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Gianluca Paolini

Publications and source records attributed to Gianluca Paolini.

At least 19 recordsLinked to original sources
arXiv2026

Homogeneity in Coxeter groups and split crystallographic groups

We prove that affine Coxeter groups, even hyperbolic Coxeter groups and one-ended hyperbolic Coxeter groups are homogeneous in the sense of model theory. More generally, we prove that many (Gromov) hyperbolic groups generated by torsion elements are homogeneous. In contrast, we construct split crystallographic groups that are not homogeneous, and hyperbolic (in fact, virtually free) Coxeter groups that are not homogeneous (or, to be more precise, not $\mathrm{EAE}$-homogeneous). We also prove that, on the other hand, irreducible split crystallographic groups and torsion-generated hyperbolic groups are almost homogeneous. We also prove that finitely generated abelian-by-finite groups are homogeneous if and only if they are profinitely homogeneous, i.e., any tuple of words from the group is profinitely rigid. We use this to deduce that affine Coxeter groups are profinitely homogeneous, a result of independent interest in the profinite context.

math.GR↗
arXiv2026

Around first-order rigidity of Coxeter groups

By the work of Sela, for any free group $F$, the Coxeter group $W_3 = \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$ is elementarily equivalent to $W_3 \ast F$, and so Coxeter groups are not closed under elementary equivalence among finitely generated groups. We study what happens after restricting to models generated by finitely many torsion elements, which we call finitely torsion-generated. We prove that if $(W,S)$ is a Coxeter system whose irreducible components are finite, affine, or non-elementary hyperbolic, and $G$ is finitely torsion-generated and elementarily equivalent to $W$, then $G$ is a Coxeter group. This combines results from [22, 25] with the following hyperbolic result: if $W$ is a hyperbolic Coxeter group and $G$ is finitely torsion-generated and $\mathrm{AE}$-equivalent to $W$, then $G$ is isomorphic to one of finitely many Coxeter groups. We also construct two non-isomorphic hyperbolic Coxeter groups that are $\mathrm{AE}$-equivalent. We then consider first-order torsion-rigidity, meaning that $W$ is the only finitely torsion-generated model of its theory, and prove it for even hyperbolic Coxeter groups and for free products of one-ended or finite hyperbolic Coxeter groups. We conjecture that analogous phenomena hold for all Coxeter groups. Finally, we prove that elementarily equivalent even Coxeter groups are isomorphic, generalizing the analogous result for right-angled Coxeter groups from [11].

math.GR↗
arXiv2026

Acylindrical Hyperbolicity and Pure Conjugating Automorphisms in Graph Products of Groups

We study graph products of groups over defining graphs of arbitrary cardinality from two closely related viewpoints. First, we characterize acylindrical hyperbolicity. If the defining graph is irreducible, has at least two vertices, and has a finite star base, then every parabolically full subgroup is either virtually cyclic or acylindrically hyperbolic. For vertex-full subgroups the finite star base condition is also necessary. In particular, the graph product itself is acylindrically hyperbolic exactly when the graph has a finite star base and the group is not virtually cyclic, or equivalently is not the infinite dihedral group. We also give the corresponding classification for reducible defining graphs. Second, motivated by our earlier work on countable right-angled Coxeter groups, we study pure conjugating automorphisms in the topology of pointwise convergence. We prove that they are topologically generated by finite-support factorwise automorphisms and partial conjugations, and establish closedness, density, exact-equality, and discreteness results. The same finite star base condition that appears in the acylindrical hyperbolicity criterion governs closedness and discreteness in the star-connected case; for arbitrary graphs, its natural refinement is the existence of a finite component witness set. Finally, for graph products of abelian groups we obtain a canonical topological semidirect decomposition of the pure conjugating automorphism group.

math.GR↗
arXiv2026

The lattice of abstract elementary classes of modules

Let $R$ be a ring. We organize the abstract elementary classes whose underlying class is the class of all $R$-modules and whose strong submodel relation lies between the submodule and direct summand relations into a lattice $\mathscr{L}_{R}$, ordered by reverse inclusion. We establish the basic lattice-theoretic properties of $\mathscr{L}_{R}$ and investigate its two natural sublattices, below and above purity. Below purity, we isolate relations defined by first-order pp-formulas for which amalgamation, tameness, and stability hold. Above purity, we introduce relations defined by infinitary pp-formulas and prove a broad stability result. Specializing to abelian groups, we show that the lattice $\mathscr{L}_{\mathbb{Z}}$ has the following properties: it has a strong submodel relation that is not positive syntactic, it contains an uncountable antichain and a strictly increasing proper-class-sized chain, and it has a broad region above purity where amalgamation fails.

math.LO↗
arXiv2026

$μ$-abstract elementary classes of modules

We prove several new results in the theory of $μ$-AECs, focusing mainly on (almost) stability, with the primary objective of undertaking a systematic study of $μ$-AECs of $R$-modules. Our main results are the following. 1. We show that, under suitable syntactic assumptions, all tame $μ$-AECs of $R$-modules (where $R$ is a ring) are almost stable, and are stable if they additionally satisfy a strong amalgamation property. This extends the work of the second author and Shelah [49] to the setting of $μ$-AECs. 2. We then turn to applications to concrete $μ$-AECs of $R$-modules. Our main result in this direction is that $(R$-Mod$, \leq_{pp}^μ)$ has a stable independence relation and is a stable and tame $μ$-AEC, where $\leq_{pp}^μ$ denotes the $μ$-pure submodule relation. We also prove similar stability results for various classes of abelian groups, including the $\aleph_1$-AEC of torsion-free abelian groups with the balanced subgroup relation. Moreover, we prove the almost stability of all $μ$-AECs of modules of the form $(R$-Mod$, \preccurlyeq)$, where $\preccurlyeq$ refines the direct summand relation and satisfies a strong form of coherence. 3. Finally, we study $μ$-AECs of the form $(K, \leq_\oplus)$, where $K$ is a class of pure-injective $R$-modules (note that this is, in general, not an AEC), and use our results to show that, for many natural choices of $K$, the class $(K, \leq_\oplus)$ has a stable independence relation and is therefore stable and tame. We use these results to give a sufficient condition for abstract classes of modules of the form $(K, \leq_{pp})$ to be stable when $K$ is closed under pure-injective envelopes. This generalizes, by a substantially different proof, results of Mazari-Armida [45].

math.LO↗
arXiv2026

Realizing Non-Archimedean Polish Groups as Outer Automorphism Groups

We prove that every non-Archimedean Polish group is topologically isomorphic to the outer automorphism group of a countable discrete group. Furthermore, our construction is Borel. This implies that determining whether two countable discrete groups have topologically isomorphic outer automorphism groups is at least as complicated as classifying non-Archimedean Polish groups up to topological isomorphism, and so in particular not classifiable by countable structures. Our construction is based on the theory of right-angled Coxeter groups. Along the way, we prove results of independent interest on the topological group $\mathrm{Aut}(W)$, for $W$ a countable right-angled Coxeter group. This has purely group-theoretic applications; in particular, we give a graph-theoretic characterization of when the group $\mathrm{Spe}(W)$ of special automorphisms of $W$ coincides with $\mathrm{Inn}(W)$. Combined with previous work, this gives a characterization of when $\mathrm{Aut}(W)$ is inner-by-graph, for $W$ countable.

math.GR↗
arXiv2026

On the Model Theory of Open Incidence Structures: The Rank 2 Case

Taking inspiration from [1, 21, 24], we develop a general framework to deal with the model theory of open incidence structures. In this first paper we focus on the study of systems of points and lines (rank $2$). This has a number of applications, in particular we show that for any of the following classes all the non-degenerate free structures are elementarily equivalent, and their common theory is decidable, strictly stable, and with no prime model: $(k, n)$-Steiner systems (for $2 \leq k < n$); generalised $n$-gons (for $n \geq 3$); $k$-nets (for $k \geq 3$); affine planes; projective Möbius, Laguerre and Minkowski planes.

math.LO↗
arXiv2026

A Galois correspondence for automorphism groups of structures with the Lascar Property

Generalizing the $ω$-categorical context, we introduce a notion, which we call the Lascar Property, that allows for a fine analysis of the topological isomorphisms between automorphism groups of countable saturated structures satisfying this property. In particular, under these assumptions, we exhibit a Galois correspondence between pointwise stabilizers of finitely generated algebraically closed subsets of $M$ and finitely generated algebraically closed subsets of $M$. We use this to characterize the group of automorphisms of $\mathrm{Aut}(M)$, for $M$ a countable saturated model of $\mathrm{ACF}_0$ or an infinite-dimensional $\mathbb{K}$-vector space with $\mathbb{K}$ countable, generalizing a classical result of Evans $\&$ Lascar (1997), while at the same time subsuming the analysis of Paolini (2024) for $ω$-categorical structures with weak elimination of imaginaries.

math.LO↗
arXiv2026

Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism

The paper establishes results following two interconnected directions. 1. Let $G$ be a Roelcke precompact closed subgroup of the group $\mathrm{Sym}(ω)$ of permutations of the natural numbers. Let $\mathrm{Aut}(G)$ denote the group of continuous automorphisms of $G$. Then $\mathrm{Inn}(G)$ is closed in $\mathrm{Aut}(G)$, where $\mathrm{Aut}(G)$ carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~$G$ is oligomorphic, $\+ N_G/G$ is profinite, where $\+ N_G$ denotes the normaliser of~$G$ in $\mathrm{Sym}(ω)$, and the topological group $\mathrm{Out}(G)= \mathrm{Aut}(G)/\mathrm{Inn}(G)$ is totally disconnected, locally compact. 2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy. 2b. Using this method we also show that if $G$ is in such a Borel class, then $\mathrm{Aut}(G)$ is topologically isomorphic to an oligomorphic group, and $\mathrm{Out}(G)$ is profinite.

math.LO↗
arXiv2026

A New Construction Principle

We use the framework of Abstract Elementary Classes ($\mathrm{AEC}$s) to introduce a new Construction Principle $\mathrm{CP}(\mathbf{K},\ast)$, which generalises the Construction Principle of Eklof, Mekler and Shelah and allows for many novel applications beyond the setting of universal algebra. From this we derive, in ZFC, that several uncountably categorical classes of structures are not axiomatisable in the logic $\mathfrak{L}_{\infty,ω_1}$, and, under $V=L$, that they are not axiomatisable in $\mathfrak{L}_{\infty,\infty}$. In particular, our methods apply to: free products of cyclic groups of fixed order, direct sums of a fixed torsion-free abelian group of rank $1$ which is not $\mathbb{Q}$, free $(k,n)$-Steiner systems, and free generalised $n$-gons.

math.LO↗
arXiv2026

On the problem of stability of abstract elementary classes of modules

It is an open problem of Mazari-Armida whether every abstract elementary class of $R$-modules $(\mathbf{K}, \leq_{\mathrm{pure}})$, with $\leq_{\mathrm{pure}}$ the pure submodule relation, is stable. We answer this question in the negative by constructing unstable abstract elementary classes $(\mathbf{K}, \leq_{\mathrm{pure}})$ of torsion-free abelian groups. On the other hand, we prove (in $\mathrm{ZFC}$) that if $R$ is any ring and $(\mathbf{K}, \preccurlyeq)$ is an abstract elementary class of $R$-modules which is $κ$-local (also called $κ$-tame) for some $κ\geq \mathrm{LS}(\mathbf{K}, \preccurlyeq)$, then $(\mathbf{K}, \preccurlyeq)$ is almost stable, where almost stability is a new notion of independent interest that we introduce in this paper, and which is equivalent to the usual notion of stability under the assumption of amalgamation. As a consequence, assuming the existence of a strongly compact cardinal $κ$, we have that every abstract elementary class $(\mathbf{K}, \preccurlyeq)$ of $R$-modules with amalgamation satisfying $κ> \mathrm{LS}(\mathbf{K}, \preccurlyeq)$ is stable.

math.LO↗
arXiv2026

$\aleph_1$-free abelian non-Archimedean Polish groups

An uncountable $\aleph_1$-free group cannot admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed, for example, by the Baer-Specker group $\mathbb{Z}^ω$; more strongly, $\mathbb{Z}^ω$ is separable. In this paper we investigate $\aleph_1$-free abelian non-Archimedean Polish groups. We prove two main results. The first is that there are continuum many separable (and so torsionless, and so $\aleph_1$-free) abelian non-Archimedean Polish groups which are pairwise not topologically isomorphic. The second is that the following four properties are complete co-analytic subsets of the space of closed abelian subgroups of $S_\infty$: separability, torsionlessness, $\aleph_1$-freeness and $\mathbb{Z}$-homogeneity.

math.LO↗
arXiv2026

Procountable groups are not classifiable by countable structures

We prove that topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ expressing that two sequences of reals have a bounded difference is Borel reducible to it. This marks substantial progress on an open problem of Kechris, Nies and Tent (2018): to determine the exact complexity of the isomorphism relation among all non-archimedean Polish groups.

math.LO↗
arXiv2026

The Construction Principle and superstability of free objects in varieties of algebras

We investigate the relationship between the Eklof-Mekler-Shelah Construction Principle for a variety of algebras $\mathbf{V}$ and the question of superstability of the free objects in $\mathbf{V}$, denoted as $\mathcal{F}_\mathbf{V}$. We consider this question in the general setting of AEC-coverings of $\mathcal{F}_\mathbf{V}$, with applications to first-order logic and beyond. Our main result is that if a strong form of the Construction Principle is satisfied, then almost all AEC-covering of $\mathcal{F}_\mathbf{V}$ are unsuperstable. Concrete applications to $R$-modules and varieties of groups are also considered.

math.LO↗
arXiv2026

Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order

In [9] we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper we show that our construction from [9] satisfies several additional properties of interest. We deduce from this that countable torsion-free abelian groups are faithfully Borel complete, in fact, more strongly, we can $\mathfrak{L}_{ω_1, ω}$-interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (equiv., elementary embeddability) among countable models of $\mathrm{Th}(\mathbb{Z}^{(ω)})$ is a complete analytic quasi-order.

math.LO↗
arXiv2025

Elementary properties of free lattices II: Decidability of the universal theory

We continue our work on the model theory of free lattices, solving two of the main open problems from our first paper on the subject. Our main result is that the universal (existential) theory of infinite free lattices is decidable. Our second main result is a proof that finitely generated free lattices are positively distinguishable, as for each $n \geq 1$ there is a positive $\exists \forall$-sentence true in $\mathbf{F}_n$ and false in $\mathbf{F}_{n+1}$. Finally, we show that free lattices are first-order rigid in the class of finitely generated projective lattices, and that a projective lattice has the same existential (universal) theory of an infinite free lattice if and only if it has breadth $> 4$ (i.e., a single existential sentence is sufficient).

math.LO↗
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↗