arXiv subjects

Luca Motto Ros

Publications and source records attributed to Luca Motto Ros.

At least 19 recordsLinked to original sources

arXiv2026

Isometry groups of Polish ultrametric spaces

We solve a long-standing open problem, formulated by Krasner in the 1950's, in the context of Polish (i.e. separable complete) ultrametric spaces by providing a characterization of their isometry groups using suitable forms of generalized wreath products of full permutation groups. Since our solution is developed in the finer context of topological (Polish) groups, it also solves a problem of Gao and Kechris from 2003. Furthermore, we provide an exact correspondence between the isometry groups of Polish ultrametric spaces belonging to some natural subclasses and various kinds of generalized wreath products proposed in the literature by Hall, Holland, and Malicki.

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arXiv2025

Generalized Descriptive Set Theory at Singular Cardinals of Countable Cofinality

We provide a comprehensive development of the basics of descriptive set theory for non-separable complete metric spaces whose weight is a singular cardinal $λ$ of countable confinality. Somewhat unexpectedly, the resulting theory is remarkably similar to the classical one, although the methods used are necessarily fairly different and combine ideas and results from general topology, infinite combinatorics, and set theory. More in detail, we study $λ$-Polish spaces and standard $λ$-Borel spaces (characterization of the generalized Cantor and Baire spaces, analogues of the Cantor-Bendixson theorem, classification up to $λ$-Borel isomorphism, etc.), their $λ$-Borel hierarchy (structural properties, changes of topologies, and so on), $λ$-analytic sets (including generalizations of the Lusin separation theorem and of the Souslin theorem), $λ$-coanalytic sets (including $λ$-$\boldsymbolΠ^1_1$-ranks and alike), and $λ$-projective sets. We also consider more advanced topics, and provide e.g. various uniformization results for $λ$-Borel set; these in turn lead to fundamental applications to the study of $λ$-Borel equivalence relations, such as a generalization of the celebrated Feldman-Moore theorem. Finally, we study a natural generalization of the classical Perfect Set Property, and develop tools to show that all definable sets enjoy such property under suitable large cardinal assumptions, most notably including Woodin's $\mathsf{I0}(λ)$.

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arXiv2025

Generalized Borel Sets

Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable cardinal $κ$, possibly singular, satisfying $2^{<κ}=κ$. We provide fundamental properties of the $κ^+$-Borel hierarchy of any regular Hausdorff space of weight at most $κ$, and establish sufficient conditions for its non-collapse. We highlight a unique phenomenon that arises in the case of singular cardinals, namely, the existence of a second, distinct Borel hierarchy, the $κ$-Borel hierarchy: we prove that it is strictly finer than the $κ^+$-Borel hierarchy, and then characterize the precise relationship between the two. Finally, for regular cardinals, we resolve three questions about the behavior of the $κ^+$-Borel hierarchy on subspaces of the generalized Baire space ${}^κκ$, constructing various models via forcing where several nontrivial constellations for the length of the $κ^+$-Borel hierarchy on the space are realized.

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arXiv2025

Structural results on idealistic equivalence relations

We address some fundamental problems concerning the structure of idealistic equivalence relations. In particular, we show that, under analytic determinacy, there are continuum many idealistic analytic equivalence relations that are not classwise Borel isomorphic to an orbit equivalence relation. Moreover, we also discuss alternative formulations of the E1 conjecture, and present an elementary counterexample to one of its earliest variants proposed by Hjorth and Kechris in 1997.

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arXiv2024

Generalized Baire class functions

Let $λ$ be an uncountable cardinal such that $2^{< λ} = λ$. Working in the setup of generalized descriptive set theory, we study the structure of $λ^+$-Borel measurable functions with respect to various kinds of limits, and isolate a suitable notion of $λ$-Baire class $ξ$ function. Among other results, we provide higher analogues of two classical theorems of Lebesgue, Hausdorff, and Banach, namely: (1) A function is $λ^+$-Borel measurable if and only if it can be obtained from continuous functions by iteratively applying pointwise $D$-limits, where $D$ varies among directed sets of size at most $λ$. (2) A function is of $λ$-Baire class $ξ$ if and only if it is $\boldsymbolΣ^{0}_{ξ+1}$-measurable.

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arXiv2024

Continuous logic in a classical setting

Let $\mathcal{L}$ be a first-order two-sorted language and consider a class of $\mathcal{L}$-structures of the form $\langle M, X \rangle$ where $M$ varies among structures of the first sort, while $X$ is fixed in the second sort, and it is assumed to be a compact Hausdorff space. When $X$ is a compact subset of the real line, one way to treat classes of this kind model-theoretically is via continuous-valued logic, as in [Ben Yaacov-Berenstein-Henson-Usvyatsov 2010]. Prior to that, Henson and Iovino proposed an approach based on the notion of positive formulas [Henson-Iovino 2002]. Their work is tailored to the model theory of Banach spaces. Here we show that a similar approach is possible for a more general class of models. We introduce suitable versions of elementarity, compactness, saturation, quantifier elimination and other basic tools, and we develop basic model theory.

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arXiv2023

Piecewise convex embeddability on linear orders

Given a nonempty set $\mathcal{L}$ of linear orders, we say that the linear order $L$ is $\mathcal{L}$-convex embeddable into the linear order $L'$ if it is possible to partition $L$ into convex sets indexed by some element of $\mathcal{L}$ which are isomorphic to convex subsets of $L'$ ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in arXiv:2309.09910), which are the special cases $\mathcal{L} = \{\mathbf{1}\}$ and $\mathcal{L} = \mathsf{Fin}$. We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders.

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arXiv2023

Convex Embeddability and Knot Theory

We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of circular orders. These results are then applied to the study of arcs and knots, establishing combinatorial properties and lower bounds (in terms of Borel reducibility) for the complexity of some natural relations between these geometrical objects.

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arXiv2023

Generalized Polish spaces at regular uncountable cardinals

In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard $κ$-Borel spaces for $κ$ an uncountable (regular) cardinal satisfying $κ^{<κ} = κ$. As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work from [Coskey-Schlicht 2016, Galeotti 2019, Luecke-Schlicht 2015], and answer some questions contained therein.

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arXiv2023

A classification of the Wadge hierarchies on zero-dimensional Polish spaces

We provide a complete classification, up to order-isomorphism, of all possible Wadge hierarchies on zero-dimensional Polish spaces using (essentially) countable ordinals as complete invariants. We also observe that although our assignment of invariants is very simple and there are only $\aleph_1$-many equivalence classes, the above classification problem is quite complex from the descriptive set-theoretic point of view: in particular, there is no Borel procedure to determine whether two zero-dimensional Polish spaces have isomorphic Wadge hierarchies. All results are based on a complete and explicit description of the Wadge hierarchy on an arbitrary zero-dimensional Polish space, depending on its topological properties.

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arXiv2023

Anti-classification results for groups acting freely on the line

We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings $\mathrm{Ar}(G)$ for a given countable group $G$, and prove that the equivalence relation induced by the natural action of $\mathrm{GL}_2(\mathbb{Q})$ on $\mathrm{Ar}(\mathbb{Q}^2)$ is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau. In particular, we show that its potential class is not $\boldsymbolΠ^0_3$. This topological constraint prevents classifying Archimedean groups using countable subsets of reals. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups, and o-minimal structures such as ordered divisible Abelian groups, and real closed fields. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice.

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arXiv2021

Classification problems from the descriptive set theoretical perspective

Twenty years have passed since Kechris' seminal survey paper [A. S. Kechris, New directions in descriptive set theory. Bull. Symbolic Logic, 5(2):161-174, 1999]. As a follow-up of that work, we review some ot the (anti-)classification results that have been obtained in the last decade using Borel reducibility and its generalizations to uncountable cardinals.

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arXiv2020

A descriptive Main Gap Theorem

Answering one of the main questions of [FHK14, Chapter 7], we show that there is a tight connection between the depth of a classifiable shallow theory $T$ and the Borel rank of the isomorphism relation $\cong^κ_T$ on its models of size $κ$, for $κ$ any cardinal satisfying $κ^{< κ} = κ> 2^{\aleph_0}$. This is achieved by establishing a link between said rank and the $\mathcal{L}_{\infty κ}$-Scott height of the $κ$-sized models of $T$, and yields to the following descriptive set-theoretical analogue of Shelah's Main Gap Theorem: Given a countable complete first-order theory $T$, either $\cong^κ_T$ is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is $κ^+ > \aleph_1$), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah's theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of $\cong^κ_T$, and provide a characterization of categoricity of $T$ in terms of the descriptive set-theoretical complexity of $\cong^κ_T$.

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arXiv2020

Polish metric spaces with fixed distance set

We study Polish spaces for which a set of possible distances $A \subseteq \mathbb{R}^+$ is fixed in advance. We determine, depending on the properties of $A$, the complexity of the collection of all Polish metric spaces with distances in $A$, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that $A$ must have in order that all Polish spaces with distances in that set belong to a given class, such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of $A$, of the relations of isometry and isometric embeddability between these Polish spaces.

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arXiv2019

Souslin quasi-orders and bi-embeddability of uncountable structures

We provide analogues of the results from [FMR11, CMMR13] in the reference list (which correspond to the case $κ= ω$) for arbitrary $κ$-Souslin quasi-orders on any Polish space, for $κ$ an infinite cardinal smaller than the cardinality of $\mathbb{R}$. These generalizations yield a variety of results concerning the complexity of the embeddability relation between graphs or lattices of size $κ$, the isometric embeddability relation between complete metric spaces of density character $κ$, and the linear isometric embeddability relation between (real or complex) Banach spaces of density $κ$.

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arXiv2019

Uncountable structures are not classifiable up to bi-embeddability

Answering some of the main questions from [MR13], we show that whenever $κ$ is a cardinal satisfying $κ^{< κ} = κ> ω$, then the embeddability relation between $κ$-sized structures is strongly invariantly universal, and hence complete for ($κ$-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [LR05,FMR11,Wil14,CMR17].

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arXiv2017

On isometry and isometric embeddability between ultrametric Polish spaces

We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of $D$. When $D$ contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If $D$ is well-ordered the situation is more complex: for isometry we have an increasing sequence of Borel equivalence relations of length $ω_1$ which are cofinal among Borel equivalence relations classifiable by countable structures, while for isometric embeddability we have an increasing sequence of analytic quasi-orders of length at least $ω+3$. We then apply our results to solve various open problems in the literature. For instance, we answer a long-standing question of Gao and Kechris by showing that the relation of isometry on locally compact ultrametric Polish spaces is Borel bireducible with countable graph isomorphism.

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arXiv2017

Universality of group embeddability

Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the isometric embeddability between separable groups with a bounded bi-invariant complete metric are all invariantly universal analytic quasi-orders. This strengthens some results from [Wil14] and [FLR09].

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