In each dimension \(d\), there exists a canonical compact, second countable space, called the \(d\)-dimensional Menger space, with certain universality and homogeneity properties. For \(d = 0\), it is the Cantor set, for \(d = \infty\), it is the Hilbert cube. I will concentrate on the \(1\)-dimensional Menger space. I will prove that it a quotient of a projective Fraïssé limit. I will show how a model-theoretic property of projective Fraïssé limits, called the projective extension property, can be used to prove high homogeneity of the \(1\)-dimensional Menger space.
This is a joint work with Aristotelis Panagiotopoulos.
This talk presents an extension of the theory of turbulent Polish group actions introduced by Hjorth to certain classes of equivalence relations that cannot be induced by Polish group actions. The main reason for this extension is to analyze the complexity and dynamical structure of the "quasi-isometry" and "finite Gromov-Hausdorff distance" relations in the Gromov space of isometry classes of pointed proper metric spaces. This analysis is in turn motivated by questions pertaining to the generic geometry of orbits of general dynamical systems.
This is joint work with Jesús A. Álvarez López (USC-Spain).
Under suitable hypothesis, all analytic equivalence relations with all classes Borel are Borel equivalence relations when restricted to some non-trivial Borel sets according to a large class of ideals.
The Lelek fan \(L\) is a compact and connected space with many symmetries, which can be constructed from a projective Fraïssé limit, and hence it has a very rich homeomorphism group \(H(L)\).
In the talk, I will first show a number of properties of \(H(L)\) - it is totally disconnected, generated by every neighbourhood of the identity, has a dense conjugacy class, and is simple. We then focus on the dynamics of \(H(L)\). Using the Graham-Rothschild theorem, the Kechris-Pestov-Todorčević correspondence, as well as some new ideas, we describe the universal minimal flow of \(H(L)\). If time permits, we show a generalization of the finite Gowers' Ramsey theorem to multiple tetris-like operations and apply it to conclude that a group of homeomorphisms that preserves a "typical" linear order of branches of \(L\) is extremely amenable.
This is joint work with Dana Bartošová.
The notion of ample generics has been actively investigated in recent years, mainly because of its very interesting implications, e.g. the automatic continuity property or the small index property. However, all the known so far groups with ample generics are non-Archimedean. I will present simple examples of Polish groups with ample generics which are not of this form.
This talk will present some results from the paper "Structurable equivalence relations" by Kechris. We will define the notion of an equivalence relation which is structurable with respect to some class of first-order structures. By a result of Miller, every Borel class of first-order structures has a universal structurable equivalence relation. We will sketch the result that characterizes non-smoothness of this universal relation for the class of structures isomorphic to a single fixed structure, and if time permits, describe the generalization to more general classes of structures.
Bowen has defined the naive entropy of a measure-preserving action of a general group as a direct generalization of entropy for amenable groups. We introduce a topological analogue of this definition. For nonamenable groups this invariant always takes values zero or infinity, and we will present results illustrating the resulting dichotomy between systems with zero naive entropy and systems with infinite naive entropy.
This talk will present some applications of infinitary logic to admissible ordinals and the analytic equivalence relation associated with countable admissible ordinals.
This talk will present some applications of infinitary logic to admissible ordinals and the analytic equivalence relation associated with countable admissible ordinals.
Given a measure-preserving Borel graph \(G\) on a standard probability space, we characterize when \(G\) has a Borel a.e. one-ended spanning subforest. We apply our characterization to show that planar groups are strongly treeable. This is joint work with Clinton Conley, Damien Gaboriau, and Robin Tucker-Drob.
I will present a characterization of Polish ultrametric spaces whose isometry groups have a neighborhood basis at the identity consisting of clopen subgroups with ample generics. I will also discuss some other applications of the arguments used to prove this characterization.
We describe algebras of clopen subsets of the greatest ambit and the universal minimal flow of groups of automorphisms of structures. We gather questions that we believe are relevant to the problem of finding Ramsey expansions of classes of finite structures.
Recently, Justin Moore proved results connecting the phenomena of amenability and Ramsey theory. In this talk, we will consider the case where \(G\) is the automorphism group of a Fraïssé structure. We will define a convex Ramsey object and show that \(G\) is amenable iff the Fraïssé class has the convex Ramsey property. Our method of proof will be different; we first define and characterize the greatest affine ambit, then show exactly what the fixed points of this flow are. If time permits, we will also consider the case when \(G\) is amenable and \(G\) has metrizable universal minimal flow.
Abert and Elek have defined a topology on the space of weak equivalence classes of measure-preserving actions of a countable group. Tucker-Drob has extended this to the space of stable weak equivalence classes. Both of these spaces carry a well-defined notion of convex combination which is compatible with the topology. We will discuss topological and geometric properties of this convex structure and present recent results of Bowen, Tucker-Drob and the speaker.
Introduced by Levitt in 1995, the notion of cost turned out to be a useful invariant for countable Borel equivalence relations in the presence of a Borel probability measure which is invariant for the given equivalence relation \(E\). One prominent result is that when the cost of \(E\) is greater than \(1\), then there is a Borel subequivalence relation induced by a free Borel action of free group in two generators.
The goal of this talk is to present a generalization of this concept for an arbitrary Borel cocycle. Among other results it will be shown that, given a Borel probability measure which is invariant with respect to a given cocycle, a countable Borel equivalence relation \(E\) is hyperfinite on a conull \(E\)-invariant Borel set iff the cost is attained and equals \(1\).
Abert and Elek have defined a topology on the space of weak equivalence classes of measure-preserving actions of a countable group. Tucker-Drob has extended this to the space of stable weak equivalence classes. Both of these spaces carry a well-defined notion of convex combination which is compatible with the topology. We will discuss topological and geometric properties of this convex structure and present recent results of Bowen, Tucker-Drob and the speaker.
The Lelek fan \(L\) is the unique non-degenerate subcontinuum of the Cantor fan with a dense set of endpoints. We denote by \(G\) the group of homeomorphisms of \(L\) with the compact-open topology. We describe \(L\) as a natural quotient of a topological structure, which provides us with a different veiwpoint on \(L\) and \(G\). Studying the dynamics of \(G\), we generalize the finite Gowers' Theorem to a variety of operations. This is joint work with Aleksandra Kwiatkowska.
Abert and Elek have defined a topology on the space of weak equivalence classes of measure-preserving actions of a countable group. Tucker-Drob has extended this to the space of stable weak equivalence classes. Both of these spaces carry a well-defined notion of convex combination which is compatible with the topology. We will discuss topological and geometric properties of this convex structure and present recent results of Bowen, Tucker-Drob and the speaker.
Full groups were introduced in Dye's visionary paper of 1959 as subgroups of \(\operatorname{Aut}(X, \mu)\) stable under cutting and pasting their elements along a countable partition of the probability space \((X, \mu)\). Two well-known examples are \(\operatorname{Aut}(X, \mu)\) itself and full groups of countable measure-preserving equivalence relations, whose Borel complexity (as subsets of \(\operatorname{Aut}(X, \mu)\)) was determined by Wei in 2005.
We will discuss other full groups, see that a lot of them are Borel subgroups of \(\operatorname{Aut}(X, \mu)\), and find a compatible Polish group topology on orbit full groups, which are full groups of equivalence relations induced by Borel actions of Polish groups. This talk is based on a joint work with Carderi.