# Borel combinatorics and ergodic theory conference

#### EPFL, CIB

November 18, 2019

#### Point processes, cost and weak containment of locally compact groups

A point process on a locally compact second countable group G is an invariant random discrete subset of G, or of its symmetric spaces. Point processes can be effectively used to analyze pmp actions of these groups. For a discrete group, the iid processes are the freest in many senses, and as such, they reflect the properties of the group the best. As we show, this role is (partially) fulfilled by Poisson processes for big groups.

The cost of a vertex transitive graph is the infimum of the expected degree of an invariant random rewiring of the graph. Similarly, one can define the cost of a point process on a homogeneous space. It turns out that the cost of a Poisson process is maximal among point processes of the same density and that it is related to the growth of the minimal number of generators of lattices in Lie groups. We expect that for semisimple Lie groups, the minimal number of generators is sublinear in the volume except for SL(2,R). We outline partial results in this direction and pose some open problems. One of them is to compute the cost of the Poisson on hyperbolic 3-space.
This is joint work with Samuel Mellick.

#### Asymptotic invariants of lattices in locally compact groups

The aim of our work is to understand some of the asymptotic properties of sequences of lattices in a fixed locally compact group. As an example we are interested in the asymptotic growth of the Betti numbers of the lattices renormalized by the covolume or the rank-gradient, the minimal number of generator also renormalized by the covolume. For doing so we will consider the ultraproduct of the actions of the locally compact group on the coset spaces and we will show how the properties of one of its cross sections are related to the asymptotic properties of the lattices.

#### Borel and measurable versions of the Lovasz Local Lemma

Lovasz Local Lemma is one of the most fundamental tools in classical combinatorics. I will describe a joint work with E. Csoka, A. Mathe, O. Pikhurko and K. Tyros in which we develop measurable and Borel versions of the local lemma for Borel graphs and graphings

#### Almost automorphisms of sofic approximations

We study sofic approximations of Kazhdan Property (T) groups by expander graphs. We prove that every almost automorphism can be improved to a better automorphism. Adding ideas of Simon Thomas we show that clusters of almost automorphisms form a group in a natural way. Hence the Bernoulli shift of the direct product of an infinite Kazhdan Property (T) group and a finitely presented, not residually finite group is not a local-global limit of finite graphs. Joint work with Andreas Thom.

#### Highly faithful actions and dense free groups in full groups

In this talk I willI present a natural strengthening of faithfulness which is somehow dual to high transitivity, and explain how it can be used to build faithful non-free actions of some free products. We will then apply this construction to produce a dense free subgroup of optimal rank in the full group of every ergodic measure-preserving equivalence relation.

#### Invariant measures of minimal homeomorphisms of a Cantor space

I'll recall a description of sets of invariant probability measures of minimal homeomorphisms of a Cantor space (obtained in joint work with T. Ibarlucia) and explain a connection with Fraïssé theory, and how it leads to a new proof of a result of Downarowicz stating that any nonempty metrizable Choquet simplex is affinely homeomorphic to the space of invariant probability measures of some minimal homeomorphism of a Cantor space. If time permits I'll discuss some open problems related to this work.

#### Discrete groups of rank 7/4

We will explain how to construct families of countable discrete groups using surgery techniques, and illustrate these constructions on a class of examples of groups of intermediate rank, called the groups of rank 7/4. This talk is based on joint work with Sylvain Barre'.

#### The Ihara zeta function for measured graphs

Ihara defined his zeta function in the 60ies to count prime elements in finite groups. In recent years, people became interested in analogs for infinite objects. In a joint project with Daniel Lenz and Marcel Schmidt, we defined the Ihara zeta function for measured graphs. This general realm extends and unifies many notions from the literature.

The new construction will be outlined in the talk and illustrated for specific classes of graphs, such as periodic or vertex percolation graphs. Focussing on the examples, we present some new formulas and approximation results for their zeta functions.

#### Riemannian volume and ergodic group actions

We discuss relations between Riemannian and simplicial volume and L2-Betti numbers of aspherical manifolds. The connecting bridge is an ergodic-theoretical invariant of the fundamental group that has some similarities with the cost.

#### Bernoullicity and treeability

A classic theorem of Ornstein states that all factors of Bernoulli shifts over Z are again isomorphic to Bernoulli shifts. On the other hand, a surprising result of Popa and Popa-Sasyk shows that property (T) groups admit factors of Bernoulli shifts which are not Bernoulli. It remains an intriguing open question to determine, in the realm of non-amenable groups, which factors of Bernoulli shifts are Bernoulli. In this talk I will draw connections between this problem and the notion of treeability for group actions. This is joint work with Damien Gaboriau.

#### A pointwise ergodic theorem for quasi-pmp graphs

We prove a pointwise ergodic theorem for quasi-pmp locally countable graphs, which states that the global condition of ergodicity amounts to locally approximating the means of $L^1$-functions via increasing subgraphs with finite connected components. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is constructive and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant and a simple method of exploiting nonamenability. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.

#### Superrigidity and measure equivalence

We show that the collection of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions is invariant under measure equivalence. In the case of orbit equivalence, the proof is mainly an application of a cocycle untwisting lemma of A. Furman and S. Popa, along with the well-known fact that any orbit equivalence of free p.m.p. actions can be lifted to an orbit equivalence of the corresponding Bernoulli extensions of those actions.

#### Invariant random subgroups of groups acting on rooted trees

An $\textit{invariant random subgroup}$ (IRS) of a countable group is a conjugation invariant probability distribution on the space of subgroups. There has been a number of recent papers studying IRS's in various types of groups: lattices in Lie groups, the group of finitary permutations of a countable set, free groups, lamplighters and more.

In this talk I will consider IRS's in the group of finitary automorphisms of a $d$-ary rooted tree. We exploit the action of this group on the boundary of the tree to understand fixed point sets of ergodic IRS. We show that in the fixed point free case IRS's behave like the ones in lattices in Lie groups, but if there are fixed points they resemble the ones in the finitary permutation group. Joint work with Ferenc Bencs.

#### The $\tau$-invariant and classification of factors of type III

I will present some recent work with Roman Sasyk and Stefaan Vaes about the classification of so-called free Araki-Woods factors in von Neumann algebra theory.

These factors arise from orthogonal representations $t\mapsto U_t$ of $R$, and a somewhat peculiar but highly useful invariant used to distinguish these factors is the tau-invariant, which corresponds to the weakest topology that makes $t\mapsto U_t$ continuous. By controlling the $\tau$-invariant, it is possibly to construct a family of free Araki-Woods factors which is not classifiable by countable structures.

#### Finding basis for quasi-orders related to graphs

Given a quasi-order on a set A, and C a subclass closed by extension, a base B of C is a subset C that contains, for every x in C another element of X smaller than C in the quasi-order. Finding one of the smallest of these basis give us a powerful characterization for C, if it is actually small. For example, Kechris, Solecki and Todorcevic found a smallest analytic graph under Borel homomorphism for the class of analytic graphs that cannot be colored by a countable Borel coloring. Exploring the same question for the class of analytic directed graphs that cannot be Borel-colored with 2 colors, Miller found that no basis can be small.

On the other hand, coloring graphs can be seen in a wider context as separation results of analytic sets on a plane. In this wider context, there are several new quasi-orders that have been explored by Lecomte, Zamora and others. We will give a new tool to transfer results from the graph context to the square one and vice-versa. Using this tool, we obtain the analogue of Miller’s result for analytic sets on a plane, as well as new proofs of several results.