arXiv subjects

Gianluca Basso

Publications and source records attributed to Gianluca Basso.

6 recordsLinked to original sources

arXiv2025

Topological groups with tractable minimal dynamics

A Polish group $G$ has the generic point property if any minimal $G$-flow admits a comeager orbit, or equivalently if the universal minimal flow (UMF) does. The class $\mathsf{GPP}$ of such Polish groups is a proper extension of the class $\sf{PCMD}$ of Polish groups with metrizable UMF. Motivated by analogous results for $\mathsf{PCMD}$, we define and explore a robust generalization of $\sf{GPP}$ which makes sense for all topological groups, thus defining the class $\mathsf{TMD}$ of topological groups with tractable minimal dynamics. These characterizations yield novel results even for $\mathsf{GPP}$; for instance, a Polish group is in $\mathsf{GPP}$ iff its UMF has no points of first countability. Motivated by work of Kechris, Pestov, and Todorčević that connects topological dynamics and structural Ramsey theory, we state and prove an abstract KPT correspondence which characterizes the class $\mathsf{TMD}$ and shows that $\mathsf{TMD}$ is $Δ_1$ in the Lévy hierarchy. We then develop set-theoretic methods which allow us to apply forcing and absoluteness arguments to generalize numerous results about $\mathsf{GPP}$ to all of $\mathsf{TMD}$. We also apply these new set-theoretic methods to first generalize parts of Glasner's structure theorem for minimal, metrizable tame flows to the non-metrizable setting, and then to prove the revised Newelski conjecture regarding definable NIP groups. We conclude by discussing some tantalizing connections between definable NIP groups and $\mathsf{TMD}$ groups.

math.DS↗

arXiv2025

Surfaces and other Peano Continua with no Generic Chains

The space of chains on a compact connected space encodes all the different ways of continuously growing out of a point until exhausting the space. A chain is \emph{generic} if its orbit under the action of the underlying homeomorphism group is comeager. In this paper we show that a large family of topological spaces do not have a generic chain: in addition to all manifolds of dimension at least 3, for which the result was already known, our theorem covers all compact surfaces except for the sphere and the real projective plane - for which the question remains open - as well as all other homogeneous Peano continua, circle excluded. If the spaces are moreover strongly locally homogeneous, which is the case for any closed manifold and the Menger curve, we prove that chains cannot be classified up to homeomorphism by countable structures, and that the underlying homeomorphism groups have non-metrizable universal minimal flows, with all orbits meager, in contrast to the case of 1-dimensional manifolds. The proof of the main result is of combinatorial nature, and it relies on the creation of a dictionary between open sets of chains on one side, and walks on finite connected graphs on the other.

math.DS↗

arXiv2023

Topological dynamics of kaleidoscopic groups

Kaleidoscopic groups are a class of permutation groups recently introduced by Duchesne, Monod, and Wesolek. Starting with a permutation group $Γ$, the kaleidoscopic construction produces another permutation group $\mathcal{K}(Γ)$ which acts on a Ważewski dendrite (a densely branching tree-like compact space). In this paper, we study how the topological dynamics of $\mathcal{K}(Γ)$ can be expressed in terms of the one of $Γ$, when the group $Γ$ is transitive. By proving a Ramsey theorem for decorated rooted trees, we show that the universal minimal flow (UMF) of $\mathcal{K}(Γ)$ is metrizable iff $Γ$ is oligomorphic and the UMF of $Γ$ is metrizable. More generally, we give concrete calculations, in an appropriate model-theoretic framework, of the UMF of $\mathcal{K}(Γ)$ when the UMF of a point stabilizer $Γ_c$ has a comeager orbit. Our results also give a large class of examples of non-metrizable UMFs with a comeager orbit. These results extend previous work of Kwiatkowska and Duchesne about the full homeomorphism groups.

math.DS↗

arXiv2021

Topological dynamics beyond Polish groups

When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of $G$. We introduce a class of groups, the CAP groups, which provides a neat generalization of this dividing line to all topological groups. We prove a number of characterizations of this class, having very different flavors, and use these to prove that the class of CAP groups enjoys a number of nice closure properties. As a concrete application, we compute the universal minimal flow of the homeomorphism groups of several scattered topological spaces, building on recent work of Gheysens.

math.DS↗

arXiv2020

Fences, their endpoints, and projective Fraïssé theory

We introduce a new class of compact metrizable spaces, which we call fences, and its subclass of smooth fences. We isolate two families $\mathcal F, \mathcal F_0$ of Hasse diagrams of finite partial orders and show that smooth fences are exactly the spaces which are approximated by projective sequences from $\mathcal F_0$. We investigate the combinatorial properties of Hasse diagrams of finite partial orders and show that $\mathcal F, \mathcal F_0$ are projective Fraïssé families with a common projective Fraïssé limit. We study this limit and characterize the smooth fence obtained as its quotient, which we call a Fraïssé fence. We show that the Fraïssé fence is a highly homogeneous space which shares several features with the Lelek fan, and we examine the structure of its spaces of endpoints. Along the way we establish some new facts in projective Fraïssé theory.

math.LO↗

arXiv2016

Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits

We establish some basic properties of quotients of projective Fraïssé limits and exhibit some classes of compact metric spaces that are the quotient of a projective Fraïssé limit of a projective Fraïssé family in a finite language. We prove the result for the arcs directly, and by applying some closure properties we obtain all hypercubes and graphs as well.

math.LO↗