arXiv subjects

Gianluca Paolini

Publications and source records attributed to Gianluca Paolini.

At least 37 records · Page 2Linked to original sources

arXiv2025

Borel completeness of Tits buildings with no rank 3 residues of spherical type

We prove that, for every Coxeter diagram $D$ with no rank $3$ residues of spherical type and such that $D$ has not only edges labelled by $2$, the space of countable (Tits) buildings of type $D$ is Borel complete, that is, classifying countable buildings of type $D$ up to isomorphism is as hard as classifying countable graphs up to isomorphism. In particular, for every $n\geq 3$, the space of countable generalised $n$-gons is Borel complete.

math.LO↗

arXiv2025

Elementary properties of free lattices III: Undecidability of the full theory

In [6] we proved that the universal theory of infinite free lattices is (algorithmically) decidable, leaving open the problem of decidability of the full theory of an (infinite) free lattice. We solve this problem by proving that, for every cardinal $κ\geq 3$, the first-order theory of the free lattice $\mathbf{F}_κ$ is undecidable.

math.LO↗

arXiv2025

First-order aspects of Artin groups

We prove several results on the model theory of Artin groups, focusing on Artin groups which are ``far from right-angled Artin groups''. The first result is that if $\mathcal{C}$ is a class of Artin groups whose irreducible components are acylindrically hyperbolic and torsion-free, then the model theory of Artin groups of type $\mathcal{C}$ reduces to the model theory of its irreducible components. The second result is that the problem of superstability of a given non-abelian Artin group $A$ reduces to certain dihedral parabolic subgroups of $A$ being $n$-pure in $A$, for certain large enough primes $n \in \mathbb{N}$. The third result is that two spherical Artin groups are elementary equivalent if and only if they are isomorphic. Finally, we prove that the affine Artin groups of type $\tilde{A}_n$, for $n \geq 4$, can be distinguished from the other simply laced affine Artin groups using existential sentences; this uses homology results of independent interest relying on the recent proof of the $K(π, 1)$ conjecture for affine Artin groups.

math.LO↗

arXiv2025

On the existence of uncountable Hopfian and co-Hopfian abelian groups

We deal with the problem of existence of uncountable co-Hopfian abelian groups and (absolute) Hopfian abelian groups. Firstly, we prove that there are no co-Hopfian reduced abelian groups $G$ of size $< \mathfrak{p}$ with infinite $\mathrm{Tor}_p(G)$, and that in particular there are no infinite reduced abelian $p$-groups of size $< \mathfrak{p}$. Secondly, we prove that if $2^{\aleph_0} < λ< λ^{\aleph_0}$, and $G$ is abelian of size $λ$, then $G$ is not co-Hopfian. Finally, we prove that for every cardinal $λ$ there is a torsion-free abelian group $G$ of size $λ$ which is absolutely Hopfian, i.e., $G$ is Hopfian and $G$ remains Hopfian in every forcing extensions of the universe.

math.GR↗

arXiv2025

Profinite rigidity of crystallographic groups arising from Lie theory

We prove that every finite direct product of crystallographic groups arising from an irreducible root system (in the sense of Lie theory) is profinitely rigid (equiv. first-order rigid). This is a generalization of recent proofs of profinite rigidity of affine Coxeter groups [1, 7, 22]. Our proof uses model theory.

math.GR↗

arXiv2025

The construction principle and non homogeneity of uncountable relatively free groups

In [11] Sklinos proved that any uncountable free group is not $\aleph_1$-homogenenous. This was later generalized by Belegradek in [1] to torsion-free residually finite relatively free groups, leaving open whether the assumption of residual finiteness was necessary. In this paper we use methods arising from the classical analysis of relatively free groups in infinitary logic to answer Belegradek's question in the negative. Our methods are general and they also applications in varieties with torsion, for example we show that if $V$ contains a non-solvable group, then any uncountable $V$-free group is not $\aleph_1$-homogenenous.

math.LO↗

arXiv2024

Profinite rigidity of affine Coxeter groups

We prove that the irreducible affine Coxeter groups are first-order rigid and deduce from this that they are profinitely rigid in the absolute sense. We then show that the first-order theory of any irreducible affine Coxeter group does not have a prime model. Finally, we prove that universal Coxeter groups of finite rank are homogeneous, and that the same applies to every hyperbolic (in the sense of Gromov) one-ended right-angled Coxeter group.

math.GR↗

arXiv2024

The Isomorphism Problem for Oligomorphic Groups with Weak Elimination of Imaginaries

In [21] it was asked if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups. We prove that under the assumption of weak elimination of imaginaries this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of $\mathrm{Aut}(\mathrm{GL}(V))$ for any vector space $V$ of dimension $\aleph_0$ over a finite field, in affinity with the classical description for finite dimensional spaces due to Schreier and van der Waerden.

math.LO↗

arXiv2024

Elementary Properties of Free Lattices

We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive $\exists \forall$-sentence true in $\mathbf F_3$ and false in $\mathbf F_4$. Secondly, we show that every model of $\mathrm{Th}(\mathbf F_n)$ admits a canonical homomorphism into the profinite-bounded completion $\mathbf H_n$ of $\mathbf F_n$. Thirdly, we show that $\mathbf H_n$ is isomorphic to the Dedekind-MacNeille completion of $\mathbf F_n$, and that $\mathbf H_n$ is not positively elementarily equivalent to $\mathbf F_n$, as there is a positive $\forall\exists$-sentence true in $\mathbf H_n$ and false in $\mathbf F_n$. Finally, we show that $\mathrm{DM}(\mathbf F_n)$ is a retract of $\mathrm{Id}(\mathbf F_n)$ and that for any lattice $\mathbf K$ which satisfies Whitman's condition $\mathrm{(W)}$ and which is generated by join prime elements, the three lattices $\mathbf K$, $\mathrm{DM}(\mathbf K)$, and $\mathrm{Id}(\mathbf K)$ all share the same positive universal first-order theory.

math.LO↗

arXiv2024

Computable Scott sentences and the weak Whitehead problem for finitely presented groups

We prove that if $A$ is a computable Hopfian finitely presented structure, then $A$ has a computable $d$-$Σ_2$ Scott sentence if and only if the weak Whitehead problem for $A$ is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable $d$-$Σ_2$ Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its $\exists^+$-types, a question which arose in a different context.

math.LO↗

arXiv2023

Anti-Classification Results for Rigidity Conditions in Abelian and Nilpotent Groups

Relying on the techniques and ideas from our recent paper [13], we prove several anti-classification results for various rigidity conditions in countable abelian and nilpotent groups. We prove three main theorems: (1) the rigid abelian groups are complete co-analytic in the space of countable torsion-free abelian groups ($\mathrm{TFAB}_ω$); (2) the Hopfian groups are complete co-analytic in $\mathrm{TFAB}_ω$; (3) the co-Hopfian groups are complete co-analytic in the space of countable $2$-nilpotent groups. In combination with our result from [13, S5], which shows that the endo-rigid abelian groups are complete co-analytic in $\mathrm{TFAB}_ω$, this shows that four major notions of rigidity from (abelian) group theory are as complex as possible as co-analytic problems. Further, the second and third theorem above solve two open questions of Thomas from [18], who asked this for the space of all countable groups. We leave open the question of whether the co-Hopfian mixed abelian groups are complete co-analytic in the space of countable abelian groups, but we reduce the problem to a concrete question on profinite groups, showing that if $G$ is a countable co-Hopfian abelian reduced group, then, for every prime number $p$, the torsion subgroup $\mathrm{Tor}_p(G)$ of $G$ is finite and $G$ embeds in the profinite group $ \prod_{p \in \mathbb{P}} \mathrm{Tor}_p(G)$.

math.LO↗

arXiv2023

Torsion-Free Abelian Groups are Borel Complete

We prove that the Borel space of torsion-free Abelian groups with domain $ω$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.

math.LO↗

arXiv2022

First-Order Aspects of Coxeter Groups

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show to be essentially the Coxeter groups of affine type. Secondly, we characterize the Coxeter groups of finite rank which are domains, a central assumption in the theory of algebraic geometry over groups, which in many respects (e.g. $λ$-stability) reduces the model theory of a given Coxeter system to the model theory of its associated irreducible components. In the second part of the paper we move to specific definability questions in right-angled Coxeter groups (RACGs) and $2$-spherical Coxeter groups. In this respect, firstly, we prove that RACGs of finite rank do not have proper elementary subgroups which are Coxeter groups, and prove further that reflection independent ones do not have proper elementary subgroups at all. Secondly, we prove that if the monoid $Sim(W, S)$ of $S$-self-similarities of $W$ is finitely generated, then $W$ is a prime model of its theory. Thirdly, we prove that in reflection independent RACGs of finite rank the Coxeter elements are type-determined. We then move to $2$-spherical Coxeter groups, proving that if $(W, S)$ is irreducible, $2$-spherical even and not affine, then $W$ is a prime model of its theory, and that if $W_Γ$ and $W_Θ$ are as in the previous sentence, then $W_Γ$ is elementary equivalent to $W_Θ$ if and only if $Γ\cong Θ$, thus solving the elementary equivalence problem for most of the $2$-spherical Coxeter groups. In the last part of the paper we focus on model theoretic applications of the notion of reflection length from Coxeter group theory, proving in particular that affine Coxeter groups are not connected.

math.LO↗

arXiv2022

Computable Scott Sentences for Quasi-Hopfian Finitely Presented Structures

We prove that every quasi-Hopfian finitely presented structure $A$ has a $d$-$Σ_2$ Scott sentence, and that if in addition $A$ is computable and $Aut(A)$ satisfies a natural computable condition, then $A$ has a computable $d$-$Σ_2$ Scott sentence. This unifies several known results on Scott sentences of finitely presented structures and it is used to prove that other not previously considered algebraic structures of interest have computable $d$-$Σ_2$ Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable $d$-$Σ_2$ Scott sentence, as well as any strongly rigid Coxeter group of finite rank. Finally, we show that the free projective plane of rank $4$ has a computable $d$-$Σ_2$ Scott sentence, thus exhibiting a natural example where the assumption of quasi-Hopfianity is used (since this structure is not Hopfian).

math.LO↗

arXiv2020

Invariant universality for projective planes

We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the second author to show that these equivalence relations are invariantly universal, in the sense of [3], and thus in particular complete analytic. We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly considered in category theory.

math.LO↗

arXiv2020

First-Order Model Theory of Free Projective Planes

We prove that the theory of open projective planes is complete and strictly stable, and infer from this that Marshall Hall's free projective planes $(π^n : 4 \leq n \leq ω)$ are all elementary equivalent and that their common theory is strictly stable and decidable, being in fact the theory of open projective planes. We further characterize the elementary substructure relation in the class of open projective planes, and show in particular that $(π^n : 4 \leq n \leq ω)$ is an elementary chain. We then prove that the theory of open projective planes does not have a prime model, that it has elimination of quantifiers down to Boolean combinations of existential formulas, and that it is not model complete. Finally, we characterize the forking independence relation in models of the theory and prove that the $π^n$'s ($4 \leq n \leq ω)$ are strongly type-homogeneous.

math.LO↗

arXiv2020

Strongly Minimal Steiner Systems I: Existence

A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner $k$-system (for $k \geq 2$) is a linear space such that each line has size exactly $k$. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (bi-interpretable) vocabulary $τ$ with a single ternary relation $R$. We prove that for every integer $k$ there exist $2^{\aleph_0}$-many integer valued functions $μ$ such that each $μ$ determines a distinct strongly minimal Steiner $k$-system $\mathcal{G}_μ$, whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.

math.LO↗

arXiv2020

A New $ω$-Stable Plane

We use a variation on Mason's $α$-function as a pre-dimension function to construct a not one-based $ω$-stable plane $P$ (i.e. a simple rank $3$ matroid) which does not admit an algebraic representation (in the sense of matroid theory) over any field. Furthermore, we characterize forking in $Th(P)$, we prove that algebraic closure and intrinsic closure coincide in $Th(P)$, and we show that $Th(P)$ fails weak elimination of imaginaries, and has Morley rank $ω$.

math.LO↗