arXiv subjects

J. de la Nuez González

Publications and source records attributed to J. de la Nuez González.

12 recordsLinked to original sources

arXiv2026

Locally approximating groups of homeomorphisms of manifolds

Let $M$ be a compact, connected manifold of positive dimension, and let $\mathcal G\leq\mathrm{Homeo}(M)$ be locally approximating. If $\mathrm{dim} M\geq 2$, we construct a uniform parameter-free interpretation of standard first-order arithmetic in $\mathcal G$. The same construction applies in dimension one under a component-selection hypothesis, which holds for the usual piecewise-linear and Thompson-group examples, as well as for full homeomorphism and diffeomorphism groups. We also define finite word evaluation and membership in finitely generated subgroups, and derive consequences for finite generation, prime models, and quasi-finite axiomatizability. In dimension one, we construct countable recursively saturated locally approximating groups of homeomorphisms of the interval and circle which do not interpret standard arithmetic, even with parameters. Finally, we prove an action rigidity result: if $M$ is closed and admits a combinatorial triangulation, $\mathcal G\leq\mathrm{Homeo}(M)$ is locally approximating, and $H\equiv\mathcal G$ admits a locally approximating action on a compact connected topological manifold $N$, then $N$ is homeomorphic to $M$.

math.GR↗

arXiv2026

Minimal and intrinsic topologies on monoids of elementary embeddings

To every $ω$-categorical structure $\mathrm{Aut}M$ 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 $τ_{\mathrm{pw}}$. We investigate the relation of $τ_{\mathrm{pw}}$ to other topologies on these spaces: in particular, when $τ_{\mathrm{pw}}$ is minimal, i.e. does not admit any strictly coarser Hausdorff semigroup topology. A common method to prove minimality of $τ_{\mathrm{pw}}$ on $\mathrm{EEmb}(M)$ is to show that it coincides with the algebraically defined semigroup Zariski topology $τ_{\mathrm{Z}}$. We show that $τ_{\mathrm{pw}}$ differs from $τ_{\mathrm{Z}}$ 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, $τ_{\mathrm{pw}}$ 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 $τ_{\mathrm{pw}}$ 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 $τ_{\mathrm{mp}}$ is minimal on $\mathrm{EEmb}(M)$, equals $τ_{\mathrm{Z}}$, and is strictly coarser than $τ_{\mathrm{pw}}$.

math.LO↗

arXiv2026

Average Numbers of Homomorphisms to Random Modules over Free Group Algebras

Let $F$ be a finitely generated free group, and let $L$ be a finitely presented $\mathbb{F}_q[F]$-module. We study the average number $Λ_L(n)$ of $\mathbb{F}_q[F]$-module homomorphisms from $L$ to an $\mathbb{F}_q[F]$-module of dimension $n$ over $\mathbb{F}_q$. We show that, for all sufficiently large $n$, the quantity $Λ_L(n)$ is given by a rational function of $q^n$ and satisfies \[ Λ_L(n) = q^{χ(L)n} + \sum_{N\in A(L)} q^{χ(L/N)n}\bigl(1+O(q^{-n})\bigr), \] where $χ(L)$ denotes the Euler characteristic of $L$, and $A(L)$ is the set of nonzero $\mathbb{F}_q[F]$-submodules $N$ of $L$ that have no nonzero free quotients. Our proof is based on a theory of partial modules that may be of independent interest and have further applications.

math.GR↗

arXiv2026

Two generalisations of sharp k-transitivity

An action $U \curvearrowleft G$ of a group $G$ on a set $U$ is sharply $k$-transitive if, for any two $k$-tuples $\bar{a}, \bar{b} \in U^k$ of distinct elements, there is a unique $g \in G$ with $\bar{a} \cdot g = \bar{b}$. We consider two generalisations of this. Firstly, given $Θ\leq \mathbb{S}_k$, we define a sharply $Θ$-transitive action $U \curvearrowleft G$ to be a $k$-set-transitive action where the restricted action on each $k$-set of its setwise-stabiliser is isomorphic to the permutation action $\mathbf{k} \curvearrowleft Θ$. An action is sharply $\mathbb{S}_k$-transitive iff it is sharply $k$-transitive. We characterise for which $Θ\leq \mathbb{S}_k$ there is a sharply $Θ$-transitive action on an infinite set, and show that if such an action exists, then the acting group $G$ can be taken to be a finitely generated non-abelian virtually free group. As a consequence, we obtain for $k = 2, 3$ the first examples of non-split finitely-presented groups admitting sharply $k$-transitive actions on an infinite set, answering a question of André and Tent, and we obtain a strengthening of the well-known result of Tits that no group admits a sharply $k$-transitive action on an infinite set for $k \geq 4$. Secondly, we generalise sharp $k$-transitivity to relational structures. Given an action $\mathcal{M} \curvearrowleft G$ of a group $G$ on a relational structure $\mathcal{M}$, we say that the action is sharply $k$-homogeneous if, for any two $k$-tuples $\bar{a}, \bar{b}$ of distinct elements of $\mathcal{M}$ where $\bar{a} \mapsto \bar{b}$ is an isomorphism, there is a unique $g \in G$ with $\bar{a} \cdot g = \bar{b}$. We show that, for $1 \leq k \leq 3$, a wide range of countable ultrahomogeneous structures admit sharply $k$-homogeneous actions by finitely generated non-abelian virtually free groups, answering a question of Cameron from 1990.

math.GR↗

arXiv2026

Set theory, logic, and homeomorphism groups of manifolds

We investigate the relationship between axiomatic set theory and the first-order theory of homeomorphism groups of manifolds in the language of group theory, concentrating on first-order rigidity and type versus conjugacy. We prove that under the axiom of constructibility (i.e.~{V=L}), homeomorphism groups of arbitrary connected manifolds are first-order rigid, and that the conjugacy class of a homeomorphism of a manifold is determined by its type. In contradistinction, under the regularity hypothesis that every projective set of reals has the Baire property, we show that in all dimensions greater than one there exist pairs of noncompact, connected manifolds whose homeomorphism groups are elementarily equivalent but which are not homeomorphic. We also show, under the same Baire-property hypothesis, that every manifold of positive dimension admits pairs of homeomorphisms with the same type which are not conjugate to each other. Projective determinacy implies the Baire-property hypothesis, so the corresponding consequences under PD follow immediately. Finally, we show that infinitary formulas do determine conjugacy classes of homeomorphisms and homeomorphism types of manifolds; specifically, the conjugacy class of a homeomorphism of an arbitrary manifold is determined by a single $L_{ω_1ω}$ formula. Similarly, the homeomorphism type of an arbitrary connected manifold is determined by a single $L_{ω_1ω}$ sentence.

math.GT↗

arXiv2026

Elementary equivalence and diffeomorphism groups of smooth manifolds

Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and $M$ and $N$ are $C^r$--diffeomorphic. This strengthens a previously known result by Takens and Filipkiewicz, which asserts that for integer regularities, a group isomorphism between diffeomorphism groups of closed manifolds necessarily arises from a diffeomorphism of the underlying manifolds. We prove an analogous result for groups of diffeomorphisms preserving smooth volume forms, in dimension at least two.

math.GR↗

arXiv2025

Uniform first order interpretation of the second order theory of countable groups of homeomorphisms

We show that the first order theory of the homeomorphism group of a compact manifold interprets the full second order theory of countable groups of homeomorphisms of the manifold. The interpretation is uniform across manifolds of bounded dimension. As a consequence, many classical problems in group theory and geometry (e.g.~the linearity of mapping classes of compact $2$--manifolds) are encoded as elementary properties of homeomorphism groups of manifolds. Furthermore, the homeomorphism group uniformly interprets the Borel and projective hierarchies of the homeomorphism group, which gives a characterization of definable subsets of the homeomorphism group. Finally, we prove analogues of Rice's Theorem from computability theory for homeomorphism groups of manifolds. As a consequence, it follows that the collection of sentences that isolate the homeomorphism group of a particular manifold, or that isolate the homeomorphism groups of manifolds in general, is not definable in second order arithmetic, and that membership of particular sentences in these collections cannot be proved in ZFC.

math.GR↗

arXiv2025

Group topologies on groups of bi-absolutely continuous homeomorphisms

The group of homeomorphisms of the closed interval that are absolutely continuous and have an absolutely continuous inverse was shown by Solecki to admit a natural Polish group topology $τ_{ac}$. We show that, under mild conditions on a compact space endowed with a finite Borel measure such a topology can be defined on the subgroup of the homeomorphism group consisting of those elements $g$ such that $g$ and $g^{-1}$ preserve the class of null sets. We use a probabilistic argument to show that in the case of a compact topological manifold equipped with an Oxtoby-Ulam measure, as well as in that of the Cantor space endowed with some natural Borel measures there is no group topology between $τ_{ac}$ and the restriction $τ_{co}$ of the compact-open topology. In fact, we show that any separable group topology strictly finer than $τ_{co}$ must be also finer than $τ_{ac}$. For one-dimensional manifolds we also show that $τ_{co}$ and $τ_{ac}$ are the only Hausdorff group topologies coarser than $τ_{ac}$, and one can read our result as evidence for the non-existence of a good notion of regularity between continuity and absolute continuity. We also show that while Solecki's example is not Roelcke precompact, the group of bi-absolutely continuous homeomorphisms of the Cantor space endowed with the measure given by the Fräissé limit of the class of measured boolean algebras with rational probability measures is Roelcke precompact.

math.GN↗

arXiv2025

First order rigidity of homeomorphism groups of manifolds

For every compact, connected manifold $M$, we prove the existence of a sentence $ϕ_M$ in the language of groups such that the homeomorphism group of another compact manifold $N$ satisfies $ϕ_M$ if and only if $N$ is homeomorphic to $M$. We prove the analogous statement for groups of homeomorphisms preserving an Oxtoby--Ulam probability measure.

math.GR↗

arXiv2024

The compact-open topology on the diffeomorphism or homeomorphism group of a smooth manifold without boundary is minimal in almost all dimensions

We show that for any connected smooth manifold $M$ of dimension different from $3$ the restriction of the compact-open topology to the diffeomorphism group of $M$ is minimal, i.e. the group does not admit a strictly coarser Hausdorff group topology. This implies the minimality of the compact-open topology on the homeomorphism group of $M$ in all dimensions different from $3$ and $4$. In those cases for which in addition to all of this automatic continuity is known to hold, such as when $M$ is closed, one can conclude that the compact-open topology is the unique separable Hausdorff group topology on the homeomorphism group.

math.GT↗

arXiv2023

The model theory of the curve graph

In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. In particular, we investigate the Ivanov Metaconjecture from the point of view of model theory, and more broadly we seek to answer the general question: why does the curve graph of a surface play such a central role in the study of surfaces and mapping class groups? More specifically, we consider a surface $Σ$ of finite type and its curve graph $\mathcal C(Σ)$, and we investigate its first-order theory in the language of graph theory. Crucially, $\mathcal C(Σ)$ is bi-interpretable with a certain object called the augmented Cayley graph of the mapping class group of the surface. We use this bi-interpretation to prove that the theory of the curve graph is $ω$--stable, to compute its Morley rank, and to show that it has quantifier elimination with respect to the class of $\forall\exists$--formulae. We also show that many of the complexes which are naturally associated to a surface are interpretable in $\mathcal C(Σ)$. This shows that these complexes are all $ω$--stable and admit certain a priori bounds on their Morley ranks. We are able to use Morley ranks to prove that various complexes are not bi--interpretable with the curve graph. As a consequence of quantifier elimination, we show that algebraic intersection number is not definable in the first order theory of the curve graph. Finally, we prove that the curve graph of a surface enjoys a novel phenomenon that we call interpretation rigidity. That is, if surfaces $Σ_1$ and $Σ_2$ admits curve graphs that are mutually interpretable, then $Σ_1$ and $Σ_2$ are homeomorphic to each other. Along the way, numerous technical results are obtained.

math.GT↗

arXiv2022

The compact-open topology on the homeomorphism group of a surface without boundary is minimal

We show that the homeomorphism group of a surface without boundary does not admit a Hausdorff group topology strictly coarser than the compact-open topology. In combination with known automatic continuity results, this implies that the compact-open topology is the unique Hausdorff separable group topology on the group if the surface is closed or the complement in a closed surface of either a finite set or the union of a finite set and a Cantor set.

math.GT↗