arXiv subjects

Paolo Marimon

Publications and source records attributed to Paolo Marimon.

8 recordsLinked to original sources

arXiv2026

All mixed identities are singular in groups with no algebraicity

We show that if a group admits an action with no algebraicity then all of its mixed identities are singular. Previously, such groups were only known to be lawless by a theorem of Ab\'{e}rt. Our result confirms, in particular, a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups. It thereby not only subsumes numerous results from the literature in a simple uniform theorem, but also settles the question for prominent groups for which the conjecture was an open problem, such as the automorphism group of $(\mathbb{Q}; <)$. It also applies outside the oligomorphic context, e.g. to much-investigated groups such as Thompson's groups $F, T$, and $V$, to Grigorchuk's group, and to the homeomorphism groups of any manifold of dimension $\geq 1$. More generally, we prove that all mixed identities of a group are singular as long as it has an action satisfying certain geometric conditions. This additionally covers, for example, the infinite-dimensional general and projective linear groups.

math.GR↗

arXiv2026

Minimal and intrinsic topologies on monoids of elementary embeddings

To every $\omega$-categorical structure $M$ one can associate two spaces of symmetries which determine the structure up to first-order bi-interpretability: the topological group $\mathrm{Aut}(M)$ of its automorphisms and the topological monoid $\mathrm{EEmb}(M)$ of its elementary embeddings, both equipped with the topology of pointwise convergence $\tau_{\mathrm{pw}}$. We investigate the relation of $\tau_{\mathrm{pw}}$ to other topologies on these spaces: in particular, when $\tau_{\mathrm{pw}}$ is minimal, i.e. does not admit any strictly coarser Hausdorff semigroup topology. A common method to prove minimality of $\tau_{\mathrm{pw}}$ on $\mathrm{EEmb}(M)$ is to show that it coincides with the algebraically defined semigroup Zariski topology $\tau_{\mathrm{Z}}$. We show that $\tau_{\mathrm{pw}}$ differs from $\tau_{\mathrm{Z}}$ on $\mathrm{EEmb}(M)$ whenever $\mathrm{Aut}(M)$ has a non-trivial centre. In spite of this, we then prove that whenever algebraic closure on $M$ is modular, then $\tau_{\mathrm{pw}}$ is minimal on $\mathrm{EEmb}(M)$. This covers, for example, countable vector spaces and projective spaces over finite fields. Turning to $\mathrm{Aut}(M)$, we describe the semigroup topologies coarser than $\tau_{\mathrm{pw}}$ on the automorphism groups of structures for which algebraic independence satisfies independent 3-amalgamation. We conclude by proving that for the real and the rational Urysohn space and sphere, the metric pointwise topology $\tau_{\mathrm{mp}}$ is minimal on $\mathrm{EEmb}(M)$, equals $\tau_{\mathrm{Z}}$, and is strictly coarser than $\tau_{\mathrm{pw}}$.

math.LO↗

arXiv2025

Taking model-complete cores

A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; a core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, if they exist, they are unique. Moreover, $\omega$-categorical theories always have a core companion, which is also $\omega$-categorical. We show that many model-theoretic properties, such as stability, $\mathrm{NIP}$, simplicity, and $\mathrm{NSOP}_k$ for ${k\in\mathbb{N}_{>0}}$, are preserved by moving to the core companion of a complete theory. On the other hand, we show that the classes of theories of structures interpretable over $({\mathbb N};=)$ and over $({\mathbb Q};<)$ are both not closed under taking core companions. The first class is contained in the class of theories of $\omega$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We conjecture the two classes to be equal. To support our conjecture we prove that all structures in Lachlan's class are trace definable in $(\mathbb{N}; =)$, confirming a conjecture of Walsberg.

math.LO↗

arXiv2025

A guide to topological reconstruction on endomorphism monoids and polymorphism clones

Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. For example, the automorphism group of a structure is, on top of being a group, a topological group when equipped with the topology of pointwise convergence. In some cases, the algebraic structure of such space alone is sufficiently rich to determine its topology (under some requirements on the topology). For automorphism groups, the problem of when this happens has been actively pursued over the last 40 years. With the exception of some early work of Lascar, the analogue of this problem for endomorphism monoids and polymorphism clones has only received attention in the past 15 years. In this guide, we survey the current state of affairs in this relatively young line of research. We moreover use this opportunity to polish several existing results and to extend them beyond what was hitherto known.

math.LO↗

arXiv2025

Minimal operations over permutation groups

We classify the possible types of minimal operations above an arbitrary permutation group. Above the trivial group, a theorem of Rosenberg yields that there are five types of minimal operations. We show that above any non-trivial permutation group there are at most four such types. Indeed, except above Boolean groups acting freely on a set, there are only three. In particular, this is the case for oligomorphic permutation groups, for which we improve a result of Bodirsky and Chen by showing one of the types in their classification does not exist. Building on these results, we answer three questions of Bodirsky that were previously open.

math.RA↗

arXiv2025

When invariance implies exchangeability (and applications to invariant Keisler measures)

We study the problem of when, given a countable homogeneous structure $M$ and a space $S$ of expansions of $M$, every $\mathrm{Aut}(M)$-invariant probability measure on $S$ is exchangeable (i.e. invariant under all permutations of the domain). We show, for example, that if $M$ is a finitely bounded homogeneous $3$-hypergraph with free amalgamation (including the generic tetrahedron-free $3$-hypergraph), all $\mathrm{Aut}(M)$-invariant random expansions by graphs are exchangeable. Moreover, we extend and recover both the work of Angel, Kechris, and Lyons on invariant random orderings and some of the work of Crane and Towsner, and Ackerman on relative exchangeability. In the second part of the paper, we apply our results to the study of invariant Keisler measures, which we prove to be particular invariant random expansions. Thus, we describe the spaces of invariant Keisler measures of various homogeneous structures, obtaining the first results of this kind since the work of Albert and Ensley. We also show there are $2^{\aleph_0}$ supersimple homogeneous ternary structures for which there are non-forking formulas which are universally measure zero.

math.LO↗

arXiv2023

Invariant Keisler measures for omega-categorical structures

A recent article of Chernikov, Hrushovski, Kruckman, Krupinski, Moconja, Pillay and Ramsey finds the first examples of simple structures with formulas which do not fork over $\emptyset$ but are universally measure zero. In this article we give the first known simple $ω$-categorical counterexamples. These happen to be various $ω$-categorical Hrushovski constructions. Using a probabilistic independence theorem from Jahel and Tsankov, we show how simple $ω$-categorical structures where the forking ideal and the universally measure zero ideal coincide must satisfy a stronger version of the independence theorem.

math.LO↗

arXiv2022

On the non-measurability of $ω$-categorical Hrushovski constructions

We study $ω$-categorical $MS$-measurable structures. Our main result is that a class of $ω$-categorical Hrushovski constructions, supersimple of finite $SU$-rank is not $MS$-measurable. These results complement the work of Evans on a conjecture of Macpherson and Elwes. In contrast to Evans' work, our structures may satisfy independent $n$-amalgamation for all $n$. We also prove some general results in the context of $ω$-categorical $MS$-measurable structures. Firstly, in these structures, the dimension in the $MS$-dimension-measure can be chosen to be $SU$-rank. Secondly, non-forking independence implies a form of probabilistic independence in the measure. The latter follows from more general unpublished results of Hrushovski, but we provide a self-contained proof.

math.LO↗