Title: The Kirchberg Embedding Problem
Abstract: With the recent landmark result MIP*=RE, a negative solution to the Connes embedding problem followed. An analogous problem for C*-algebras remains open, namely the Kirchberg embedding problem (KEP), which asks whether or not every C*-algebra embeds into an ultrapower of the Cuntz algebra O_2. In these tutorials, I will introduce the Cuntz algebra and explain its importance in C*-algebra theory. I will then discuss a number of older results (joint with Sinclair) explaining the model-theoretic content of the KEP, as well as a couple of recent results, joint with Fox and Hart, which describe some computability-theoretic consequences of a positive solution to the KEP. I will conclude with a model-theoretic question about the Cuntz algebra for which a positive solution would yield a negative resolution to the KEP.
Title: Homeomorphism groups of manifolds through logic
Abstract: I will discuss applications of model theory to the study of homeomorphism groups, motivated by some concrete open questions in the theory of group actions on manifolds. I will discuss first order rigidity, complexity of theories of groups of homeomorphisms, and questions at the interface of set theory and the first order theory of homeomorphism groups. This minicourse will discuss joint work with Sang-hyun Kim, J. De la Nuez González, Christian Rosendal, and James Hanson.
Title: Treeability of groups via planarity and one-ended spanning subforests
Abstract: We survey the theory of treeability of groups and equivalence relations. We discuss perspectives from Gromov's notion of measure equivalence of groups, and orbit equivalence and Borel reducibility of equivalence relations. We show how tools from measurable graph combinatorics can be used to give combinatorial proofs of treeability for new classes of groups including surface groups. These arguments use measurable constructions of one-ended spanning subforests of Borel graphs, and planar duality. This is joint work with Clinton Conley, Damien Gaboriau, and Robin Tucker-Drob.
Title: On sharply 2-transitive groups, the Cherlin-Zilber-Conjecture and the Burnside problem
Abstract: Until recently the class of sharply 2-transitive groups seemed very rigid, all known examples being of the form $K\rtimes K^*$ for some (near-) field $K$. However, work with Rips, Segev, Ande, Amelio et al shows that the class is in fact wild: any group can be embedded into a sharply 2-transitive group. In particular, the class also contains simple groups and thus, potentially, counterexamples to the Cherlin-Zilber-Conjecture.
I will introduce the necessary background on sharply 2-transitive groups and will present recent result around classification and non-classification results and the connection to the Burnside problem. I will then explain the relevance of this problem in the context of the Cherlin-Zilber conjecture.
Title: The Complexity of the Quasi-Isometry Relation for Finitely Generated Groups
Abstract: Gromov's geometric group theory seeks to classify finitely generated groups in terms of the ``large scale geometry'' of their Cayley graphs. In this tutorial, we will discuss this program from the perspective of the theory of Borel equivalence relations; and we will present a number of results which strongly suggest that the quasi-isometry relation for finitely generated groups is considerably more complex than the isomorphism relation.
Title: On the positive theory of groups acting on trees
Abstract: In this talk, we will discuss how the dynamics of an action of a group on a tree determine model-theoretical properties of the group. More precisely, we introduce the notion of weakly acylindrical action on a tree and show that groups with such actions have the same positive theory as a nonabelian free group. The class of groups that admit a weakly acylindrical action on a tree is very wide and contains, among others, non-virtually solvable fundamental groups of 3-manifolds and generalised Baumslag-Solitar groups, almost all one-relator groups, and more surprisingly, finitely generated simple groups. In order to prove these results, we describe a uniform way for constructing (weak) small cancellation tuples. This result is of its own interest and is fundamental to obtain corollaries of a general nature, such as a quantifier reduction for positive sentences and the preservation of the non-trivial positive theory under extensions of groups.
Title: Canonical topology on end spaces of countable, Borel, and measured graphs
Abstract: In a graph, tree, or any other type of combinatorial structure, there is a correspondence between being able to perform a combinatorial construction (e.g., finding a coloring, or picking a point at infinity) "definably", versus being able to do it "equivariantly/measurably" on a locally countable Borel structure on a group(oid) action. This talk will give an overview of this correspondence. We will then discuss several instances of such "canonical" constructions on graphs that concern their large-scale topology, i.e., space of ends. For example, we show that a measurable selection of a closed proper subset of non-vanishing ends in a measured graph implies amenability, generalizing a result of Epstein–Hjorth in the measure-preserving case. We also show that every graph admits a canonical sequence of subgraphs exhibiting the Cantor–Bendixson analysis on their end spaces, excepting the degenerate case that the Cantor–Bendixson rank is a limit ordinal and there is a unique end of that rank. This is based on joint works with Rishi Banerjee, Alexander Kechris, Greg Terlov, Anush Tserunyan, and Robin Tucker-Drob.
Title: Large Artinian Groups
Abstract: I will review some classical examples of groups whose strictly descending chains of subgroups are finite. Some very recent constructions of such groups (having large cardinality) will also be explained. Set-theoretic consistency will play a role in some of the stated results. This is joint work with Saharon Shelah.
Title: The first-order theory of torsion-free Tarski monsters
Abstract: In this talk, we will discuss a method for controlling the first-order theory of certain small cancellation quotients of a torsion-free hyperbolic group. As an application, we construct a torsion-free Tarski monster (a group all of whose proper subgroups are infinite cyclic or trivial) that has trivial positive theory (namely, the same positive theory as a free group), answering questions of Casals-Ruiz, Garreta, and de la Nuez Gonzalez. Furthermore, the Tarski monster we construct has unbounded w-length but vanishing stable w-length for every non-trivial, non-surjective word w; and it does not admit an action on a hyperbolic space with a loxodromic element. As another application, we show that a torsion-free hyperbolic group satisfies a one-quantifier sentence if and only if its few relator random quotient also satisfies the sentence. This is joint work with Rémi Coulon and Francesco Fournier-Facio.
Title: Cayley-Abels-Rosendal Graphs and Homeomorphism Groups of Stone spaces
Abstract: We will discuss the expansion of the theory of Cayley graphs and Cayley-Abels graphs to the setting of Polish groups due to Rosendal. An expansion of the Milnor-Schwarz lemma applies, giving a quasi-isometry between a group with a coarsely bounded generating set and its associated Cayley-Abels-Rosendal graph. We will then apply these concepts to homeomorphism groups of Stone spaces, including a complete classification of countable Stone spaces whose homeomorphism groups are coarsely bounded, are locally bounded, and admit a Cayley--Abels--Rosendal graph, producing a coarsely bounded generating set.
Title: First-order sentences in random groups
Abstract: We prove that a random group, in Gromov's density model with d<1/2, satisfies an AE sentence (in the language of groups) if and only if this sentence is true in a nonabelian free group. This is a joint work with R. Sklinos. We will also discuss arbitrary sentences.
Title: Comeager isomorphism classes in zero-dimensional dynamics
Abstract: It is often possible to parametrize a given class of dynamical systems by elements of a Polish space, and then it becomes natural to ask what properties hold "generically", i.e., on a comeager set of systems. The most extreme situation is when there is a single comeager isomorphism class: that is, the generic properties are captured by a single system. This does not usually happen in ergodic theory but is quite common in zero-dimensional topological dynamics. For example, it is a result of Kechris and Rosendal that there is a generic action of Z on the Cantor space and of Kwiatkowska that there is such a generic action of the free group F_n. In this work, we are interested in minimal dynamical systems and show that there is a generic minimal action of F_n and also a generic minimal action of F_n that preserves a probability measure. We also develop a model-theoretic framework to study this and related questions. This is joint work with Michal Doucha and Julien Melleray.
Title: Measure Equivalence of Baumslag-Solitar Groups
Abstract: We show that all non-amenable non-unimodular Baumslag-Solitar groups BS(r,s), 2≤|r|<s, are measure equivalent to each other, thereby completing the measure equivalence classification of Baumslag-Solitar groups. Consequently, each BS(r,s) belongs to one of three measure equivalence classes according to whether it is amenable, virtually isomorphic to F_n x Z, or has non-amenable and non-unimodular Bass-Serre tree. This is joint work with Damien Gaboriau, Antoine Poulin, Anush Tserunyan, and Konrad Wrobel.
Title: The Hrushovski property and the pro-odd topology
Abstract: In this talk, we will define the Hrushovski property (also known as the extension property for partial automorphisms or EPPA), tournaments, and the pro-odd topology. We will then draw the connection between the Hrushovski property for finite tournaments and the pro-odd topology.
Title: Automatic Continuity for Homeomorphism Groups
Abstract: I will talk about recent work with Mladen Bestvina and Kasra Rafi on classifying when homeomorphism groups of surfaces and second countable Stone spaces have the automatic continuity property.
Title: Hyperfiniteness of boundary actions of hyperbolic-like groups
Abstract: An action of a countable group on a Polish space by Borel automorphisms is "hyperfinite" if, roughly speaking, the orbits can be arranged into the structure of a line in a Borel manner. Several classes of hyperbolic-like groups have recently been shown to exhibit hyperfinite actions on the Gromov boundaries of their hyperbolic Cayley graphs. We survey these results and present joint work with Damian Osajda and Koichi Oyakawa on hyperfiniteness of actions of graphical small cancellation groups on boundaries of their natural hyperbolic Cayley graphs.
Title: Conjugation in MCG(graphs)
Abstract: I will define a family of topological groups known as mapping class groups of locally finite graphs, and discuss when they contain dense conjugacy classes.
Title: Homeomorphism groups and (non-)classifiability by countable structures
Abstract: While it is provable that self-homeomorphisms of the interval [0,1] can be classified up to conjugacy by using countable structures as invariants, it is a classical result of Hjorth that there is no definable way to do so for self-homeomorphisms of the square [0,1]^2. That is, the conjugacy equivalence relation on the homeomorphism group of the square reduces a turbulent action. In this lightning talk, we take a step towards understanding the broader interplay between dimension and turbulence by describing a one-dimensional compact Polish space whose homeomorphisms also cannot be classified up to conjugacy in this manner.
Title: Borel asymptotic dimension of the Roller boundary of finite dimensional CAT(0) cube complexes
Abstract: Borel median graphs are a natural generalization of Borel trees. I'll talk about my recent work on the Borel complexity of Borel median graphs that appear naturally in geometry. More precisely, for any countable finite dimensional CAT(0) cube complex, the Borel median graph on its Roller compactification has the Borel asymptotic dimension bounded from above by its dimension.
Title: Boundary dynamics and mixed identities in weakly hyperbolic groups
Abstract: A group G is mixed identity free (MIF) if no nontrivial element w in G * F_n belongs to the kernel of every homomorphism from G * F_n to G that restricts to the identity on G. For groups admitting general-type actions on hyperbolic spaces, the purely algebraic condition of being MIF is equivalent to a dynamical condition on the induced G-action on the Gromov boundary. This equivalence reveals a rigidity phenomenon for general type group actions on hyperbolic spaces.
Title: Homogeneity in Toral Relatively Hyperbolic Groups
Abstract: Homogeneity is one of the fundamental concepts in first-order logic so it is natural to ask which structures are homogeneous. Byron and Perin classified all homogeneous torsion-free hyperbolic groups using JSJ decompositions. We try to generalize one direction of their results to toral relatively hyperbolic groups.
Title: Tree structures on countable Borel equivalence relations
Abstract: I will give a pictorial summary of a way to obtain a tree structure from tree-like structures in a Borel way on a countable Borel equivalence relation. This is based on joint work with Ruiyuan Chen, Antoine Poulin, Anush Tserunyan.