Working group "Geometric group theory and its coarse neighbourhood"

Co-organized with 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.

Room 1013 of bâtiment Sophie Germain, Tuesday (maybe Monday sometimes), around 4pm.
Online : BBB.

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• Week of 21-04-2025: Holidays!
• Tuesday 29-04-2025 (4pm): Busy schedule !
  • Due to technical problems during the last session, Juan will finish his talk about Nash Inequalities and Upper bounds on the return probability to the origin. He will need around 30 minutes.
  • Corentin Correia will then start his series of talks about 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).
• Tuesday 06-05-2025 (4pm): second talk of Corentin.
• Tuesday 13-05-2025 (4pm): third talk of Corentin.
• Tuesday 20-05-2025 (4pm): We will listen to Sasha Bontemps.

• Talks of Juan Paucar:
  • (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.

    , 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