# Structure and dynamics of Polish groups conference

May 25, 2020

#### Entropy of Invariant Measures

The entropy of a probability measure on a finite set quantifies one notion of the complexity of the measure. For a finite relational language $L$, and a measure $\mu$ on $L$-structures with underlying set the natural numbers, if $\mu$ is $S_\infty$-invariant then it is determined by its restriction to $L$-structures of the form $\{0, \ldots , n-1\}$. We can therefore canonically assign to $\mu$ its \emph{entropy function}, $En_\mu$, which sends each natural number $n$ to the entropy of the restriction of $\mu$ to $L$-structures with underlying set $\{0, \ldots , n-1\}$.

In this talk I will discuss the relationship between the growth rate of $En_\mu$ and the language $L$. If the largest arity of a relation in $L$ is $k$, then $En_\mu$ grows as $O(n^k)$. I will give a precise expression for the coefficient of $n^k$ based on the hypergraphon representation of $\mu$, and show how these notions generalize what is often called the \emph{entropy of a graphon}. When $\mu$ comes from sampling a Borel structure we will show that the growth rate of the entropy function is $o(n^k)$. We will also discuss lower bounds on the possible growth rates of the entropy function in this case. This generalizes work of Hatami-Norine from graphons to hypergraphons.

Joint work with Cameron Freer and Rehana Patel.

#### Ultraproduct of self-adjoint operators and the conjugation action of the unitary group

The ultraproduct construction has many applications in functional analysis, especially in the theory of operator algebras (both in C$^*$ and in W$^*$ settings). Because we need to treat $\textit {unbounded}$ objects (e.g. modular operators) even when we study operator algebras (=algebras of $\textit {bounded}$ operators), it is desirable to study the ultraproduct construction for unbounded operators. The notion of the ultrapower for unbounded operators was given by Derndinger in 1974 (for the generators of strongly continuous semigroups) and by Krupa and Zawisza (for a more specific class of self-adjoint operators) in the 1980s. Recently a model theoretic approach to the ultrapower of self-adjoint operators is given by Argoty. In this talk we discuss the construction by Krupa and Zawisza and relate the properites of ultrapowers to the topological property of the space ${\rm{SA}}(H)$ of self-adjoint operators on a separable Hilbert space $H$.

We observe that there is probably a unified theory, very likely to be model theoretical, hidden among our observation on self-adjoint operators, the work of the speaker with Haagerup and Winsløw (on von Neumann algebras), and the work of Conley, Kechris and Tucker--Drob (on the measure--preserving actions of a countable group).

#### Polish group actions on topometric spaces

Consider a Polish group $G$ acting continuously on a Polish space $X$. Then it is known that there is a robust notion of a generic element $x$ for this action: $Gx$ is non-meagre if and only if $Ux$ is dense in a neighbourhood of $x$ for every neighbourhood $U$ of the identity.

We give a similar characterisation when $X$ carries, in addition, a notion of distance, which may be finer than its topology (e.g., the norm on the unitary group).

The final aim is to give a necessary and sufficient condition for a Polish group $G$ to have metric ample generics, in terms of any associated Fraïssé class, similar to the one given by Kechris and Rosendal for ample genreics.

This is joint work with Julien Melleray.

#### The Dichotomy for Montone Monadic SNP via Extreme Amenability

A theorem of Cherlin, Shelah, and Shi asserts that for every finite class of finite structures F there exists an (up to isomorphism unique) model-complete structure U that is universal for the class of all countable structures that do not homomorphically embed any structure from F. Hubicka and Nesetril recently proved that the expansion of U by a generic linear order has the Ramsey property, and by a result of Kechris, Pestov, and Todorcevic it has an extremely amenable automorphism group.

We present an application of this result in complexity classification in theoretical computer science. Specifically, we give a new proof that the complexity class Monotone Monadic Strict NP (MMSNP) has a complexity dichotomy (P vs NP-complete) which does not rely on the involved expander constructions of Kun.

Joint work with Antoine Mottet and Florent Madelaine.

#### Farber sequences of lattices

A sequence of lattices of a locally compact group G is said Farber if for every neighborhood of the identity U in G the proportion of conjugates of the lattices which intersect trivially U tends to 1. In this talk I will explain how one can use ultraproducts of actions of locally compact groups to obtain an easy proof that sequences of lattices whose covolume tends to infinity are automatically Farber for many property (T) groups for which the Nevo-Stuck-Zimmer theorem holds.

#### Generic representations of countable groups

For a fixed countable discrete group Gamma and a Polish group G we consider the Polish space of all homomorphisms (representations) from Gamma into G. We are mainly interested in the problem whether there exists a generic homomorphism" from Gamma into G; that is, a homomorphism whose orbit (in this space) under the natural action of G by conjugation is comeager. Questions of this type go back to Halmos and we follow some recent research of e.g. Rosendal, and of Glasner, Kitroser and Melleray.

We shall present some results when the Polish group in question is e.g. the unitary group of a separable infinite-dimensional Hilbert space, the isometry, resp. linear isometry, group of the Urysohn space, resp. the Gurarii space. Or the automorphism group of some homogeneous countable structure, such as the random K_n-free graph, the random tournament, or the random poset.
The talk will be based mainly on the paper with Maciej Malicki and also partially on the paper with Malicki and Alain Valette.

#### Topological dynamics of automorphism groups of sparse graphs.

An infinite graph is sparse if there is a positive integer k such that for every finite subgraph, the number of edges is bounded above by k times the number of vertices. Such graphs arise in model theory via Hrushovski’s predimension constructions. In joint work with J. Hubicka and J. Nesetril, we study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorcevic correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of k-orientations' of the graph. We show that Hrushovski’s example of an omega-categorical sparse graph has no omega-categorical expansion with an amenable automorphism group and that all orbits on a minimal subflow of the space of k-orientations are meagre.

#### A metric minimal PI cascade with 2^c minimal ideals

Given a topological group G we consider the invariant mi(X,G) which assigns to a minimal compact dynamical flow (X, G) the number of minimal left ideals in its enveloping semigroup E(X,G).
It measures the complexity of the proximal relation on (X,G).
We first improve an old result of McMahon and show that a metric minimal flow with mi(X,G) < 2^c (where c = 2^{\aleph_0}) is necessarily PI (proximal-isometric). (The class of PI flows contains the class of distal flows and, for strongly amenable acting groups, is disjoint from the class of weakly mixing ones.)
We then show the existence of various minimal PI flows with many minimal left ideals, as follows.
For the acting group G=SL_2(R)^N, we construct a metric minimal PI G-flow with c minimal left ideals.
Next we use this example and results established in [GW-79] to construct a metric minimal PI cascade (X,T) with c minimal left ideals.
We go on and construct an example of a minimal PI-flow (Y, H) on a compact manifold Y and a suitable path-wise connected group H of homeomorphism of Y, such that the flow (Y, H) is PI and has c minimal left ideals.
Finally, we use this latter example and a theorem of Dirbak to construct a cascade (X, T) which is PI (of order 3) and has c minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication less than 2^c minimal left ideals implies PI", fails.
To sum up, we show that on the domain of minimal cascades, the range of mi(.) includes the set of cardinals {2^n : n = 1, 2, … , \aleph_0, \aleph_1, … , c}. We don't know whether any other cardinal number is in this range.

This is a joint work with Yair Glasner

#### On permutation representations of some higher rank lattices.

This is a joint work with Tsachik Gelander,

We define the notion of a weak convergence action of a group G on a compact space X. This notion generalises (the already wide) notion of convergence actions. For example any irreducible lattice in a product GxH, where G rank-1 simple group acts as a weak convergence group on the boundary of the symmetric space of G. A group is called a weak convergence group if it admits a faithful weak-convergence action.

We prove that every weak convergence group admits a dense embedding into S(infinity).

#### Universal ergodic compact extensions, and distal systems of property (T) groups

I will discuss the existence and uniqueness of universal ergodic compact extensions of ergodic systems of countable groups, and use them to prove that distal ergodic systems of property (T) groups have discrete spectrum. This is joint work with Todor Tsankov.

#### Homeomorphism groups of Ważewski dendrites

We study universal minimal flows of homeomorphism groups of Ważewski dendrites $W_P$, where $P\subset \{3,4,\ldots,\omega\}$. In the case when $P$ is finite, we prove that the universal minimal flow of ${\rm Homeo}(W_P)$ is metrizable and we compute it explicitly. This answers a question of B. Duchesne. If $P$ is infinite, we show that the universal minimal flow of ${\rm Homeo}(W_P)$ is not metrizable. Then ${\rm Homeo}(W_P)$ turn out to be a source of interesting examples. In particular, in that case ${\rm Homeo}(W_P)$ are examples of groups which are at the same time Roelcke precompact and have non-metrizable universal minimal flow.

#### Order structures and large conjugacy classes

A Polish group G has ample generics if every action of G on its finite power G^n by diagonal conjugation has a comeager orbit. This property is shared by automorphism groups of many naturally arising countable structures, e.g. the random graph or the rational Urysohn space. However, there are no examples of groups with ample generics that are at the same time extremely amenable, i.e. are such that all their flows have a fixed point.
It is known that structures whose automorphism groups are extremely amenable are typically equipped with an ordering, e.g. the ordered random graph or the ordered rational Urysohn space. During the talk, I will present some results about the existence of large diagonal conjugacy classes and topological similarity classes for automorphism groups of order structures, obtained jointly with Aleksandra Kwiatkowska.

#### Actions of topological full groups and homomorphisms between them

Topological full groups are group naturally associated to topological dynamical systems, and more generally to étale groupoids on the Cantor set. These groups are countable discrete, but their introduction was motivated by a prominent class of non locally compact Polish groups, namely Dye's measurable full groups associated to a pmp equivalent relation. In analogy with their measurable cousins, any isomorphism between topological full groups is implemented by an isomorphism of the underlying groupoids.
In this talk I will focus on the dynamical properties of actions of topological full groups on compact spaces. I will explain a theorem giving a necessary and sufficient condition for an action of the topological full group to be `induced'' by an action of a power of the underlying groupoid. As a consequence, we will see that not only isomorphisms but arbitrary embeddings between full groups often arise from the groupoids, and discuss other applications.
These results are proven by analysing the conjugation action of t.f.g.'s on their compact space of subgroups.

#### Generalized (extremal) amenability of topological groups and circular orders

We study topological groups $G$ for which the universal minimal $G$-system $M(G)$ or/and the universal irreducible affine $G$-system $IA(G)$ are tame $(dynamically$ $small)$. We analyze ultra-homogeneous actions on circularly ordered sets and prove a $circular$ $analog$ of V. Pestov's classical result about ultra-homogeneous actions on linearly ordered sets.

Joint work with Eli Glasner.

#### Dense subgroups of isometry groups of Urysohn spaces

It is in general difficult to understand which countable groups are isomorphic to a dense subgroup of a given Polish group $G$. I will discuss some results and questions related to that problem - for instance, the fact that any non trivial free product with at least one infinite factor is isomorphic to a dense subgroup of the isometry group of any countable Urysohn space; and an example of a countable, dense subgroup of $S_\infty$ which is not isomorphic to a dense subgroup of any nontrivial countable Urysohn space. Related open questions are numerous and I hope to have time to mention some of them.
This is joint work with P. Fima, F. Le Maître, S. Moon and Y. Stalder.

#### Kaleidoscopic groups

The kaleidoscope is a device that produces a vast gallery of examples of polish groups. Specifically, it is a unified method to construct a continuum of subgroups of $\mathrm{Sym}(\infty)$.

In joint work with Duchesne and Wesolek, we study some properties of these groups: algebraic, permutational and homological.

#### Fixed points in compactifications and combinatorial counterparts

The Kechris-Pestov-Todorcevic correspondence connects extreme amenability of non-Archimedean Polish groups with Ramsey properties of classes of finite structures. The purpose of this talk will be to recast it as one of the instances of a more general construction, allowing to show that Ramsey-type statements actually appear as natural combinatorial expressions of the existence of fixed points in certain compactifications of groups, and that similar correspondences in fact exist in various dynamical contexts.

#### Coarse structure of covering maps

We will present a basic theory describing the coarse structure of Polish groups obtained by group extensions. This involves analysing the coarse structure of the associated cocycles and sections. We apply this to the structure of group extensions induced by covering maps of compact manifolds.

#### Equivariant concentration in topological groups

In their 1983 work on measure concentration phenomena, Gromov and Milman showed that every Lévy group, i.e., any second-countable topological group G containing an increasing sequence of compact subgroups whose union is dense in G and whose normalized Haar measures form a Lévy family with respect to some right-invariant compatible metric on G, is extremely amenable. Since the Lévy property for sequences of measured metric spaces is equivalent to the sequence converging to a single point with respect to Gromov's concentration topology, it is natural to ask for similar consequences in more general situations where the limit object may be a non-trivial space. In my talk, I will address this question and present a generalization of the result by Gromov and Milman: if G is a second-countable topological group containing an increasing sequence of compact subgroups with dense union in G such that the compact subgroups, equipped with their normalized Haar measures and the restrictions of a fixed right-invariant compatible metric on G, concentrate to some compact, fully supported measured metric space X, then X is homeomorphic to a G-invariant subspace of the Samuel compactification of G. In particular, this confirms a 2006 conjecture by Pestov. If time permits, I will also talk about known examples and related questions.

#### Characters of inductive limits of finite alternating groups

If $G \ncong Alt(\mathbb{N})$ is an inductive limit of finite alternating groups, then the indecomposable characters of $G$ are precisely the associated characters of the ergodic invariant random subgroups of $G$.

#### Some applications of near ultrafilter constructions

Given $G$ a topological group, we consider various near ultrafilter constructions one can perform on a $G$-space $X$. When $X$ is a minimal $G$-flow, we can construct the \emph{universal highly proximal extension} of $X$. This produces a new proof of the Generic Point Problem recently solved by Ben--Yaacov, Melleray, and Tsankov. When $X$ is a non-compact $G$-space, we can construct the \emph{maximal equivariant compactification} of $X$ and use this construction to address some open problems recently asked by Pestov.