arXiv subjects

Gianluca Paolini

Publications and source records attributed to Gianluca Paolini.

At least 55 records · Page 3Linked to original sources

arXiv2019

Some Results on Polish Groups

We prove that no quantifier-free formula in the language of group theory can define the $\aleph_1$-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power $λ$ is the group of automorphism of a locally finite group of power $λ$; secondly, we conjecture that the group of automorphisms of a locally finite group of power $λ$ has a locally finite subgroup of power $λ$, and reduce the problem to a problem on $p$-groups, thus settling the conjecture in the case $λ= \aleph_0$.

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arXiv2019

On a Cardinal Invariant Related to the Haar Measure Problem

In [6], given a metrizable profinite group $G$, a cardinal invariant of the continuum $\mathfrak{fm}(G)$ was introduced, and a positive solution to the Haar Measure Problem for $G$ was given under the assumption that $\mathrm{non}(\mathcal{N}) \leq \mathfrak{fm}(G)$. We prove here that it is consistent with ZFC that there is a metrizable profinite group $G_*$ such that $\mathrm{non}(\mathcal{N}) > \mathfrak{fm}(G_*)$, thus demonstrating that the strategy of [6] does not suffice for a general solution to the Haar Measure Problem.

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arXiv2018

Automorphism Groups of Countable Stable Structures

For every countable structure $M$ we construct an $\aleph_0$-stable countable structure $N$ such that $Aut(M)$ and $Aut(N)$ are topologically isomorphic. This shows that it is impossible to detect any form of stability of a countable structure $M$ from the topological properties of the Polish group $Aut(M)$.

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arXiv2018

The Class of Countable Projective Planes is Borel Complete

We observe that Hall's free projective extension $P \mapsto F(P)$ of partial planes is a Borel map, and use a modification of the construction introduced in [9] to conclude that the class of countable non-Desarguesian projective planes is Borel complete. In the process, we also rediscover the main result of [7] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.

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arXiv2018

The Strong Small Index Property for Free Homogeneous Structures

We show that in algebraically locally finite countable homogeneous structures with a free stationary independence relation the small index property implies the strong small index property. We use this and the main result of [15] to deduce that countable free homogeneous structures in a locally finite relational language have the strong small index property. We also exhibit new continuum sized classes of $\aleph_0$-categorical structures with the strong small index property whose automorphism groups are pairwise non-isomorphic.

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arXiv2018

A Universal Homogeneous Simple Matroid of Rank $3$

We construct a $\wedge$-homogeneous universal simple matroid of rank $3$, i.e. a countable simple rank~$3$ matroid $M_*$ which $\wedge$-embeds every finite simple rank $3$ matroid, and such that every isomorphism between finite $\wedge$-subgeometries of $M_*$ extends to an automorphism of $M_*$. We also construct a $\wedge$-homogeneous matroid $M_*(P)$ which is universal for the class of finite simple rank $3$ matroids omitting a given finite projective plane $P$. We then prove that these structures are not $\aleph_0$-categorical, they have the independence property, they admit a stationary independence relation, and that their automorphism group embeds the symmetric group $Sym(ω)$. Finally, we use the free projective extension $F(M_*)$ of $M_*$ to conclude the existence of a countable projective plane embedding all the finite simple matroids of rank $3$ and whose automorphism group contains $Sym(ω)$, in fact we show that $Aut(F(M_*)) \cong Aut(M_*)$.

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arXiv2018

Polish Topologies for Graph Products of Groups

We give strong necessary conditions on the admissibility of a Polish group topology for an arbitrary graph product of groups $G(Γ, G_a)$, and use them to give a characterization modulo a finite set of nodes. As a corollary, we give a complete characterization in case all the factor groups $G_a$ are countable.

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arXiv2018

The Class of Non-Desarguesian Projective Planes is Borel Complete

For every infinite graph $Γ$ we construct a non-Desarguesian projective plane $P^*_Γ$ of the same size as $Γ$ such that $Aut(Γ) \cong Aut(P^*_Γ)$ and $Γ_1 \cong Γ_2$ iff $P^*_{Γ_1} \cong P^*_{Γ_2}$. Furthermore, restricted to structures with domain $ω$, the map $Γ\mapsto P^*_Γ$ is Borel. On one side, this shows that the class of countable non-Desarguesian projective planes is Borel complete, and thus not admitting a Ulm type system of invariants. On the other side, we rediscover the main result of [15] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.

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arXiv2018

Reconstructing Structures with the Strong Small Index Property up to Bi-Definability

Let $\mathbf{K}$ be the class of countable structures $M$ with the strong small index property and locally finite algebraicity, and $\mathbf{K}_*$ the class of $M \in \mathbf{K}$ such that $acl_M(\{ a \}) = \{ a \}$ for every $a \in M$. For homogeneous $M \in \mathbf{K}$, we introduce what we call the expanded group of automorphisms of $M$, and show that it is second-order definable in $Aut(M)$. We use this to prove that for $M, N \in \mathbf{K}_*$, $Aut(M)$ and $Aut(N)$ are isomorphic as abstract groups if and only if $(Aut(M), M)$ and $(Aut(N), N)$ are isomorphic as permutation groups. In particular, we deduce that for $\aleph_0$-categorical structures the combination of strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin's well-known $\forall \exists$-interpretation technique of [7]. Finally, we show that every finite group can be realized as the outer automorphism group of $Aut(M)$ for some countable $\aleph_0$-categorical homogeneous structure $M$ with the strong small index property and no algebraicity.

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arXiv2018

Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup. We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria for such classes to be abstract elementary classes, for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translate into a machinery to build concrete examples of $\mathrm{AECs}$ satisfying given model-theoretic properties. We exhibit the power of our method constructing three concrete examples of finitary classes. We show that the first and third class are non-homogeneous, and that the last two are tame, uncountably categorical and axiomatizable by a single $L_{ω_{1}, ω}$-sentence. We also observe that the isomorphism relation of any countable complete first-order theory is $κ$-Borel reducible (in the sense of generalized descriptive set theory) to the isomorphism relation of the theory of right-angled Coxeter groups whose Coxeter graph is an infinite random graph.

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arXiv2018

A Finite Axiomatization of G-Dependence

We show that a form of dependence known as G-dependence (originally introduced by Grelling) admits a very natural finite axiomatization, as well as Armstrong relations. We also give an explicit translation between functional dependence and G-dependence.

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arXiv2018

Polish Topologies for Graph Products of Cyclic Groups

We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from [5], [7] and [4].

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arXiv2017

Reduction of Database Independence to Dividing in Atomless Boolean Algebras

We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov [4], the former case of independence can be seen as the discrete version of the latter.

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arXiv2017

Beyond Abstract Elementary Classes: On The Model Theory of Geometric Lattices

Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, $(\mathbf{K}^3, \preccurlyeq)$, is $ω$-stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in $(\mathbf{K}^3, \preccurlyeq)$ the Smoothness Axiom fails, and so $(\mathbf{K}^3, \preccurlyeq)$ is not an $\mathrm{AEC}$.

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arXiv2017

Group Metrics for Graph Products of Cyclic Groups

We complement the characterization of the graph products of cyclic groups $G(Γ, \mathfrak{p})$ admitting a Polish group topology of [9] with the following result. Let $G = G(Γ, \mathfrak{p})$, then the following are equivalent: (i) there is a metric on $Γ$ which induces a separable topology in which $E_Γ$ is closed; (ii) $G(Γ, \mathfrak{p})$ is embeddable into a Polish group; (iii) $G(Γ, \mathfrak{p})$ is embeddable into a non-Archimedean Polish group. We also construct left-invariant separable group ultrametrics for $G = G(Γ, \mathfrak{p})$ and $Γ$ a closed graph on the Baire space, which is of independent interest.

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arXiv2017

The Automorphism Group of Hall's Universal Group

We study the automorphism group of Hall's universal locally finite group $H$. We show that in $Aut(H)$ every subgroup of index $< 2^ω$ lies between the pointwise and the setwise stabilizer of a unique finite subgroup $A$ of $H$, and use this to prove that $Aut(H)$ is complete. We further show that $Inn(H)$ is the largest locally finite normal subgroup of $Aut(H)$. Finally, we observe that from the work of [Sh:312] it follows that for every countable locally finite $G$ there exists $G \cong G' \leq H$ such that every $f \in Aut(G')$ extends to an $\hat{f} \in Aut(H)$ in such a way that $f \mapsto \hat{f}$ embeds $Aut(G')$ into $Aut(H)$. In particular, we solve the three open questions of Hickin on $Aut(H)$ from [3], and give a partial answer to Question VI.5 of Kegel and Wehrfritz from [6].

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arXiv2017

No Uncountable Polish Group Can be a Right-Angled Artin Group

We prove that no uncountable Polish group can admit a system of generators whose associated length function satisfies the following conditions: (i) if $0 < k < ω$, then $lg(x) \leq lg(x^k)$; (ii) if $lg(y) < k < ω$ and $x^k = y$, then $x = e$. In particular, the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes results from [3] and [5], where this is proved for free and free Abelian uncountable groups.

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arXiv2016

Independence Logic and Abstract Independence Relations

We continue the work on the relations between independence logic and the model-theoretic analysis of independence, generalizing the results of [15] and [16] to the framework of abstract independence relations for an arbitrary AEC. We give a model-theoretic interpretation of the independence atom and characterize under which conditions we can prove a completeness result with respect to the deductive system that axiomatizes independence in team semantics and statistics.

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