Working group "Geometric group theory and its coarse neighbourhood"

Organizers: Corentin Correia and Vincent Dumoncel.

Goal: Get a better understanding on topics interacting with geometric group theory (dynamics, large-scale geometry, representations, operator algebras, logic, among others).
Format: "Mini-courses" intended to be elementary and accessible to a wide audience. They should last between one and two hours.

The working group takes place in bâtiment Sophie Germain (usual room to specify), on Tuesday (sometimes Thursday), around 4pm.
You can also attend it online.

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Previsional schedules:

• 10-06-2026 (on Wednesday!), 4pm, room to be specified of bâtiment Sophie Germain: we will listen to the second talk of Hermès Lajoinie-Dodel, about Property (T) and strong Property (T) with an emphasis on fixed points.
  Abstract

  In 1967, Kazhdan introduced Property (T) for locally compact groups in order to show that a large class of lattices are finitely generated. One characterization of Property (T), due to Delorme and Guichardet, states that every action by affine isometries on a Hilbert space has a fixed point.
  
  Another characterization of Property (T), established by Chatterji, Druţu, and Haglund, states that every action on a median space has bounded orbits. Examples of median spaces include CAT(0) cube complexes, and in particular trees. However, this bounded orbit property fails for general Gromov- hyperbolic spaces, which naturally generalize the geometry of trees.
  
  In his work on the Baum–Connes conjecture, Lafforgue introduced in 2007 a strengthening of Property (T), called strong Property (T). Instead of considering only unitary representations, strong Property (T) deals with representations whose norms exhibit controlled growth. Moreover, strong Property (T) has found numerous applications, the most notable is probably the spectacular progress on the Zimmer Program, which aims to understand actions of higher-rank lattices on manifolds.
  
  Lafforgue also proved that every action of a group with strong Property (T) on a uniformly locally finite hyperbolic space has bounded orbits. In particular, this implies that infinite hyperbolic groups cannot have strong Property (T).
  
  According to the work of Lafforgue, de Laat, and de la Salle, higher-rank Lie groups and their lattices constitute fundamental examples of groups with strong Property (T). On the other hand, independent works of Bader, Furman, and Haettel show that actions of these groups on hyperbolic spaces are highly rigid. We therefore conjecture that hyperbolicity, in a very broad sense, constitutes an obstruction to strong Property (T). We provide a partial answer to this question in the setting of relatively hyperbolic groups.
  
  In this series of lectures, we will discuss the following topics:

  • Basic properties of Property (T) and Property (FH)
  • Property (T) implies Property (FA)
  • Property (T) and median spaces
  • Definition of strong Property (T)
  • Hyperbolic groups do not have strong Property (T)
  • (Personal work) relatively hyperbolic groups do not have strong Property (T).

Previous talks:

• Talks of Hermès Lajoinie-Dodel, about Property (T) and strong Property (T) with an emphasis on fixed points
  Abstract,

  In 1967, Kazhdan introduced Property (T) for locally compact groups in order to show that a large class of lattices are finitely generated. One characterization of Property (T), due to Delorme and Guichardet, states that every action by affine isometries on a Hilbert space has a fixed point.
  
  Another characterization of Property (T), established by Chatterji, Druţu, and Haglund, states that every action on a median space has bounded orbits. Examples of median spaces include CAT(0) cube complexes, and in particular trees. However, this bounded orbit property fails for general Gromov- hyperbolic spaces, which naturally generalize the geometry of trees.
  
  In his work on the Baum–Connes conjecture, Lafforgue introduced in 2007 a strengthening of Property (T), called strong Property (T). Instead of considering only unitary representations, strong Property (T) deals with representations whose norms exhibit controlled growth. Moreover, strong Property (T) has found numerous applications, the most notable is probably the spectacular progress on the Zimmer Program, which aims to understand actions of higher-rank lattices on manifolds.
  
  Lafforgue also proved that every action of a group with strong Property (T) on a uniformly locally finite hyperbolic space has bounded orbits. In particular, this implies that infinite hyperbolic groups cannot have strong Property (T).
  
  According to the work of Lafforgue, de Laat, and de la Salle, higher-rank Lie groups and their lattices constitute fundamental examples of groups with strong Property (T). On the other hand, independent works of Bader, Furman, and Haettel show that actions of these groups on hyperbolic spaces are highly rigid. We therefore conjecture that hyperbolicity, in a very broad sense, constitutes an obstruction to strong Property (T). We provide a partial answer to this question in the setting of relatively hyperbolic groups.
  
  In this series of lectures, we will discuss the following topics:

  • Basic properties of Property (T) and Property (FH)
  • Property (T) implies Property (FA)
  • Property (T) and median spaces
  • Definition of strong Property (T)
  • Hyperbolic groups do not have strong Property (T)
  • (Personal work) relatively hyperbolic groups do not have strong Property (T).

  • (1/?) 28-05-2026: recording
    
• Talks of Antonio López Neumann:
  
  From sums of squares to higher Property (T), via the GNS construction
  Abstract,

  Property (T) is a central rigidity property in analytic group theory. It can be formulated in many equivalent ways, for instance as a spectral gap property or in terms of group cohomology. Ozawa showed a remarkable characterization: Property (T) can be detected by an equation in the group algebra involving sums of squares. This was fundamental in proving that Aut(F_n) has Property (T) for n at least 4 via computer assisted methods.
  
  Recently, there has been increasing interest in higher versions of property (T). Using group cohomology, there are at least two possible non-equivalent definitions: ona via vanishing and one via reducedness of cohomology. It is thus natural to explore whether these higher analogues can be characterized in terms of equations à la Ozawa. In this series of lectures we want to discuss (not in 1-to-1 correspondence with lectures):

  • Basic definitions of Property (T) and group cohomology in degree 1
  • The GNS construction for functions of conditionally negative type and the Delorme-Guichardet theorem
  • *-algebras and Ozawa's theorem
  • More group cohomology and finiteness properties
  • Higher (T) as vanishing of cohomology: the Bader-Nowak characterization
  • (joint work with Piotr Nowak) Higher (T) as reducedness of cohomology and the higher GNS construction.

  • (3/3) 22-04-2026: recording
    
  • (2/3) 14-04-2026: recording
    
  • (1/3) 07-04-2026: notes and recording
    
• Talks of Vincent Dumoncel:
  
  Scaling quasi-isometries and wreath products
  Abstract,

  Scaling quasi-isometries are a refinment of quasi-isometries and were introduced recently by Genevois and Tessera as a crucial ingredient in their quasi-isometric classification of (amenable) lamplighters.
  The goal of this series of talks is to introduce this class of maps and to explain the main steps of the rigidity phenomenon proved by Genevois and Tessera: under mild assumptions, any quasi-isometry between lamplighters is scaling. If time permits, we will also explain elementary applications to subgroups of lamplighters and to the quasi-isometric rigidity of some iterated wreath products.

  handwritten and detailed notes
  • (3/3) 24-03-2026: recording
    
  • (2/3) 17-03-2026: recording
    
  • (1/3) 10-03-2026: recording
    
• Talks of Ivory Fronteau:
  
  Metric Fraïssé Theory
  Abstract

  Let $K$ be a class of finitely generated structures. Fraïssé isolated conditions for which we can define the Fraïssé limit of $K$, a countable structure $M$ in which all structures in $K$ embeds and which is homogeneous in the sense that any isomorphism between finite substructures of $M$ can be extended to an automorphism of $M$. For instance, the order of the rationals is the Fraïssé limit of the class of finite linear orders, the Rado graph is the Fraïssé limit of the class of finite graphs and the atomless countable Boolean algebra is the Fraïssé limit of the class of finite Boolean algebras. The goal of this talk (and probably the next one) will be to give a formulation and a proof of the Fraïssé correspondence for metric structures, which extends the classical Fraïssé theory. We will be mainly following a proof due to Tsankov, which will be the occasion to introduce central notions in (metric) model theory, like metric structures, (quantifier-free) types, type spaces, ultrahomogeneity or existential closedness.

  and notes
  • (4/4) 10-02-2026: recording
    
  • (3/4) 03-02-2026: recording
    
  • (2/4) 27-01-2026: recording (the session was cut short due to technical issues)
    
  • (1/4) 20-01-2026: recording
    
• Talks of Patrick Salhany:
  
  Median graphs
  Abstract

  Median graphs, also known as CAT(0) cube complexes, are interesting mathematical objects : they have an interesting geometry and are very useful in geometric group theory. We will start by defining what median graphs are and take some fundamental examples of median graphs. Later we will study the most important notions relative to median graphs, namely hyperplanes, convex subgraphs and convex hulls, and projections on convex subgraphs.

  • (4/4) 26-11-2025: notes and recording
    
    
  • (3/4) 19-11-2025: notes and recording
    
  • (2/4) 12-11-2025: notes and recording
    
  • (1/4) 05-11-2025: notes and recording
    
• Talks of Rémi Barritault:
  
  Unitary representations of permutation groups
  Abstract

  I will go over the basics of abstract harmonic analysis with an emphasis on the context of permutation groups. Most of the examples of our groups will come from model theory and include S_∞ (the group of all permutations of a countable set), Aut(Q) (the order preserving bijections of the rationals) and the isometry group of the Urysohn spaces (universal and homogeneous countable metric spaces for a prescribed distance set). All of the groups mentioned lie far away from the locally compact groups in the spectrum of Polish groups, hence the need for new techniques in place of tools such as the Haar measure in the classical theory. To that aim, I will develop the notion of dissociation: a definition, ways to obtain dissociated groups, rigidity results for dissociated groups. Based, in parts, on a joint work with Colin Jahel and Matthieu Joseph.

  • (2/2) 15-10-2025: notes and recording
    
  • (1/2) 08-10-2025: notes and recording
    
• Talks of Sasha Bontemps:
  
  Bass Serre theory: a journey into the forest
  Abstract

  The goal of this series of lectures is to give an overview of Bass-Serre theory and to discuss some applications of it.
  
  Roughly speaking, Bass-Serre theory gives a generalization of the following well-known theorem in geometric group theory: "A group is free iff it acts freely on a tree". It is a powerful tool to study the structure of a group acting (non necessarily freely) on a tree, and to get information on its subgroups.
  
  Groups acting on trees arise in many contexts in mathematics. If a topological space admits an open cover that satisfies suitable conditions, the action of its fundamental group on its universal cover leads to an action on a tree. On the other hand, Van Kampen's theorem stands that this group can be explicitly calculated using the fundamental groups of the given open sets, and of their pairwise intersections. In that sense, Bass-Serre theory can be thought of as a combinatorial generalisation of Van Kampen's theorem.
  
  Following the first chapter of Arbres, amalgames, SL2 from Jean-Pierre Serre, we will enter into the details of the main theorems and we will give two applications :

  • Kurosh's theorem (that gives the structure of a subgroup of an amalgamated free product);
  • Grushko's theorem on the rank of a free product.

  and notes
  • (2/2) 27-05-2025: recording
    
  • (1/2) 20-05-2025: recording
    
• Talks of Corentin Correia:
  
  Quantitative orbit equivalence for amenable groups
  Abstract

  Two groups are orbit equivalent if they admit free probability-measure preserving actions on a standard and atomless probability space which have the same orbits. This equivalence relation is a particular case of measure equivalence, introduced by Gromov as a measured analogue of quasi-isometry. Many rigidity results have been found in the non-amenable world, whereas Ornstein and Weiss proved that amenable groups are always orbit equivalent. During these sessions, I will talk about quantitative strenghtenings of orbit equivalence, introduced by Delabie, Koivisto, Le Maître and Tessera, which capture the geometry of amenable groups (in contrast to Ornstein-Weiss Theorem). I will finish by explaining some methods to build concrete orbit equivalences (using Følner tiling sequences).

  and notes
  • (3/3) 13-05-2025: recording
    

    Keywords : Odometers, Følner tiling sequences, cofinite equivalence relation

  • (2/3) 06-05-2025: recording
    

    Keywords : Isoperimetric profile, regular embedding

  • (1/3) 29-04-2025: recording (from 28:30)
    

    Keywords : Standard probability space, Ornstein-Weiss, L^1 orbit equivalence, volume growth

• Talks of Juan Paucar:
  • (3/3) 29-04-2025: Nash Inequalities and Upper bounds on the return probability to the origin
    Abstract

    In this last talk, we will see the links between upper bounds on the return probability and the L2-isoperimetric profile via Nash inequalities.

    , notes and recording (until 28:30)
  • (2.5/3) 15-04-2025: Nash Inequalities and Upper bounds on the return probability to the origin
    Abstract

    In this last talk, we will see the links between upper bounds on the return probability and the L2-isoperimetric profile via Nash inequalities.

    , notes and recording
  • (2/3) 01-04-2025: Return probability to the origin and L2-isoperimetric profile
    Abstract

    I will introduce another coarse invariant, closely related to the L2-isoperimetric profile, namely the return probability to the origin. I will introduce such an invariant and prove its links to the L2-isoperimetric profile and Nash inequalities on graphs.

    , notes and recording
  • (1/3) 18-03-2025: Isoperimetric profile
    Abstract

    I will start with a brief motivation by discussing about Sobolev inequalities. I will proceed to discuss the notion of isoperimetric profile for graphs as well as proving its invariance under quasi-isometries. I will finish by proving the relationship between the isoperimetric profile and the growth for Cayley graphs (Coulhon Saloff-Coste inequality).

    , notes and recording
• Talks of Kostyantyn Krutoy:
  • (3/3) 11-03-2025: Operator algebras: uniform Roe algebras of uniformly locally finite spaces
    Abstract

    In this talk, we will introduce the uniform Roe algebras of uniform locally finite metric spaces. We will discuss some examples of these algebras and examine their basic properties. We conclude the series of talks with a brief introduction to the rigidity problem of uniform Roe algebras.

    , notes and recording
  • (2/3) 04-03-2025: Coarse Geometry: Property A and Coarse Embeddability into a Hilbert Space
    Abstract

    In this talk, we will examine two fundamental properties of coarse spaces: property A and coarse embeddability into a Hilbert space. These properties play a crucial role in the coarse Baum-Connes conjecture and the rigidity problem for Roe algebras. We will demonstrate that they are preserved under coarse equivalences and present illustrative examples. Their significance for the rigidity problem will be addressed in the third talk.

    , notes and recording
  • (1/3) 18-02-2025: Coarse geometry: coarse spaces and coarse maps
    Abstract

    In this talk, we will explore the fundamental concepts of coarse geometry and compare them with the framework of geometric group theory introduced by Vincent in the preceding lecture series. We will introduce coarse structures as an abstraction of metric spaces viewed through the lens of large-scale geometry and discuss coarse equivalences, which are maps preserving large-scale properties.

    , notes and recording
• Talks of Vincent Dumoncel:
  
  Quasi-isometries of finitely generated groups: invariants and rigidity
  Abstract

  A finitely generated group is naturally a metric space, that can be studied from a large-scale point of view. Quasi-isometries are the maps preserving this structure, and a natural problem is then to determine whether two finitely generated groups are quasi-isometric. We introduce two invariants that provide answers in some cases: the volume growth and the number of ends. The second part will be devoted to the quasi-isometric rigidity of ℤ.

  and notes of Vincent
  • (3/3) 04-02-2025: recording
    

    Keywords : Milnor’s theorem, rigidity of wreath products

  • (2/3) 27-01-2025: recording
    

    Keywords : types of growth, polycyclic groups, wreath products

  • (1/3) 21-01-2025: recording
    

    Keywords : finitely generated groups, quasi-isometry, Milnor-Svarc Lemma, volume growth