# DESCRIPTIVE SET THEORY IN PARIS

## Titles and Abstracts of the talks

 Itai BEN YAACOV A polish group which cannot act transitively on a complete space Márton ELEKES Nullsets without $G_\delta$ hulls David EVANS Free actions of free groups on countable structures and property (T) Rafal FILIPOW Pointwise versus equal (quasinormal) convergence via ideals Sy-David FRIEDMAN More on the Descriptive Set Theory of Generalised Baire Space Vassilis GREGORIADES Effectivity and Polish group actions Tomás IBARLUCIA Separably categorical structures and Banach representations Takayuki KIHARA Recursion theoretic methods in topological dimension theory Viktor KISS Ranks on the Baire class $\xi$ functions François LE MAITRE Small index property for groups acting on trees Michael MEGRELISHVILI Eventual nonsensitivity, tameness and topological groups Arno PAULY The role of endofunctors in DST Marcin SABOK Classification of operator systems Katrin TENT Sharply $3$-transitive groups Zoltan VIDNYANSZKY Characterisation of order types representable by Baire class 1 functions Phillip WESOLEK Elementary amenable groups and descriptive set theory Miroslav ZELENY Baire classes of strongly affine functions on simplices

Itai BEN YAACOV                A polish group which cannot act transitively on a complete space

There are several natural candidates for a Polish group which does not admit a continuous transitive action by isometries on a complete metric space - all of them Roelcke pre-compact. I shall present the origins of the question, as well as a model-theoretic argument as to why $\mathrm{Aut}^*(\mu)$ has this property. This answers a question of J Melleray (and possibly others).

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Márton ELEKES                Nullsets without $G_\delta$ hulls

We answer an old question of Mycielski by showing that in every non-locally compact Polish group admitting an invariant metric there exists a Haar null Borel set (in the sense of Christensen) that is not contained in any $G_\delta$ Haar null set. We also show that $G_\delta$ can be replaced by an arbitrary Borel class here. Moreover, answering a question from Fremlin's list we prove that in every non-locally compact Polish group admitting an invariant metric there exists a co-analytic Haar null set that is not contained in any Borel Haar null set.

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David EVANS                Free actions of free groups on countable structures and property (T)

In joint work with Todor Tsankov, we extend earlier work of his and show that if $G$ is a non-archimedean, Roelcke precompact, Polish group, then $G$ has Kazhdan's property (T). The main new aspect of this is the construction of a non-abelian free subgroup of $G$ which acts freely on the cosets of every open subgroup of infinite index in $G$.

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Rafal FILIPOW                Pointwise versus equal (quasinormal) convergence via ideals

We prove a characterization showing when the ideal pointwise convergence does not imply the ideal equal (aka quasi-normal) convergence. The characterization is expressed in terms of a cardinal coefficient related to the bounding number $\mathfrak{b}$. We also prove a characterization showing when the ideal equal limit is unique. This is joint work with Marcin Staniszewski.

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Sy-David FRIEDMAN                More on the Descriptive Set Theory of Generalised Baire Space

This is joint work with Hyttinen und Kulikov. Fix an uncountable $\kappa=\kappa^{ < \kappa}$; we consider equivalence relations on $\kappa$-Baire space. 1. Any equivalence relation induced by the action of a group of size at most $\kappa$ is reducible to $E_0$. 2. There are smooth Borel equivalence relations with classes of size $2$ which are not induced by such a group. 3. E_1 is reducible to E_0. 4. Borel isomorphism relations are reducible to equality modulo the nonstationary ideal restricted to any fixed cofinality.

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Vassilis GREGORIADES                Effectivity and Polish group actions

We present some ongoing work about the applications of effective descriptive set theory to equivalence relations induced by continuous Polish group actions. While much has been done on this topic especially in connection with the Gandy-Harrington topology, here we focus rather on applications of results of Spector about the hyperjump.

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Tomás IBARLUCIA                Separably categorical structures and Banach representations

We will describe the hierarchy of Banach representations for functions on Roelcke precompact Polish groups, in terms of model-theoretic properties of the corresponding $\aleph_0$-categorical structures. We will show for example that Asplund-representable functions (and moreover strongly uniformly continuous functions, as introduced by Glasner and Megrelishvili) are reflexively-representable.

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Takayuki KIHARA                Recursion theoretic methods in topological dimension theory

By introducing the notion of generalized Turing degrees in any admissibly represented space (represented second-countable $T_0$-quotient spaces), we provide a refinement of Roman Pols solution to Pavel Alexandrovs old problem in infinite dimensional topology. Formally, we show that there is an embedding of an uncountable partial ordering into the $\sigma$-embeddability ordering of weakly infinite dimensional metrizable compacta (indeed, metrizable C-compacta, also known as selectively screenable compacta $S_c(\mathcal{O},\mathcal{O})$). This is a joint work with Arno Pauly (Univ. Cambridge)

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Viktor KISS                Ranks on the Baire class $\xi$ functions

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class 1 functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class $\xi$ functions, and generalize most of the results from the Baire class 1 case. We also show that their assumption of the compactness of the underlying space can be eliminated. As an application, we solve a problem concerning the so called solvability cardinals of systems of difference equations, arising from the theory of geometric decompositions. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in $\omega_1$. Joint work with Márton Elekes and Zoltán Vidnyánszky

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François LE MAITRE                 Small index property for groups acting on trees

We will discuss a work in progress with Wesolek where we investigate the small index property and the uncountable cofinality property for some locally compact groups acting on trees, such as Burger-Mozes universal groups.

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Michael MEGRELISHVILI                Eventual nonsensitivity, tameness and topological groups

We study eventually nonsensitive families of functions. A topological generalization of the nonsensitivity concept familiar in dynamical systems theory. This issue is closely related to Rosenthal and Bourgain-Fremlin-Talagrand dichotomies. As well as to the fragmentability -- an useful tool in Banach space theory. Among some applications we discuss: tameness of some coding (bi)sequences and symbolic systems; weakly Radon-Nikodym G-spaces; topological groups G the universal minimal G-system M(G) of which is tame. The talk is based on a joint work with E. Glasner.

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Arno PAULY                The role of endofunctors in DST

If one moves from the usual setting of Polish spaces to the larger category of represented spaces, then plenty of basic concepts are revealed as being induced by certain endofunctors. Theorems linking different concepts (such as Banach-Lebesgue-Hausdorff or Jayne-Rogers) typically relate two constructions based on the same endofunctor - and the two example theorems have exactly the same gestalt in this setting. Fundamental properties of the endofunctors are often sufficient to understand the derived concepts - eg if an endofunctor preserves countable products, the derived pointclass behaves like a Sigma-class. The aim of the talk would be a brief introduction of the main themes, and to show how this simultaneously may shed a new light on classic DST, and can inform its extension to more general spaces.

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Marcin SABOK                Classification of operator systems

I will discuss the problem of classification of separable operator systems up to complete order isomorphism. While the relation for arbitrary separable operator systems is a complete orbit equivalence relation, the restriction to the class of finite-dimensional ones gives a smooth equivalence relation. This is joint work with M. Argerami, S. Coskey, M. Kennedy, M. Kalantar and M. Lupini realized within the program of Banff focused research groups.

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Katrin TENT                Sharply $3$-transitive groups

We construct the first sharply $3$-transitive groups not arising from a near field, i.e. point stabilizers have no nontrivial abelian normal subgroup.

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Zoltan VIDNYANSZKY                Characterisation of order types representable by Baire class 1 functions

Let $X$ be a Polish space. The pointwise limits of continuous functions defined on $X$ are called Baire class $1$ functions (denoted by $\mathcal{B}_1(X)$). A natural partial ordering on $\mathcal{B}_1(X)$ is the pointwise ordering, that is, we say that $f < _pg$ if for every $x \in X$ we have $f(x) \leq g(x)$ and there exists an $x$ so that $f(x) < g(x)$. The description of the linearly ordered subsets of $(\mathcal{B}_1(X), < _p)$ reveals lots of information about the poset $(\mathcal{B}_1(X), < _p)$. We say that a linearly ordered set $(L, < _L)$ is representable in a poset $(P, < )$ if $P$ contains an order isomorphic copy of $L$. It was shown by Kuratowski that $\omega_1$ is not representable in $\mathcal{B}_1(X)$. In the 80s Laczkovich posed the following problem: Characterise the linearly ordered subsets of the poset $(\mathcal{B}_1(X), < _p)$. Partial results were proved by Komjath, Steprans, Kunen and Elekes concerning this problem. In a joint work with Marton Elekes we solved Laczkovich's problem proving that there exists a concrete, combinatorially describable universal linearly ordered set $(U, < _U)$, that is, a linearly ordered set so that a linearly ordered set is representable in $(\mathcal{B}_1(X), < _p)$ iff it is representable in $(U, < _U)$. Using this result we answered all of the known open questions concerning the linearly ordered subsets of the poset $(\mathcal{B}_1(X), < _p)$.

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Phillip WESOLEK                Elementary amenable groups and descriptive set theory

The space of marked groups is a natural Borel parameterization of countable groups. We show the set of elementary amenable marked groups is a $\Pi^1_1$ non-Borel set. As a corollary to our proof, we obtain a chain condition equivalent to elementary amenability. We additionally obtain a new non-constructive proof of the existence of finitely generated amenable groups that are not elementary amenable.

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Miroslav ZELENY                Baire classes of strongly affine functions on simplices

For each $n \in \mathbb N$, $n \ge 2$, we construct a metrizable simplex $X$ and a strongly affine function of Baire class $n$ on $X$ that is not of affine class $n$.

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