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Todor Tsankov

Publications and source records attributed to Todor Tsankov.

At least 19 recordsLinked to original sources
arXiv2026

Dense and comeager conjugacy classes in zero-dimensional dynamics

Given a countable group $G$, we initiate a systematic study of the Polish spaces of all minimal and topologically transitive actions of $G$ on the Cantor space by homeomorphisms, with a focus on the existence of comeager conjugacy classes in these spaces. We develop a general model-theoretic framework to study this and related questions, recovering on the way many existing results from the literature. A substantial part of the paper is devoted to actions of free groups. We show that in that case, there is a comeager conjugacy class in the space of minimal actions, as well as in the space of minimal, probability measure-preserving actions. The first one is the Fraïssé limit of all sofic minimal subshifts and the second, the universal profinite action. The case of the integers was already treated by Hochman and there the two actions coincide with the universal odometer. In the non-abelian case, they are substantially different and new techniques are required. In the opposite direction, if $G$ is an amenable group which is not finitely generated, we show that there is no comeager conjugacy class in the space of all actions, and if $G$ is locally finite, also in the space of minimal actions. Finally, we study the question of existence of a dense conjugacy class in the space of topologically transitive actions. We show that if $G$ is hyperbolic or virtually polycyclic, then such a dense conjugacy class exists iff $G$ is virtually cyclic, suggesting that the case of the integers may be exceptional.

math.DS↗
arXiv2026

Locally $\aleph_0$-categorical theories and locally Roelcke precompact groups

It is well-known that Polish Roelcke precompact groups are the groups that can be represented as automorphism groups of $\aleph_0$-categorical structures in continuous logic and that there is a precise correspondence between properties of the group and properties of the structure. The goal of this paper is to extend this correspondence to the classes of locally Roelcke precompact groups and locally $\aleph_0$-categorical structures, the latter of which we define here. We characterise locally Roelcke precompact groups in terms of their isometric actions. We define locally $\aleph_0$-categorical theories and structures, prove an appropriate version of the Ryll-Nardzewski theorem, and identify the Polish locally Roelcke precompact groups as the automorphism groups of such structures. In all locally $\aleph_0$-categorical structures, there is a definable metric, which we call localising, and which captures the coarse geometric structure of the corresponding automorphism group. We show that two locally $\aleph_0$-categorical structures are bi-interpretable if and only if their automorphism groups are isomorphic. Finally, we show that (the unit ball of) a Banach space is $\aleph_0$-categorical if and only if the corresponding affine space is locally $\aleph_0$-categorical (as a metric space).

math.LO↗
arXiv2025

Non-singular and probability measure-preserving actions of infinite permutation groups

We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(Ω, μ)$ admits an invariant $σ$-finite measure equivalent to $μ$. Second, we prove the following de Finetti type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^M$, where $Z$ is a Polish space, is a product measure.

math.DS↗
arXiv2024

Extremal models and direct integrals in affine logic

Affine logic is a fragment of continuous logic, introduced by Bagheri, in which only affine functions are allowed as connectives. This has the effect of endowing type spaces with the structure of compact convex sets. We study extremal models of affine theories (those that only realize extreme types), and the ways and conditions under which all models can be described from the extremal ones. We introduce and develop the general theory of measurable fields of metric structures and their direct integrals. One of our main results is an extremal decomposition theorem for models of simplicial theories, that is, affine theories whose type spaces form Choquet simplices. We prove that every model of a simplicial theory can be (uniquely) decomposed as a direct integral of extremal models. This generalizes known decomposition results (ergodic decomposition, tracial von Neumann factor decomposition), and moreover, holds without any separability hypothesis. Two extreme kinds of simplicial theories are Bauer theories, whose extreme types form a closed set, and Poulsen theories, whose extreme types form a dense set. We show that Keisler randomizations of continuous theories are, essentially, the same thing as affine Bauer theories. We establish a dichotomy result: a complete simplicial theory is either Bauer or Poulsen. As part of our analysis, we adapt many results and tools from continuous logic to the affine or extremal contexts (definability, saturation, type isolation, categoricity, etc.). We also provide a detailed study of the relations between continuous logic and affine logic. Finally, we present several examples of simplicial theories arising from theories in discrete logic, Hilbert spaces, probability measure-preserving systems, and tracial von Neumann algebras.

math.LO↗
arXiv2024

On compact uniformly recurrent subgroups

Let a group $Γ$ act on a paracompact, locally compact, Hausdorff space $M$ by homeomorphisms and let $2^M$ denote the set of closed subsets of $M$. We endow $2^M$ with the Chabauty topology, which is compact and admits a natural $Γ$-action by homeomorphisms. We show that for every minimal $Γ$-invariant closed subset $\mathcal Y$ of $2^M$ consisting of compact sets, the union $\bigcup \mathcal{Y}\subset M$ has compact closure. As an application, we deduce that every compact uniformly recurrent subgroup of a locally compact group is contained in a compact normal subgroup. This generalizes a result of Ušakov on compact subgroups whose normalizer is compact.

math.GR↗
arXiv2023

Continuity of the stabilizer map and irreducible extensions

Let $G$ be a locally compact group. For every $G$-flow $X$, one can consider the stabilizer map $x \mapsto G_x$, from $X$ to the space $\mathrm{Sub}(G)$ of closed subgroups of $G$. This map is not continuous in general. We prove that if one passes from $X$ to the universal irreducible extension of $X$, the stabilizer map becomes continuous. This result provides, in particular, a common generalization of a theorem of Frolík (that the set of fixed points of a homeomorphism of an extremally disconnected compact space is open) and a theorem of Veech (that the action of a locally compact group on its greatest ambit is free). It also allows to naturally associate to every $G$-flow $X$ a stabilizer $G$-flow $\mathrm{S}_G(X)$ in the space $\mathrm{Sub}(G)$, which generalizes the notion of stabilizer uniformly recurrent subgroup associated to a minimal $G$-flow introduced by Glasner and Weiss.

math.GR↗
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

Invariant measures on products and on the space of linear orders

Let $M$ be an $\aleph_0$-categorical structure and assume that $M$ has no algebraicity and has weak elimination of imaginaries. Generalizing classical theorems of de Finetti and Ryll-Nardzewski, we show that any ergodic, $\operatorname{Aut}(M)$-invariant measure on $[0, 1]^M$ is a product measure. We also investigate the action of $\operatorname{Aut}(M)$ on the compact space $\mathrm{LO}(M)$ of linear orders on $M$. If we assume moreover that the action $\operatorname{Aut}(M) \curvearrowright M$ is transitive, we prove that the action $\operatorname{Aut}(M) \curvearrowright \mathrm{LO}(M)$ either has a fixed point or is uniquely ergodic.

math.DS↗
arXiv2021

Continuous logic and Borel equivalence relations

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbfΣ^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth--Kechris about discrete structures. As a different application, we also give a new proof of Kechris's theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable.

math.LO↗
arXiv2020

Bernoulli disjointness

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions $\mathfrak{A}(G)$ is a proper subalgebra of $\ell^\infty(G)$ and that the enveloping semigroup of the universal minimal flow $M(G)$ is a proper quotient of the universal enveloping semigroup $βG$. When $G$ is countable, we also prove that for any metrizable, minimal $G$-flow, there exists a free, minimal flow disjoint from it and that there exist continuum many mutually disjoint minimal, free, metrizable $G$-flows. Finally, improving a result of Frisch, Tamuz, and Vahidi Ferdowsi and answering a question of theirs, we show that if $G$ is a countable icc group, then it admits a free, minimal, proximal flow.

math.DS↗
arXiv2019

A model-theoretic approach to rigidity of strongly ergodic, distal actions

We develop a model-theoretic framework for the study of distal factors of strongly ergodic, measure-preserving dynamical systems of countable groups. Our main result is that all such factors are contained in the (existential) algebraic closure of the empty set. This allows us to recover some rigidity results of Ioana and Tucker-Drob as well as prove some new ones: for example, that strongly ergodic, distal systems are coalescent and that every two such systems that are weakly equivalent are isomorphic. We also prove the existence of a universal distal, ergodic system that contains any other distal, ergodic system of the group as a factor.

math.DS↗
arXiv2018

Polish groups with metrizable universal minimal flows

We prove that if the universal minimal flow of a Polish group $G$ is metrizable and contains a $G_δ$ orbit $G \cdot x_0$, then it is isomorphic to the completion of the homogeneous space $G/G_{x_0}$ and show how this result translates naturally in terms of structural Ramsey theory. We also investigate universal minimal proximal flows and describe concrete representations of them in a number of examples.

math.DS↗
arXiv2018

Realizing uniformly recurrent subgroups

We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the Chabauty space is the family of stabilizers of an action on a compact space on which the stabilizer map is continuous everywhere. This answers a question of Glasner and Weiss. We also introduce the notion of a universal minimal flow relative to a uniformly recurrent subgroup and prove its existence and uniqueness.

math.DS↗
arXiv2017

Eberlein oligomorphic groups

We study the Fourier--Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups whose Fourier--Stieltjes algebra is dense in the algebra of weakly almost periodic functions: those are exactly the automorphism groups of $\aleph_0$-stable, $\aleph_0$-categorical structures. This analysis is then extended to all semitopological semigroup compactifications $S$ of such a group: $S$ is Hilbert-representable if and only if it is an inverse semigroup. We also show that every factor of the Hilbert compactification is Hilbert-representable.

math.LO↗
arXiv2017

Metric Scott analysis

We develop an analogue of the classical Scott analysis for metric structures and infinitary continuous logic. Among our results are the existence of Scott sentences for metric structures and a version of the Lopez-Escobar theorem. We also derive some descriptive set theoretic consequences: most notably, that isomorphism on a class of separable structures is a Borel equivalence relation iff their Scott rank is uniformly bounded below $ω_1$. Finally, we apply our methods to study the Gromov-Hausdorff distance between metric spaces and the Kadets distance between Banach spaces, showing that the set of spaces with distance $0$ to a fixed space is a Borel set.

math.LO↗
arXiv2015

Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

We investigate the automorphism groups of $\aleph\_0$-categorical structures and prove that they are exactly the Roelcke precompact Polish groups. We show that the theory of a structure is stable if and only if every Roelcke uniformly continuous function on the automorphism group is weakly almost periodic. Analysing the semigroup structure on the weakly almost periodic compactification, we show that continuous surjective homomorphisms from automorphism groups of stable $\aleph\_0$-categorical structures to Hausdorff topological groups are open. We also produce some new WAP-trivial groups and calculate the WAP compactification in a number of examples.

math.LO↗