arXiv subjects

Jeroen Winkel

Publications and source records attributed to Jeroen Winkel.

11 recordsLinked to original sources

arXiv2026

An unusual example of a universal automorphism group

Let $M$ be a Fra\"{i}ss\'{e} structure (a countably infinite ultrahomogeneous structure). We refer to the class of structures embeddable in $M$ as the $\omega$-age of $M$. We consider the following two properties of $M$: we say that $M$ has a universal automorphism group if, for each $A$ in the $\omega$-age of $M$, there is an embedding $\textrm{Aut}(A) \to \textrm{Aut}(M)$, and we say that $M$ has group-extensible $\omega$-age if, for each $A$ in the $\omega$-age of $M$, there is an embedding $A \to M$ such that each automorphism of the image extends to an automorphism of $M$ and the extension map preserves group composition. It is immediate that if $M$ has group-extensible $\omega$-age, then $M$ has a universal automorphism group. We give an example of a Fra\"{i}ss\'{e} structure with a universal automorphism group whose $\omega$-age is not group-extensible, showing that the above two properties are not equivalent.

math.LO↗

arXiv2026

Determining the normal subgroups of the automorphism groups of ultrahomogeneous structures via stabilisers

We show the simplicity of the automorphism groups of the generic $n$-hypertournament and the semigeneric tournament, and determine the normal subgroups of the automorphism groups of several other ultrahomogeneous oriented graphs. We also give a new proof of the simplicity of the automorphism group of the dense $\frac{2\pi}{n}$-local order $\mathbb{S}(n)$ for $n \geq 2$ (a result due to Droste, Giraudet and Macpherson). Previous techniques of Li, Macpherson, Tent and Ziegler involving stationary weak independence relations (SWIRs) cannot be applied directly to these structures; our approach involves applying these techniques to a certain expansion of each structure, where the expansion has a SWIR and its automorphism group is isomorphic to a stabiliser subgroup of the automorphism group of the original structure.

math.LO↗

arXiv2026

Structured sunflowers and canonical Ramsey properties

A first-order structure $M$ is said to have the infinite sunflower property if, for each $k \in \mathbb{N}_+$ and each structure $M' \cong M$ whose elements are $k$-sets, there is $S \subseteq M'$, $S \cong M$, such that $S$ is a sunflower: a collection of sets such that each pair of elements has the same intersection. A class $\mathcal{K}$ of finite structures is said to have the finite sunflower property if for all $k \in \mathbb{N}_+$ and $B \in \mathcal{K}$, there is $C \in \mathcal{K}$ such that any structure $C' \cong C$ whose elements consist of $k$-sets contains a copy of $B$ which is a sunflower. These two notions were introduced by Ackerman, Karker and Mirabi in a recent paper, and give a structural generalisation of the well-known Erd\H{o}s-Rado sunflower lemma for sets. We show two results for countable ultrahomogeneous relational structures with strong amalgamation: first, the infinite sunflower property is equivalent to the canonical infinite point-Ramsey property; second, a certain strengthening of the canonical finite point-Ramsey property implies the finite sunflower property. (Here, "canonical" refers to statements analogous to the Erd\H{o}s-Rado canonical Ramsey theorem, involving colourings with infinitely many colours.) We also show that all free amalgamation classes with a single vertex isomorphism-type have the finite sunflower property, as do many classes of finite metric spaces, and we give a variety of further examples and observations.

math.CO↗

arXiv2025

Embeddings into the generic poset

Let M be the generic poset, defined as the Fraïssé limit of the class of finite posets. We show that every countably infinite poset A can be embedded with coinfinite image into M so that each automorphism of the image of A extends uniquely to an automorphism of M.

math.LO↗

arXiv2025

Group-extensive embeddings into Fra\"iss\'e structures and stationary weak independence relations

Let $M$ be a Fra\"iss\'e structure (a countably infinite ultrahomogeneous structure). We call an embedding $f : A \to M$ group-extensive if each automorphism of its image extends to an automorphism of $M$, where the extension map respects composition. We say that $M$ has group-extensible $\omega$-age if each substructure admits a group-extensive embedding into $M$. We investigate the relationship between the following two properties: the presence of a stationary weak independence relation (SWIR) on $M$, and group-extensibility of the $\omega$-age of $M$. We show that linearly ordered Fra\"iss\'e structures with a SWIR have group-extensible $\omega$-age, but also we give examples of Fra\"iss\'e structures where only one of the two properties holds. Finally, we consider whether a wide range of examples of Fra\"iss\'e structures have group-extensible $\omega$-age or a finite SWIR expansion, including all countably infinite ultrahomogeneous oriented graphs (with one exception).

math.LO↗

arXiv2024

Infinite Hat Problems and Large Cardinals

Picture countably many logicians all wearing a hat in one of $κ$-many colours. They each get to look at finitely many other hats and afterwards make finitely many guesses for their own hat's colour. For which $κ$ can the logicians guarantee that at least one of them guesses correctly? This will be the archetypical hat problem we analyse and solve here. We generalise this by varying the amount of logicians as well as the number of allowed guesses and describe exactly for which combinations the logicians have a winning strategy. We also solve these hat problems under the additional restriction that their vision is restrained in terms of a partial order. Picture e.g.~countably many logicians standing on the real number line and each logician is only allowed to look at finitely many others in front of them. In many cases, the least $κ$ for which the logicians start losing can be described by an instance of the free subset property which in turn is connected to large cardinals. In particular, $\mathrm{ZFC}$ can sometimes not decide whether or not the logicians can win for every possible set of colours.

math.LO↗

arXiv2023

Dynamical propagation and Roe algebras of warped spaces

Given a non-singular action $\Gamma \curvearrowright (X,\mu)$, we define the $*$-algebra $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and only depends on the measure class of $\mu$. We prove that the algebraic crossed product $L^{\infty}X \rtimes_{\rm alg} \Gamma$ surjects onto $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ and that this surjection is a $\ast$-isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones.

math.OA↗

arXiv2022

Coarse fixed point properties

We investigate fixed point properties for isometric actions of topological groups on a wide class of metric spaces, with a particular emphasis on Hilbert spaces. Instead of requiring the action to be continuous, we assume that it is ``controlled", i.e. compatible with respect to some natural left-invariant coarse structure. For locally compact groups, we prove that these coarse fixed point properties are equivalent to the usual ones, defined for continuous actions. We deduce generalisations of two results of Gromov originally stated for discrete groups. For Polish groups with bounded geometry (in the sense of Rosendal), we prove a version of Serre's theorem on the stability of coarse property FH under central extensions. As an application we prove that the group $\text{Homeo}^+_{\mathbb Z}(\mathbb R)$ has property FH. Finally, we characterise geometric property (T) for sequences of finite Cayley graphs in terms of coarse property FH of a certain group.

math.GR↗

arXiv2022

Cycles in graphs with geometric property (T)

We show that a sequence of graphs with geometric property (T) has many small cycles. We also show that when a small part of a sequence of graphs with geometric property (T) is changed, it still has geometric property (T), provided that it is still an expander. We use this to give an example of a sequence of graphs with geometric property (T) that has large cycle-free balls.

math.FA↗

arXiv2021

Geometric property (T) for non-discrete spaces

Geometric property (T) was defined by Willett and Yu, first for sequences of graphs and later for more general discrete spaces. Increasing sequences of graphs with geometric property (T) are expanders, and they are examples of coarse spaces for which the maximal coarse Baum-Connes assembly map fails to be surjective. Here, we give a broader definition of bounded geometry for coarse spaces, which includes non-discrete spaces. We define a generalisation of geometric property (T) for this class of spaces and show that it is a coarse invariant. Additionally, we characterise it in terms of spectral properties of Laplacians. We investigate geometric property (T) for manifolds and warped systems.

math.FA↗

arXiv2019

The fundamental group of a noncommutative space

We introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded algebras and show that the category of flat connections on such modules forms a Tannakian category. As such this category can be realised as the category of representations of an affine group scheme $G$, which in the classical case is (the pro-algebraic completion of) the usual fundamental group. This motivates us to define $G$ to be the fundamental group of the noncommutative space under consideration. The needed assumptions on the differential graded algebra are rather mild and completely natural in the context of noncommutative differential geometry. We establish the appropriate functorial properties, homotopy and Morita invariance of this fundamental group. As an example we find that the fundamental group of the noncommutative torus can be described as the algebraic hull of the topological group $(\mathbb Z+θ\mathbb Z)^2$.

math.QA↗