The videos of the talks are available on the following webpage

## Graphical small cancellation groups

### Goulnara Arzhantseva

Graphical small cancellation theory is a generalization of classical small cancellation theory. The main application is an embedding of a desired sequence of graphs into the Cayley graph of a finitely generated group. The group properties are thus derived from the combinatorial or asymptotic properties of the embedded subgraphs. The theory was introduced by Gromov in the description of groups now known as Gromov monsters, which are finitely generated groups that contain sequences of expander graphs in their Cayley graphs. In this expository talk, we discuss our results on the geometry and analysis of graphical small cancellation groups, indicate crucial technical points and formulate a few open problems.

## Subgroups of direct products of surface groups

### Martin Bridson

After reviewing what is known about subgroups of direct products of surface groups and their significance in the story of which groups are Kähler, I shall describe a new construction that provides infinite families of finitely presented subgroups. These subgroups have varying higher-finiteness properties and are not of Stallings-Bieri type.

Random foldings of pentagons

### Serge Cantat

Start with a pentagon in the euclidean plane, and consider the space of all pentagons with the same side lengths up to euclidean motion. This space is the real part of some K3 surface. Folding the pentagons along their diagonals, one obtains involutive automorphism of this K3 surface. I will describe the main dynamical properties of the group generated by these involutions. This is based on a joint work with Romain Dujardin.

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## Connections between hyperbolic geometry and median geometry

### Cornelia Drutu

The interest of median geometry comes from its connections with property (T) and a-T-menability and, in its discrete version, with the solution to the virtual Haken conjecture. In this talk I shall explain how groups endowed with various forms of hyperbolic geometry, from lattices in rank one simple groups to acylindrically hyperbolic groups, present various degrees of compatibility with median geometry. This is on joint work with Indira Chatterji and joint work with John Mackay.

## Action of groups on the Poisson boundary

### Anna Erschler

joint works with Vadim Kaimanovich and Josh Frisch

## Still unsolved problems

### Misha Gromov

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## Positivity in flag manifolds and positive representations

### Olivier Guichard

This talk is based on joint works with Anna Wienhard and François Labourie.

In a number of situations, an action on a flag manifold is well controlled via a structure mimicking the orientability of the circle. After reviewing the classical examples of such structures, we will propose an axiomatization of them leading to a complete classification. We will then describe some properties of corresponding positive actions or representations.

## Measure equivalence rigidity for \({\rm Out}(F_N)\) and dynamical decomposition

### Vincent Guirardel

Measure equivalence is a measurable analogue of quasi-isometry. For instance, two lattices (co-compact or not) in a same Lie group are measurably equivalent by definition. We prove that for \(N\geq 3\), any countable group that is measure equivalent to \({\rm Out}(F_N)\) is virtually isomorphic to it. I will discuss some of the tools introduced for this proof, and in particular, a notion of a canonical dynamic decomposition associated to a subgroup of \({\rm Out}(F_N)\) which somehow generalizes the dynamical decomposition of a surface associated to a subgroup of the mapping class group.

This is a joint work with Camille Horbez.

Asymptotic dimension of graphs of polynomial growth and systolic inequalities

### Panagiotis Papasoglu

Asymptotic dimension and n-Uryson width are useful notions of dimension in coarse and systolic geometry respectively. I will explain how using similar techniques one obtains:

1. Sharp estimates for the asymptotic dimension of graphs of polynomial growth

2. A new proof of a theorem of Guth on the n-Uryson width of Riemannian manifolds of small volume growth. This leads to new proof of Gromov's systolic inequality.

## Complex geometry and higher finiteness properties of groups

### Pierre Py

Following C.T.C. Wall, we say that a group G is of type \(F_n\) if it has a classifying space (a \(K(G,1)\)) whose \(n\)-skeleton is finite. When \(n=1\) (resp. \(n=2\)) one recovers the condition of finite generation (resp. finite presentation). The study of examples of groups which are of type \(F_{n-1}\) but not of type \(F_n\) has a long history (Stallings, Bestvina-Brady, ...). One says that these examples of groups have exotic finiteness properties. In this talk I will explain how to use complex geometry to build new examples of groups with exotic finiteness properties. This is part of a joint work with F. Nicolás, which generalizes earlier works by Dimca, Papadima and Suciu, Llosa Isenrich and Bridson and Llosa Isenrich.

## Automorphisms of groups and a higher rank JSJ decomposition

### Zlil Sela

The JSJ (for groups) was originally constructed to study the automorphisms and the cyclic splittings of a (torsion-free) hyperbolic group. Such a structure theory was needed to complete the solution of the isomorphism problem for (torsion-free) hyperbolic groups.

Later, the JSJ was generalized to all finitely presented groups. In this generality it encodes the splittings but not all the automorphisms.

We further generalize the JSJ decomposition to study automorphisms of groups that act on products of hyperbolic spaces, and more generally to study automorphisms of (some) hierarchically hyperbolic groups (e.g. right angled Artin groups). The object that we construct can be viewed as a higher rank JSJ decomposition.