

gr-qc↗
arXiv2022

Every CBER is smooth below the Carlson-Simpson generic partition

Let $E$ be a countable Borel equivalence relation on the space $\mathcal{E}_{\infty}$ of all infinite partitions of the natural numbers. We show that $E$ coincides with equality below a Carlson-Simpson generic element of $\mathcal{E}_{\infty}$. In contrast, we show that there is a hypersmooth equivalence relation on $\mathcal{E}_{\infty}$ which is Borel bireducible with $E_1$ on every Carlson-Simpson cube. Our arguments are classical and require no background in forcing.

math.LO↗
arXiv2022

Universality vs Genericity and $C_4$-free graphs

We show that the existence of a universal structure implies the existence of a generic structure for any approximable class $\mathcal{C}$ of countable structures. We also show that the converse is not true. As a consequence, we provide several new examples of weak Fraïssé classes of finite graphs. Finally, we show that the class of all countable $C_4$-free graphs does not contain a generic structure, strengthening a result of A. Hajnal and J. Pach.

math.LO↗
arXiv2021

Higher dimensional obstructions for star reductions

A $*$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a $*$-reduction between orbit equivalence relations induces generically an embedding between the associated Becker graphs. We introduce a notion of dimension for Polish $G$-spaces which is generically preserved under $*$-reductions. For every natural number $n$ we define a free action of $S_{\infty}$ whose dimension is $n$ on every invariant Baire measurable non-meager set. We also show that the $S_{\infty}$-space which induces the equivalence relation $=^{+}$ of countable sets of reals is $\infty$-dimensional on every invariant Baire measurable non-meager set. We conclude that the orbit equivalence relations associated to all these actions are pairwise incomparable with respect to $*$-reductions.

math.LO↗
arXiv2021

Examples of weak amalgamation classes

We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum-sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.

math.LO↗
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

Dynamical obstructions to classification by (co)homology and other TSI-group invariants

In the spirit of Hjorth's turbulence theory, we introduce "unbalancedness": a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two side invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace $C^*$-algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we show that the Wreath product of any two non-compact subgroups of $S_{\infty}$ admits an action whose orbit equivalence relation is generically ergodic against any action of a TSI group and we deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by actions of TSI groups.

math.LO↗
arXiv2020

On Polish groups admitting non-essentially countable actions

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

math.LO↗
arXiv2020

A combinatorial model for the Menger curve

We represent the universal Menger curve as the topological realization $|\mathbb{M}|$ of the projective Fraïssé limit ${\mathbb M}$ of the class of all finite connected graphs. We show that $\mathbb{M}$ satisfies combinatorial analogues of the Mayer-Oversteegen-Tymchatyn homogeneity theorem and the Anderson-Wilson projective universality theorem. Our arguments involve only $0$-dimensional topology and constructions on finite graphs. Using the topological realization $\mathbb{M}\mapsto|\mathbb{M}|$, we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson's finite homogeneity theorem, and prove a variant of Anderson-Wilson's theorem. The finite homogeneity theorem is the first instance of an "injective" homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions.

math.LO↗
arXiv2018

Games orbits play and obstructions to Borel reducibility

We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.

math.LO↗