### MINI-COURS

#### Elia FIORAVANTI (Max-Planck-Institut fur Mathematik)

Median spaces and algebras

## Résumé

Median spaces are a simultaneous generalisation of L^1 spaces, CAT(0) cube complexes (their discrete counterpart) and R-trees (their 1-dimensional counterpart). They naturally occur in two disparate areas of geometric group theory.

On the one hand, finite-dimensional median spaces arise as degenerations of sequences of spaces that coarsely resemble cube complexes. As such, they have applications to the study of quasi-flats, quasi-isometries and automorphisms for broad classes of groups: importantly, those with an HHS structure (such as mapping class groups) and those that can be cocompactly cubulated (such as RAAGs).

At the same time, infinite-dimensional median spaces (and their dual concept: spaces with measured walls) can also be a useful tool to test more « functional analytic » properties for a given group, particularly Kazhdan’s property (T) and the Haagerup property.

I will give an overview of these two aspects of the subject, also focusing on the connections to the theory of median algebras and its implications for the study of cocompactly cubulated groups.

#### Thomas HAETTEL (Université de Montpellier)

Helly graphs and injective metric spaces

## Résumé

We will discuss Helly graphs and injective metric spaces: basic definitions, elementary properties, simple examples. We will focus on nonpositive features, such as local properties, bicombings and classification of isometries. We will present various rich constructions of such spaces, quite often associated with groups of geometric nature, such as hyperbolic groups, braid groups, mapping class groups and higher rank lattices.

#### Nima HODA (Cornell University)

Notions of nonpositive curvature and groups

## Résumé

Combinatorial and geometric methods have a long history in group theory, often with connections or parallels to classical forms of nonpositive curvature. In this minicourse we will learn about classical small cancellation theory, its connections to systolic and quadric complexes and how these classes of combinatorially nonpositively curved spaces can be applied to study groups. We will also discuss more general classes of spaces with a view towards unification.

#### Nir LAZAROVICH (Technion Institute)

CAT(0) cube complexes

## Résumé

This minicourse will serve as an introduction to CAT(0) cube complexes.

### EXPOSÉS

#### Corey BREGMAN (University of Southern Maine)

The Nielsen realization for untwisted automorphisms

## Résumé

In joint work with Charney and Vogtmann, we constructed an outer space for the automorphism group of a right-angled Artin group (RAAG). This is a contractible, finite-dimensional space on which the automorphism group of the RAAG acts with finite stabilizers. The Nielsen realization problem asks whether every finite subgroup of automorphisms fixes a point under this action. We show that finite subgroups of the untwisted automorphism group of a RAAG satisfy Nielsen realization and discuss a strategy for extending this result to all finite subgroups.

#### Montse CASALS-RUIZ (University of the Basque Country)

Real Cubings

## Résumé

The theory of real trees and groups acting on them has had a deep impact on Group Theory by providing tools to attack new problems, by simplifying proofs of classical results, and by establishing new connections.

#### Jérémie CHALOPIN (LIS, CNRS & Aix-Marseille Université)

Graphs with convex balls

## Résumé

Graphs with convex balls have been introduced and characterized by Soltan and Chepoi (1983) and Farber and Jamison (1987). They generalize bridged/systolic graphs that have been introduced in the same papers.

In this work, we give a local-to-global characterization of graphs with convex balls, as graphs in which balls of radius 3 are convex and whose triangle-pentagon complexes are simply connected. We also establish that their Rips complexes are contractible. Finally, we showthat groups acting geometrically on graphs with convex balls arebiautomatic. (Joint work with V. Chepoi and U. Giocanti.)

#### Ruth CHARNEY (Brandeis University)

Conférences de la Chaire Aisenstadt (voir les titres et résumés)

#### María CUMPLIDO (Universidad de Sevilla)

Cube complexes to understand the fiteness properties of block mapping class groups

## Résumé

In this talk I will explain a joint work with Javier Aramayona, Julio Aroca, Rachel Skipper and Xiaolei Wu. We define a new family of groups that are subgroups of the mapping class group M ap(Σg) of asurface Σg of genus g with a Cantor set removed and we will call these groups Block Mapping ClassGroups B(H), where H is a subgroup of Σg. More visually, this family will be constructed by making a tree-like surface gluing pair of pants and taking homeomorphisms that depend on H with certainpreservation properties (it will preserve what we will call a block decomposition of this surface, hence the name of our groups). We will see that this family is closely related to Thompson’s groups and that

it has the property of being of type Fn if and only if H is. As a consequence, for every g ∈ N ∪ {0, ∞} and every n ≥ 1, we construct a subgroup G < (Σg) that is of type Fn but not of type Fn+1, and which contains the mapping class group of every compact surface of genus less or equal to and with nonempty boundary. As expected in this workshop, the techniques involve manipulating cube complexes, as the Stein-Farley cube complex.

#### Eduard EINSTEIN (Swarthmore College)

Constructing relatively geometric actions

## Résumé

Groves and I introduced relatively geometric actions on CAT(0) cube complexes to study relatively hyperbolic groups that act on CAT(0) cube complexes. In this talk, I will explain how to construct relatively geometric actions using a boundary criterion. I will discuss some of the properties of relatively geometric actions and how we hope to use these tools to develop structure theorems for groups that admit relatively geometric actions. This is primarily joint work with Daniel Groves and may also include joint work with Thomas Ng.

#### Talia FERNOS (University of North Carolina Greensboro)

An invitation to the lifting decomposition

## Résumé

The lifting decomposition associated to a consistent set of half-spaces on a CAT(0) cube complex was introduced in my joint work with Chatterji and Iozzi, extending previous work such as that of Caprace and Sageev. In this talk I answer a question of Wise by demonstrating the simplicity and utility of this tool. Specifically, I will give lifting decompositions associated to a variety of natural objects one may have on or in a CAT(0) cube complex and discuss theorems that they helped prove. In particular, I will discuss my recent joint work with Futer and Hagen connecting various boundaries together and *roughly speaking* allows one to place an Aut(X) equivariant simplicial structure on the Tits boundary of the CAT(0) cube complex X.

#### Anthony GENEVOIS (University of Montpellier)

Lamplighter-like geometry of groups

## Résumé

After a general introduction to the large-scale geometry of lamplighter groups, with an emphasis on lamplighters over one-ended finitely presented groups, I will explain how a similar geometry appears in other infinitely presented groups and how efficient invariants can be constructed to distinguish them up to quasi-isometry. Joint work with R. Tessera.

#### Jingyin HUANG (Ohio State University)

Integral measure equivalence versus quasi-isometry for some right-angled Artin groups

## Résumé

Two finitely generated groups G and H are quasi-isometric (QI), if they admit a topological coupling, i.e. an action of G times H on a locally compact topological space such that each factor acts properly and cocompactly. This topological definition of quasi-isometry was given by Gromov, and at the same time he proposed a measure theoretic analogue of this definition, called the measure equivalence, which is closely related to the notion of orbit equivalence in ergodic theory. Despite the similarity in the definition of measure equivalence and quasi-isometry, their relationship is rather mysterious and not well-understood.

We will start with a discussion of an interesting similarity between some QI invariants and ME invariants for general cubical groups. Then we look at the case of right-angled Artin groups. We show that if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a large class of right-angled Artin groups. This is perhaps the first instance a rigidity result in the ME side is obtained via establishing quasi-isometry. This is joint work with Camille Horbez.

#### Sam HUGHES (University of Oxford)

Finite quotients of generic free-by-cyclic groups

## Résumé

In this talk we will investigate to what extent we can distinguish two free-by-cyclic groups by their finite quotients. I will present recent work joint with Monika Kudlinska proving that there exist at most finitely many free-by-cyclic groups with the same finite quotients as a generic free-by-cyclic group.

#### Kasia JANKIEWICZ (University of California Santa Cruz)

Hyperbolicity and relative hyperbolicity in cubical small cancellation

## Résumé

The classical small cancellation theory studies group presentations whose relations have small overlaps with one another. The cubical small cancellation theory is a generalization where in the place of standard group presentation we consider groups defined using « cubical presentations », consisting of a non-positively curved cube complex X together with a collection of cube complexes {Y_1,…, Y_k} that locally isometrically embed in X. The associated group is the fundamental group of X with a cone attached along each Y_i. I will discuss some results about hyperbolicity and relative hyperbolicity of such groups, including a joint work with M. Arenas and D.T. Wise.

#### Ioana-Claudia LAZAR (Politehnica University of Timisoara)

Connections between 7-located complexes, weakly systolic complexes and CAT(0) spaces

## Résumé

Some results presented in the talk are joint work with Nima Hoda (Cornell University). We show that weakly systolic complexes are 7-located. A simplicial complex is 7-located if it is flag and every full loop of length at most 7 that is at the boundary of a dwheel, is contained in a 1-ball. We investigate the existence of a CAT(0) metric on a 7-located disc endowed with a certain metric. We prove the minimal filling diagrams lemma for 7-located, locally 5-large complexes. As corollary we get that the quadratic isoperimetric inequality holds on such complexes. We introduce the 5/9-condition on a simplicial complex and we show that it implies the Gromov hyperbolicity of its universal cover. This follows as an application of a version of 8-location. We prove the minimal filling diagrams lemma for 5/9-complexes. We show that the minimal displacement set in a weakly systolic simplicial complex embeds isometrically into the complex and that it is systolic.

#### Alice KERR (University of Bristol)

Loxodromic elements in right-angled Artin groups

## Résumé

The ability to quickly generate loxodromic elements in an action on a hyperbolic space is key to many statements about exponential type growth. In examples like the mapping class group, results of this type are already known. We will discuss how we can acheive similar results for the action of right-angled Artin groups on their associated extension graph (or equivalently on the contact graph).

#### Jone LOPEZ DE GAMIZ ZEARRA (Vanderbilt University)

Finitely generated normal subgroups of RAAGs

## Résumé

This minicourse will serve as an introduction to CAT(0) cube complexes.

#### Thomas NG (Technion Institute)

Small cancellation free product quotients are relatively cubulated

## Abstract

Quotients of free products are a rich source of relatively hyperbolic groups with exotic subgroups. Martin and Steenbock introduced a model for when quotients of free products can act properly and cocompactly on CAT(0) cube complexes. I will discuss joint work with Einstein that gives a new boundary criterion proof of Wise’s theorem that C’(1/6) groups are cubulated and can be naturally adapted to exhibit relatively geometric actions of C’(1/6) free products on CAT(0) cube complexes. I will also discuss how our work implies residual finiteness is preserved under C’(1/6) free products.

#### Damian OSAJDA (University of Copenhagen)

Coxeter groups are biautomatic

## Résumé

I will present our recent proof of biautomaticity of all Coxeter groups. From the construction of the biautomatic structure it follows that uniform lattices in isometry groups of buildings are biautomatic. The talk is based on a joint work with Piotr Przytycki (McGill).

#### Kim RUANE (Tufts University)

Some Questions about Boundaries of CAT(0) Groups

## Résumé

I will discuss some questions about the visual and Tits boundaries of CAT(0) groups that I have been interested in for a long time. I am guessing I will have more questions than answers.

#### Harry PETYT (University of Oxford)

Hyperbolicity from walls

## Résumé

Many groups can be effectively studied by finding an action on a hyperbolic space. I’ll describe a procedure inspired by Sageev’s construction for producing such an action, and then discuss some applications. (Joint with Davide Spriano and Abdul Zalloum.)

#### Chandrika SADANAND (Bowdoin College)

Heegaard splittings and square complexes

## Résumé

When building a finite cover of a graph of spaces/groups, one needs to construct finite-index subgroups of the vertex groups that intersect the edge groups in a controlled manner. This leads to the notion of a group commanding a collection of subgroups. I will discuss examples of this (most of which are virtually special cubulated groups) and give an application to quasi-isometric rigidity.

#### Sam SHEPHERD (Vanderbilt University)

Commanding subgroups

## Résumé

When building a finite cover of a graph of spaces/groups, one needs to construct finite-index subgroups of the vertex groups that intersect the edge groups in a controlled manner. This leads to the notion of a group commanding a collection of subgroups. I will discuss examples of this (most of which are virtually special cubulated groups) and give an application to quasi-isometric rigidity.

#### Nicolas VASCOU (University of Bristol)

The isomorphism problem for Artin groups.

## Résumé

The isomorphism problem for Artin groups is the question of determining which presentation graphs give rise to isomorphic Artin groups. In general, it is conjectured that this can only happen if the presentations graphs are isomorphic, or if they are « twist-equivalent ». There are two steps to solve the isomorphism problem for a given class of Artin groups: first, solve the problem within that class ; and then, show that Artin groups in the class cannot be isomorphic to Artin groups outside this class. So far, the only class of Artin groups for which this problem has been (fully) solved is the class of RAAGs, although other partial results exist. In this talk, I will give background around this problem and I will present a solution of the isomorphism problem for so-called « large-type » Artin groups, relating it to the isomorphism problem for Coxeter groups.

#### Abdul ZALLOUM (University of Toronto)

Effective rank-rigidity

## Résumé

I will discuss some recent work where, relying on Caprace and Sageev’s celebrated rank-rigidity, we show that each CAT(0) cube complex X with a factor system determines an integer m=m(X) where each group G acting freely cocompactly on X is either quasi-isometric to a product or contains a rank-one element of length at most m=m(X). To accomplish this, we show that (under the above hypothesis) Caprace and Sageev’s flipping and skewering tools can be accomplished uniformly quickly, depending only on the geometry of the underlying cube complex X.