Sahana Balasubramanya (University of Münster)
María Cumplido (Heriot-Watt University)
Elia Fioravanti (Max Planck Institute University of Bonn)
Ursula Hamenstädt (Rheinische Friedrich-Wilhelms-University of Bonn)
Nima Hoda (ENS Paris)
Dawid Kielak (Bielefeld University)
Ivan Levcovitz (Technion – Israel Institute of Technology)
Anne Lonjou (University of Basel)
Alex Margolis (Technion – Israel Institute of Technology)
Rose Morris-Wright (Brandeis University)
Ian Runnels (University of Virginia)
Rachel Skipper (Ohio State University)
Ignat Soroko (Louisiana State University)
Emily Stark (University of Utah)
Olga Varghese (University of Münster)
Genevieve Walsh (Tufts University)

Virtual Geometric Group Theory conference
Rencontre virtuelle en géométrie des groupes

1 – 5 June 2020

Indira Chatterji (Université Nice Sophia-Antipolis)
Luis Paris (Université de Bourgogne)
Karen Vogtmann (University of Warwick)

Geometric group theory is the study of infinite discrete groups using geometric techniques inspired broadly by classical Riemannian geometry, while also borrowing ideas from other mathematical domains such as formal language theory and algebraic topology. The field has developed rapidly and is currently very active, with many innovative projects reaching in new directions. This conference will highlight the interaction between two main themes in the field : Artin groups on the one hand, and CAT(0) geometry on the other hand. Although these two themes will be the leitmotivs of the conference, related topics such as Coxeter groups, 3-dimensional manifolds, automorphisms of free groups and mapping class groups are also included in the topics in order to enrich exchanges between participants and inspire ideas for developing new research projects.

​Dear Colleagues, dear participants,

Unfortunately, due to the unusual circumstances, we were forced to cancel the conference « Artin Groups, CAT(0) Geometry and Related Topics » in its original format. We will nevertheless organize a virtual version of it.

We will record the mini-courses and will make them available on this webpage before June 1st. 

During the week which starts on June 1st, we will organize virtual office hours in order to allow the participants to interact and discuss the content of the talks with their authors. 

Best regards,

Indira Chatterji, Luis Paris , Karen Vogtmann 

(Marseille, France local time)
This is the schedule for the live discussions on the pre-recorded talks below. Please watch the talks at your convenience before joining the sessions. To follow the live discussions,  join us in the main lecture hall (registered participants only).
You can submit questions related to the pre-recorded talks here.
Discussion rooms will be available to registered conference participants via the link below. Your access code to the rooms is provided by the organizers.

Welcome word from the organisers
Virtual coffee
Virtual coffee
Virtual coffee
Virtual coffee
Ursula Hamenstädt  ​  
Spin mapping class groups and curve graphs

Chair Tara Brendle
Sahana Balasubramanya    Quasi-Parabolic Structures on Groups

Chair Matt Cordes
Elia Fioravanti
Spaces of cubulations

Chair Nir Lazarovich

Dawid Kielak   Computing fibring of 3-manifoldsand free-by-cyclic groups

Chair Alessandra Iozzi

Mladen Bestvina and Kim Ruane
​open questions group therapy session
15:30 ​​
Anne Lonjou    
Action of the Cremona group on a CAT(0) cube complex

Chair Michah Sageev 
Maria Cumplido
Complexes of parabolic subgroups for Artin groups

Chair Ruth Charney

Nima Hoda
Shortcut Graphs and Groups

Goulnara Arzhantseva

Ignat Soroko   Intersections and joins of subgroups in free groups

Chair Gilbert Levitt

Alex Margolis
Quasi-actions and almost normal subgroups

Conchita Martinez
Rose Morris-Wright    Parabolic subgroups of Infinite type Artin groups

Chair Mike Davis 
Ivan Levcovitz  
Right-angled Coxeter groups commensurable to right-angled Artin groups

Chair Petra Schwer

Emily Stark  
Action rigidity for free products of closed hyperbolic manifold groups

Chair Tullia Dymarz

Rachel Skipper
Finiteness properties for simple groups

Tatiana Nagnibeda
Ian Runnels  
Effectively Generating RAAGs in MCGs

Chair Matthieu Calvez

Olga Varghese
Automorphism groups of Coxeter groups do not have Kazhdan’s property (T)

Chair Alain Valette

Genevieve Walsh
Incoherence of free-by-free and surface-by-free groups

Chair Talia Fernos

Social mixer
Movie night
Themed hangouts
Game night

Action of the Cremona group on a CAT(0) cube complex
By Anne Lonjou
Abstract. The Cremona group is the group of birational transformations of the projective plane. Even if this group comes from algebraic geometry, tools from geometric group theory have been powerful to study it. In this talk, based on a joint work with Christian Urech, we will build a natural action of the Cremona group on a CAT(0) cube complex. We will then explain how we can obtain new and old group theoretical and dynamical results on the Cremona group.

Complexes of parabolic subgroups for Artin groups
By María Cumplido Cabello
Abstract: One of the main examples of Artin groups are braid groups. We can use powerful topological methods on braid groups that come from the action of braid on the curve complex of the n-puctured disk. However, these methods cannot be applied in general to Artin groups. In this talk we explain how we can construct a complex for Artin groups, which is an analogue to the curve complex in the braid case, by using parabolic subgroups.

Parabolic Subgroups of Infinite Type Artin Groups
By Rose Morris-Wright
Abstract. Parabolic subgroups are the fundamental building blocks of Artin groups. These subgroups are isomorphic copies of smaller Artin groups nested inside a given Artin group. In this talk, I will discuss questions surrounding how parabolic subgroups sit inside Artin groups and how they interact with each other. I will show that, in an FC type Artin group, the intersection of two finite type parabolic subgroups is a parabolic subgroup. I will also discuss how parabolic subgroups might be used to construct a simplicial complex for Artin groups similar to the curve complex for mapping class groups. This talk will focus on using geometric techniques to generalize results known for finite type Artin groups to Artin groups of FC type.

Computing fibring of 3-manifolds and free-by-cyclic groups
By ​Dawid Kielak
Abstract. We will discuss an analogy between the structure of fibrings of 3-manifolds and free-by-cyclic groups; we will focus on effective computability. This is joint work with Giles Gardam.

Intersections and joins of subgroups in free groups
By Ignat Soroko
Abstract. The famous Hanna Neumann Conjecture (now the Friedman–Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups H and K of a non-abelian free group. It is an interesting question to `quantify’ this bound with respect to the rank of the join of H and K, the subgroup generated by H and K. In this talk I describe what is known about the set of realizable values (rank of join, rank of intersection) for arbitrary H, K, and about my recent results in this direction. In particular, we resolve the remaining open case (m=4) of Guzman’s `Group-Theoretic Conjecture’ in the affirmative. This has some interesting corollaries for the geometry of hyperbolic 3-manifolds. Our methods rely on recasting the topological pushout of core graphs in terms of the Dicks graphs introduced in the context of his Amalgamated Graph Conjecture. This allows to translate the question of existence of a pair of subgroups H,K with prescribed ranks of joins and intersections into graph theoretic language, and completely resolve it in some cases. In particular, we completely describe the locus of realizable values of ranks in the case when the rank of one of the subgroups H,K equals two.

Right-angled Coxeter groups commensurable to right-angled Artin groups  
By Ivan Levcovitz
Abstract. A well-known result of Davis-Januszkiewicz is that every right-angled Artin group (RAAG) is commensurable to some right-angled Coxeter group (RACG). In this talk we consider the converse question: which RACGs are commensurable to some RAAG? To do so, we investigate some natural candidate RAAG subgroups of RACGs and characterize when such subgroups are indeed RAAGs. As an application, we show that a 2-dimensional, one-ended RACG with planar defining graph is quasi-isometric to a RAAG if and only if it is commensurable to a RAAG. This talk is based on work joint with Pallavi Dani.

Effectively Generating RAAGs in MCGs
By ​Ian Runnels
Abstract. Given a (mostly arbitrary) collection of mapping classes on a surface S, we find an explicit constant N such that their Nth powers generate a right-angled Artin subgroup of the mapping class group MCG(S).

Spin mapping class groups and curve graphs
By Ursula Hamenstädt
Abstract. A spin structure on a closed surface S of genus g greater or equal to 2 is a covering of the unit tangent bundle of S which restricts to a standard covering of the fiber. Such a spin structure has a parity, even or odd. The spin mapping class group is the stabilizer of such a spin structure in the mapping class group of S. We use a subgraph of the curve graph to construct an explicit generating set of the spin mapping class group, consisting of Dehn twists about a system of 2g-1 simple closed curves.

Shortcut Graphs and Groups
By Nima Hoda
Abstract. Shortcut graphs are graphs in which long enough cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory including: systolic and quadric groups (in particular finitely presented C(6) and C(4)-T(4)
small cancellation groups), cocompactly cubulated groups, hyperbolic groups, Coxeter groups and the Baumslag-Solitar group BS(1,2). Most of these examples satisfy a strong form of the shortcut property. I will discuss some of these examples as well as some general constructions and properties of shortcut graphs and groups.

Quasi-parabolic structures on groups
By Sahana Balasubramanya

Abstract. ​The study of the poset of hyperbolic structures H(G) on a group G was initiated by Abbott-Balasubramanya-Osin.
However, the sub-poset of quasi- parabolic structures is still very far from being understood and several questions
remain unanswered.
In this talk, I will talk about the motivation behind our work, describe some structural results related to quasi-parabolic
structures and thus answer some of the open questions. I will end my talk by discussing ongoing work in the area.
This talk contains some joint work with C.Abbott, D.Osin and A.Rasmussen.

Action rigidity for free products of hyperbolic manifold groups
By Emily Stark
Abstract. ​​Abstract: The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group’s quasi-isometry class, a group’s abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G’ is a proper geodesic metric space on which G and G’ act geometrically. A group G is action rigid if every group G’ that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

Spaces of cubulations
By Elia Fioravanti
Abstract. « The theory of group actions on CAT(0) cube complexes has exerted a strong influence on geometric group theory and low-dimensional topology in the last two decades. Indeed, knowing that a group G acts properly and cocompactly on a CAT(0) cube complex reveals a lot of its algebraic structure. However, in general, « cubulations » are non-canonical and the group G can act on cube complexes in many different ways. It is thus natural to try and formulate a good notion of « space of all cubulations of G », which would prove useful in the study of Out(G) for quite general groups G. I will describe some results in this direction, based on joint works with J. Beyrer and M. Hagen. »

Quasi-actions and almost normal subgroups
By Alex Margolis
Abstract. If a group G acts isometrically on a metric space X and Y is any metric space that is quasi-isometric to X, then G  quasi-acts on Y.  A fundamental problem in geometric group theory is to straighten (or quasi-conjugate) a quasi-action to an isometric action on a nice space.  We  will introduce and investigate discretisable spaces, those for which every cobounded quasi-action can be quasi-conjugated to an isometric action of a locally finite graph. Work of Mosher-Sageev-Whyte shows that free groups have this property, but it holds much more generally. For instance, we show that every hyperbolic group is either commensurable to a cocompact lattice in rank one Lie group, or it is discretisable.‭We give several applications and indicate possible future directions of this ongoing work, particularly in showing that normal and almost normal subgroups are often preserved by quasi-isometries. For instance, we show that any finitely generated group quasi-isometric to a ‬Z‭-by-hyperbolic group is Z-by-hyperbolic. We also show that within the class of residually finite groups, the class of central extensions of finitely generated abelian groups by hyperbolic groups is closed under quasi-isometries.

Automorphism groups of Coxeter groups do not have Kazhdan’s property (T) 
By Olga Varghese
Abstract. We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_\Gamma$ in $\mathcal{W}$, 
the automorphism group  ${\rm Aut}(W_\Gamma)$ virtually surjects onto $W_\Gamma$. In particular, the group ${\rm Aut}(G_\Gamma)$ is virtually indicable and therefore does not satisfy Kazhdan’s property (T). Moreover, if $W_\Gamma$ is not virtually abelian, then the group ${\rm Aut}(W_\Gamma)$ is large.

Incoherence of free-by-free and surface-by-freegroups 
By Genevieve Walsh
A group is said to be coherent if every finitely generated subgroup is finitely presented.  This property is enjoyed by free groups, and the fundamental groups of surfaces and 3-manifolds.  A group that is not coherent is incoherent, and it is very interesting to try and understand which groups are coherent.  We will discuss some of the geometric and topological aspects of this question, particularly quasi-convexity and algebraic fibers.  We show that free-by-free and surface-by-free groups are incoherent, when the rank and genus are at least 2.  The proof uses an understanding of fibers and also the RFRS property. This is joint work with Robert Kropholler and Stefano Vidussi. 
REFERENCE :  « Incoherence of free-by-free and surface-by-free groups, R. Kropholler, S. Vidussi, and G. Walsh. arXiv:2005.01202 (May2020) 

Finiteness properties for simple groups
​By Rachel Skipper

Abstract. A group is said to be of type $F_n$ if it admits a classifying space with compact $n$-skeleton. We will consider the class of R\ »{o}ver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type $F_{n-1}$ but not $F_n$ for each $n$. These are the first known examples for $n\geq 3$.
As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups, the first is due to Caprace and R\'{e}my. This is joint work with Stefan Witzel and Matthew C. B. Zaremsky.