Atelier "Géométrie et dynamique des groupes de convergence"

Du 31 mai au 2 juin 2011



L'atelier est dédié à des développements récents en théorie géométrique des groupes. Il comprendra trois mini-cours, et six exposés.

Mini-cours :

Victor Gerasimov (Université de Belo Horizonte) : Visibility and convexity for convergence group actions

Mahan Mj (Université RKVM Belur) : Semiconjugacies of Convergence Actions

Dani Wise (Université McGill) : Groups with a quasiconvex hierarchy


Nicolas Bergeron (Université Paris 6) : A boundary criterion for cubulation

Brian Bowditch (Université de Warwick) : Median structures on groups

Emmanuel Breuillard (Université d'Orsay) : Word maps and the Banach-Tarski paradox

François Dahmani (Université de Grenoble) : Presentations of parabolic subgroups of relatively hyperbolic groups

Anders Karlsson (Université de Genève) : Boundaries and dynamics of isometries

John Mackay (Université d'Oxford) : Generic conformal dimension estimates for random groups

Lieu : LPP, Salle de Réunion



  • Nicolas Bergeron : A boundary criterion for cubulation.
  • I will describe a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. This has applications towards arithmetic hyperbolic lattices and hyperbolic 3-manifold groups. By combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol's criterion, this in particular shows that every subgroup separable closed hyperbolic 3-manifold is virtually fibered. This is joint work with Dani Wise.

  • Brian Bowditch : Median structures on groups.
  • A common theme in recent work on the mapping class groups has been the existence of some kind of median structure. We aim to give a brief survey of this, and suggest that some results, such as the rank theorem, can be seen efficiently in these terms. It seems to be a natural question to ask which groups admit structures of this sort.

  • Emmanuel Breuillard : Word maps and the Banach-Tarski paradox.
  • In joint work with Bob Guralnick, Ben Green and Terence Tao, we establish that most Cayley graphs of finite simple groups of bounded rank are expander graphs. A key part of the proof consists in establishing the existence of free subgroups of semisimple groups with strong density properties. Such properties are related to the study of word maps and to paradoxical decompositions of homogeneous spaces, in the spirit of the Banach Tarski paradox.

  • François Dahmani : Presentations of parabolic subgroups of relatively hyperbolic groups.
  • Joint work with V. Guirardel. Given a finitely presented relatively hyperbolic group, we prove that all its maximal parabolic subgroups are finitely presented. The proof is constructive in the sense that one can compute such presentations if one is given generators of the parabolic subgroups, and a solution to the word problem in the ambiant group.

  • Victor Gerasimov : Visibility and convexity for convergence group actions.
  • The purpose of my lectures is to demonstrate advantages of simultaneous usage of geometric and topological ideas in studying actions with convergence property. The construction of ''attractor sum'' is actually a glueing of geometry and topology. The main applications concern the relatively hyperbolic groups. However we introduce a wider class of ''geometric'' convergence actions including the limits and the quotients of relatively hyperbolic actions. For any finitely generated group G we define the ''initial geometric boundary'' ∂_{ig}G that covers the (relatively) hyperbolic boundaries, the Floyd boundaries and the space of ends. The action of G on ∂_{ig}G has convergence property. For some groups we are able to calculate this ''space of fine ends''.

  • Anders Karlsson : Boundaries and dynamics of isometries.
  • Metrics, invariant or semi invariant, under transformations arise in many contexts: differential geometry, group theory, several complex variables, operator theory, ergodic theory etc. Half-spaces and their limits, called stars, in various compactifications are notions suitable to describe the dynamics of metric preserving transformations. One fundamental type of boundary dynamics is that of convergence group actions, as it appears frequently and has well-known and strong consequences. This boundary dynamics can be given a generalized description in terms of the incidence structure at infinity that the stars of a compactification give rise to. Some examples and questions will be pointed out, in particular to Teichmuller spaces and the first L^2 cohomology.

  • John Mackay : Generic conformal dimension estimates for random groups.
  • In Gromov's density model of random groups, at densities less than one half, a random group is hyperbolic. It is known that the group also infinite and that its boundary at infinity is homeomorphic to the Menger curve, however this does not completely determine the quasi-isometry type of such groups. I will outline recent work showing how Pansu's conformal dimension can be used in the contexts of certain small cancellation groups to give a more refined geometric picture of random groups at small densities.

  • Mahan Mj : Semiconjugacies of Convergence Actions.
  • Let G be a non-elementary (relatively) hyperbolic group acting freely and properly discontinuously on a hyperbolic metric space X. Let ∂G be the (relative) boundary of G and ∂X be the boundary of X. Then G acts as a convergence group on both ∂G and ∂X. We shall discuss the following general question: Does the action of G on X extend continuously to an equivariant map between ∂G and ∂X? A complete positive answer when X is hyperbolic 3-space will be discussed. Such continuous extensions (called Cannon-Thurston maps) have been studied by Cannon and Thurston, Minsky, Bowditch, McMullen, Pal, Das and the author.

  • Dani Wise : Groups with a quasiconvex hierarchy.
  • Nonpositively curved cube complexes have become increasingly central objects in combinatorial and geometric group theory. We will give a quick introduction focusing especially on their identity as "higher dimensional graphs". We will describe cubical small-cancellation theory, and outline how to use nonpositively curved cube complexes to resolve certain outstanding group theoretical problems. These include Baumslag's conjecture on the residual finiteness of one-relator groups with torsion, as well as the virtual-fibering problem for Haken hyperbolic 3-manifolds.



    Mardi 31 Mai :

    9h30-10h30 : Dani Wise
    11h-12h : Emmanuel Breuillard
    13h30-14h30 : Victor Gerasimov
    14h45-15h45 : John Mackay
    16h15-17h15 : Mahan Mj

    Mercredi 1 juin :

    9h30-10h30 : Victor Gerasimov
    11h-12h : François Dahmani
    13h30-14h30 : Mahan Mj
    14h45-15h45 : Anders Karlsson
    16h15-17h15 : Dani Wise

    Jeudi 2 juin :

    9h-10h : Mahan Mj
    10h15-11h15 : Nicolas Bergeron
    11h30-12h30 : Dani Wise
    14h-15h : Brian Bowditch
    15h15-16h15 : Victor Gerasimov


    Les organisateurs voudraient que tous les auditeurs s'inscrivent. Prière de les contacter aussi pour vos demandes de financement éventuelles.

    For english speakers... The organisers would like all participants to register. You can also contact them in case you need financial support.


    : 03 20 43 65 95

    M. Bourdon
    F. Guéritaud   : 03 20 43 42 12
    L. Potyagailo  

    : 03 20 43 68 78



    Avec le soutien de : l'ANR "ETTT", le GDR "Tresses", l'IUF, et le laboratoire Painlevé.